RATIONAL CURVES ON HYPERSURFACES OF LOW DEGREE II

JOE HARRIS AND JASON STARR

Abstract. This is the second in a sequence of papers on the geometry of spaces of rational curves of degree e on a general hypersurface X ⊂ ℙ^{n} of degree d. In [11] it is proved that if d < \frac{n+1}{2} then for each e the space of rational curves is irreducible, reduced and has the expected dimension. In this paper it is proved that if d^{2} + d + 1 ≤ n, then for each e the space of rational curves is a rationally connected variety; in particular it has negative Kodaira dimension.

Contents

1. Statement of results 1 2. Deformation ample 5 3. Deformation theory of stable maps 11 4. Conditions on families of stable maps 26 5. The induction argument 32 6. Twistable lines on hypersurfaces 43 7. Base case of the induction for hypersurfaces 47 8. Proof of the main theorem 57 References 57

1. Statement of results

This is the second in a sequence of papers on the geometry of spaces of rational curves on a general hypersurface X ⊂ ℙ^{n}. Let d = deg(X). In [11] it is proved that if d < \frac{n+1}{2} then for each e the space of rational curves of degree e is irreducible, reduced and has the expected dimension. The main result of this paper is the following theorem.

Theorem 1.1. If X ⊂ ℙ^{n} is a general hypersurface of degree d and if n ≥ d^{2} + d + 1 then for each integer e ≥ 1 the stack \overline{M}_{0,0}(X, e) is rationally connected. More precisely, there exists a morphism f : ℙ^{1} → ((\overline{M}_{0,0}(X, e))_{fine})_{sm} such that f^{∗}T_{\overline{M}_{0,0}(X,e)} is ample.

Remark 1.2. (i) The scheme \overline{M}_{0,0}(X, e) is the coarse moduli space of \overline{M}_{0,0}(X, e), the open subset (\overline{M}_{0,0}(X, e))_{fine} is the fine moduli locus, and the open subset ((\overline{M}_{0,0}(X, e))_{fine})_{sm} is the smooth locus of the fine moduli locus.

(ii) For the cases d = 1, 2, a related and stronger theorem is proved in [15, Thm. 3]; namely the coarse moduli space \overline{M}_{0,0}(X, e) is rational. The proof relies on the fact that X is a homogeneous space and does not extend to the case d ≥ 3. Moreover, it is not clear from [15] that there exists a very free rational curve in the smooth locus of the fine moduli locus of the coarse moduli space.

Date: June 30, 2004.

2 J. HARRIS AND J. STARR

The space \overline{M}_{0,0}(X, e) is the Kontsevich moduli stack of stable maps, which will be recalled below; it is a Deligne-Mumford stack containing the parameter space of smooth rational curves in X of degree e as an open substack. A variety is rationally connected if any two closed points are contained in the image of a morphism from ℙ^{1} to the variety. Rationally connected varieties have negative Kodaira dimension, hence the schemes \overline{M}_{0,0}(X, e) have negative Kodaira dimension.

The motivation behind Theorem 1.1 is a conjectural relationship between rational connectedness of \overline{M}_{0,0}(X, e) and a theorem of Lang about rational points of varieties defined over the function field of a surface.

Theorem 1.3 (Lang, [18]). Let K be the function field of a surface over ℂ and let X ⊂ ℙ_{K}^{n} be a hypersurface of degree d. If d^{2} ≤ n, then X(K) ≠ ∅.

There is a naive parameter count that suggests that if d^{2} ≤ n then for e ≫ 0 the stack \overline{M}_{0,0}(X, e) is rationally connected, and if d^{2} ≥ n+2 then for e ≫ 0 the stack \overline{M}_{0,0}(X, e) is of general type; i.e., the Kodaira dimension of \overline{M}_{0,0}(X, e) is determined by the same inequality as in Lang’s theorem. This suggests Lang’s theorem is related to rational connectedness of spaces of rational curves. In a personal communication, A. J. de Jong has outlined an approach for proving that a K-variety X has a K-point if the stacks \overline{M}_{0,0}(X ⊗_{K} \overline{K}, e) are rationally connected, and a certain Brauer obstruction vanishes. Of course Lang’s original proof is simple and direct. But de Jong’s approach could apply to classes of varieties where Lang’s proof does not apply, i.e., to varieties that are not hypersurfaces in projective space.

The naive parameter count is not rigorous. As it seems impossible to make it rigorous, it is not recalled here. In this paper a different strategy is developed for proving the stacks \overline{M}_{0,0}(X, e) are rationally connected, and this strategy is applied to hypersurfaces in projective space. This strategy should also apply to other varieties, hence it is formulated in greater generality than strictly needed for the case of hypersurfaces.

1.1. The Kontsevich moduli space. The most natural parameter space for rational curves of degree e on X is the open subscheme of the Hilbert scheme parametrizing smooth rational curves of degree e on X, R_{X}^{e} ⊂ Hilb_{X}^{et+1}. For e > 1, R_{X}^{e} is not proper. To study the global geometry of R_{X}^{e}, e.g. to determine its Kodaira dimension, it is necessary to embed it as an open subset of a proper scheme. The simplest choice is to take the closure \overline{R_{X}^{e}} inside Hilb_{X}^{et+1}. This is a poor choice for two reasons: First, there is no simple characterization of the closed subschemes of X that correspond to points in \overline{R_{X}^{e}}. Second, the deformation theory of a closed subscheme of X is difficult to work with.

There is a better choice; one where the points of the closure have a simple geometric meaning, and where the deformation theory is easier to work with. This choice is \overline{M}_{0,0}(X, e), or more generally \overline{M}_{0,r}(X, e), the Kontsevich moduli space of degree e, r-pointed, genus 0 stable maps to X. This space has one disadvantage over \overline{R_{X}^{e}}; namely \overline{M}_{0,r}(X, e) is a Deligne-Mumford stack rather than a scheme. However, the coarse moduli space \overline{M}_{0,r}(X, e) is a projective scheme, cf. [2]. For the reader unfamiliar with stacks, most occurrences of \overline{M}_{0,r}(X, e) can safely be replaced by (\overline{M}_{0,r}(X, e))_{fine}, the fine moduli locus of \overline{M}_{0,r}(X, e).

To be precise, \overline{M}_{0,r}(X, e) is the stack whose objects are triples

ζ = ((p : Σ → B, σ_{1}, …, σ_{r}), g : Σ → X),

consisting of a flat, proper family of curves p : Σ → B, a collection of r disjoint sections σ_{i} : B → Σ, i = 1, …, r with image in the smooth locus of p, and a morphism g : Σ → X such that for each

RATIONAL CURVES ON HYPERSURFACES II 3

geometric point b ∈ B, the fiber Σ_{b} is a connected, at-worst-nodal curve of arithmetic genus 0, the morphism g_{b} : Σ_{b} → X ⊗_{ℂ} κ(b) has no infinitesimal automorphism fixing the marked points σ_{i}(b), and the degree of (g_{b})^{∗}O_{X}(1) is e, cf. [9], [5].

A refinement of this stack used in this paper is the Behrend-Manin stack, \overline{M}(X, τ), associated to a genus 0 stable A-graph τ, cf. [5]. A genus 0 stable A-graph is a tree with tails with a degree associated to each vertex that satisfies a certain stability condition. A tree is a graph that contains no cycles. A tail, or a half-edge, is an edge that originates on a vertex but does not terminate on a vertex (e.g. what one would get if one were to “cut in half” an edge of a usual graph). A degree function is an assignment of a nonnegative integer to each vertex of the graph. And the stability condition is that no vertex both has degree 0 and valence less than 3 (tails count toward the valence of a vertex).

Given an at-worst-nodal curve of genus 0, Σ, a collection of marked points on Σ, σ_{1}, …, σ_{r}, and a morphism g : Σ → X, there is an associated A-graph defined as follows. The tree is the dual graph of Σ; there is one vertex for each irreducible component of Σ and one edge for each node of Σ. For each marked point σ_{i} of Σ there is a tail attached to the obvious vertex. And the degree of a vertex is simply the degree of g^{∗}O(1) on the corresponding irreducible component. The Behrend-Manin stack \overline{M}(X, τ) (essentially) parametrizes the closure of the locally closed substack of \overline{M}_{0,r}(X, e) of stable maps whose associated A-graph equals τ. For the precise definition, cf. [5].

The boundary of the stack \overline{M}(X, τ) is a union of stacks \overline{M}(X, σ) where σ ranges over stable Agraphs such that there is a contraction from σ to τ, i.e., the graph σ is “more degenerate” than the graph τ, cf. [5]. Hence the boundary of every Behrend-Manin stack can be understood inductively starting from the “most degenerate” graphs τ, for which \overline{M}(X, τ) is a fiber product over X of the space of pointed lines on X. So questions about \overline{M}(X, τ) that can be studied by specializing to points in the boundary eventually reduce to questions about the space of pointed lines on X. Moreover, the deformation theory of a point in \overline{M}(X, τ) is straightforward; it will be recalled in Section 3.

1.2. Sketch of the proof. The proof of Theorem 1.1 uses a theorem of Kollár.

Theorem 1.4 (Kollár, Thm. IV.3.7 [16]). Let V be an irreducible, projective variety, and let V_{sm} ⊂ V denote the smooth locus of V. If there exists a very free morphism f : ℙ^{1} → V_{sm}, i.e. a morphism such that f^{∗}T_{V} is an ample vector bundle, then V is rationally connected.

The reader is warned that “very free” is the first of a multitude of definitions with “very” similar names: free, very free, deformation ample, very stable, unobstructed, twisting, very twisting, twistable, very twistable, positive, very positive, inducting pair, inductable, modification, typical, and c-generating linear system. A morphism f : ℙ^{1} → V is free if f^{∗}T_{V} is a vector bundle that is generated by global sections. The goal is to prove there exists a very free morphism to \overline{M}_{0,0}(X, e) for all e. It is difficult to directly construct a very free morphism. However, existence of a very free morphism can be studied by specializing to the boundary of \overline{M}_{0,0}(X, e), and, using an induction argument, can ultimately be reduced to a question about pointed lines on X. The induction argument does not produce a very free morphism to the boundary of \overline{M}_{0,0}(X, e), but rather a reducible rational curve B in the boundary such that the restriction of T_{\overline{M}_{0,0}(X,e)} to B is deformation ample: for a deformation of B to an irreducible rational curve B′, the restriction of T_{\overline{M}_{0,0}(X,e)} to B′ is ample, cf. Section 2 (a curve B′ is called a deformation of B if both B and B′ have a common generization). The rational curve B′ is a very free rational curve which proves that \overline{M}_{0,0}(X, e) is rationally connected.

The induction argument constructs a reducible rational curve B which itself parametrizes stable maps from reducible rational curves to X. Each of these stable maps is the union of a map of degree e − 1, g : Σ → X, and a line L ⊂ X that intersect in a node σ ∈ Σ ∩ L. Since the union is stable, the 1-pointed map (Σ, σ, g) is stable. To simplify the deformation theory, it is assumed that (Σ, σ, g) is very stable: the unmarked map (Σ, g) is stable (this assumption must be justified!). To guarantee that B is in the smooth locus of \overline{M}_{0,0}(X, e), it is assumed that (Σ, g) is unobstructed: 𝔼xt_{O_{Σ}}^{1}(L_{g}, O_{Σ}) = {0}.

The condition that T_{\overline{M}_{0,0}(X,e)}|_{B} is deformation ample can be translated into a condition on the family of pointed lines, (L, σ), together with a condition on the family of pointed maps of degree e − 1, (Σ, σ, g); these conditions are defined in Section 4. The condition on the family of pointed lines is that it is very twisting. The family is twisting (resp. very twisting) if the associated morphism ζ : B → \overline{M}_{0,1}(X, 1) pulls back the vertical tangent bundle of the projection pr : \overline{M}_{0,1}(X, 1) → \overline{M}_{0,0}(X, 1) to a bundle which is generated by global sections, and pulls back the vertical tangent bundle of the “evaluation at the marked point” morphism ev : \overline{M}_{0,1}(X, 1) → X to a bundle which is generated by global sections (resp. deformation ample). Composing ζ with ev gives a map h : B → X. Such a map which arises from a twisting (resp. very twisting) family is called twistable (resp. very twistable). The property of being twistable (resp. very twistable) is an open condition on the family of all maps.

The condition on the family of pointed maps of degree e − 1 is that it is very positive. The family is positive (resp. very positive) if the associated morphism ζ : B → \overline{M}_{0,1}(X, e − 1) pulls back the vertical tangent bundle of the projection pr : \overline{M}_{0,1}(X, 1) → \overline{M}_{0,0}(X, 1) to a bundle which is generated by global sections (resp. ample) and the pullback by pr ∘ ζ of T_{\overline{M}_{0,0}(X,1)} is deformation ample. The main observation is this: For a positive (resp. very positive) family over a smooth rational curve, the morphism ζ : B → \overline{M}_{0,1}(X, e − 1) is free (resp. very free).

An inducting pair consists of a very twisting family of pointed lines over B and a very positive family of pointed maps of degree e−1 over B which intersect along the marked points. The induction step proves that if an inducting pair exists for degree e, then an inducting pair exists for degree e+1. It is sometimes useful to “forget” the very twisting family of pointed lines, and only “remember” that the family of marked points is a very twistable map. A very positive family of pointed maps of degree e − 1 such that the family of marked points is a very twistable map is an inductable family.

The induction step begins with an inducting pair for degree e and produces an inductable family for degree e + 1. In particular, the inductable family is very positive and there is a deformation to a very free morphism from an irreducible curve to \overline{M}_{0,1}(X, e) whose projection to \overline{M}_{0,0}(X, e) is a very free morphism. The family of unmarked maps over B is obtained by gluing the very twisting family of lines and the very positive family of maps of degree e − 1 along the curve of marked points σ. To get a family of marked maps, let σ′ be an irreducible curve in the total space Σ of the very positive family which is linearly equivalent to σ. The curve σ′ fails to be a family of marked points precisely at the finitely many points in B over which the curve σ′ intersects the curve σ. The solution is to blow up each of these finitely many points on Σ; this produces a family of marked stable maps of degree e. Unfortunately blowing up destroys the “very positivity” of the family. Very positivity is restored by making a modification at each of the finitely many points of B over which Σ is blown up. The modification attaches a ℙ^{1} to B at the specified point and extends the family over this ℙ^{1} so that the stabilized family of unmarked maps is a constant family, but such that the

RATIONAL CURVES ON HYPERSURFACES II 5

marked points in this constant family vary. After modification, the family of marked stable maps is inductable. Incidentally, it is the process of modification that requires using reducible curves in \overline{M}_{0,0}(X, e) and working with the property of deformation ample bundles.

The last section establishes the base case of the induction argument: existence of a family of pointed lines that is both very twisting and very positive. The total space of a very positive family of lines is a scroll Σ. The whole argument reduces to producing a pair of a scroll Σ and a hypersurface X containing Σ such that Σ corresponds to a very positive family of lines on X. This reduces to a computation of the dimension of certain linear systems on the scroll Σ, c-generating linear systems. This is a straightforward computation in the Cox homogeneous coordinate ring of the scroll. The hypothesis that d^{2} + d + 1 ≤ n is used in this last computation.

1.3. Technical Hypotheses. In the proof of the induction step, some technical hypotheses are used.

Hypothesis 1.5. For each contraction of genus 0 stable A-graphs, ϕ : σ → τ, the image of the morphism of Behrend-Manin stacks \overline{M}(X, σ) → \overline{M}(X, τ) has codimension dim(X, τ) − dim(X, σ) in \overline{M}(X, τ).

By [11, Prop. 7.4], if d < \frac{n+1}{2} and if X ⊂ ℙ^{n} is a general hypersurface of degree d, each stack \overline{M}(X, σ) has the expected dimension; thus Hypothesis 1.5 holds for X.

Hypothesis 1.6. A general fiber of the evaluation map ev : \overline{M}_{0,1}(X, 1) → X is irreducible.

For a pair (X, p) consisting of a hypersurface X ⊂ ℙ^{n} of degree d and a point p ∈ X, the associated fiber of ev is a subvariety Z ⊂ ℙ^{n−1} which is a complete intersection of a sequence of hypersurfaces Y_{1}, …, Y_{d} in ℙ^{n} with deg(Y_{i}) = i: the defining equation of Y_{i} is the degree i homogeneous part of the Taylor expansion of the defining equation of X about the point p. If the pair (X, p) is general, the sequence of hypersurfaces Y_{1}, …, Y_{d} is general. By the Bertini theorem [14, Thms. 4.10, 6.10] the intersection Y_{1} ∩ ⋯ ∩ Y_{d} is smooth and connected if d ≤ n − 2.

Hypothesis 1.7. For each integer e ≥ 0, the locus in \overline{M}_{0,1}(X, e) parametrizing stable maps with nontrivial automorphism group has codimension at least 2.

Of course any stable map with nontrivial automorphism group has an irreducible component which is a multiple cover of its image. In light of [11, Prop. 7.4], a simple parameter count shows that if d ≤ \frac{n+1}{2} and if X ⊂ ℙ^{n} is a general hypersurface of degree d, Hypothesis 1.7 is satisfied.

1.4. Conventions. Unless stated otherwise, schemes are of finite type and separated over Spec ℂ. Absolute fiber products of schemes will be fiber products over Spec ℂ. Absolute fiber products of stacks will be 2-fibered products over Spec ℂ.

1.5. Acknowledgments. We are very grateful to A. J. de Jong and Steven Kleiman for many useful conversations. We are also grateful to the referees for their many useful comments.

2. Deformation ample

Let T be a scheme.

Definition 2.1. A family of prestable curves of genus g over T is a proper, flat morphism π : B → T such that every geometric fiber of π is a connected, at-worst-nodal curve of arithmetic genus g.

Notation 2.2. Let π : B → T be a morphism of schemes and let E be a quasi-coherent sheaf on B. Denote by μ_{E} : π^{∗}π_{∗}E → E the morphism left adjoint to the identity morphism π_{∗}E → π_{∗}E. The sheaf E is π-relatively generated by global sections if μ_{E} is surjective.

6 J. HARRIS AND J. STARR

Let π : B → T be a family of prestable curves of genus 0 and let E be a coherent sheaf on B. Let σ : T → B be a section of π and let I ⊂ O_{B} be the ideal sheaf of σ(T). Lemma 2.3.

(i) If E is π-relatively generated by global sections, then R^{1}π_{∗}E = {0}. (ii) If E is π-relatively generated by global sections, then R^{1}π_{∗}(I · E) = {0}. (iii) Let E′ and E′′ be coherent sheaves on B and let

0 ⟶ E′ ⟶ E ⟶ E′′ ⟶ 0

be a short exact sequence of coherent sheaves. If E′ and E′′ are π-relatively generated by global sections, then E is π-relatively generated by global sections.

Proof. All three statements can be proved locally over T. Hence it suffices to consider the case that T is affine.

(i): Since E is π-relatively generated by global sections and T is affine, E is generated by global sections, i.e., there is a short exact sequence of coherent sheaves,

0 ⟶ K ⟶ O_{B}^{⊕N} ⟶ E ⟶ 0.

Since π is of relative dimension 1, R^{2}π_{∗}K = {0}. Since the fibers of B are connected of arithmetic genus 0, R^{1}π_{∗}O_{B} = {0}. In the long exact sequence of higher direct images associated to the short exact sequence above, R^{1}π_{∗}E fits between R^{1}π_{∗}O_{B}^{⊕N} and R^{2}π_{∗}K; hence R^{1}π_{∗}E = {0}.

(ii): There is a short exact sequence of coherent sheaves,

0 ⟶ I · E ⟶ E ⟶ E ⊗_{O_{B}} O_{σ(T)} ⟶ 0

giving rise to a long exact sequence of cohomology groups,

π_{∗}E ⟶ π_{∗}(E ⊗_{O_{B}} O_{σ(T)}) ⟶ R^{1}π_{∗}(I_{b} · E) ⟶ R^{1}π_{∗}E.

By the last paragraph, R^{1}π_{∗}E = {0}. Since E is generated by global sections, π_{∗}E → π_{∗}(E ⊗_{O_{B}} O_{σ(T)}) is surjective. Therefore R^{1}π_{∗}(I · E) = {0}.

(iii): By (i), h^{1}(B, E′) = 0; therefore every global section of E′′ is the image of a global section of E. So the global sections of E generate E′′. And the global sections of E′ generate E′. Therefore E is generated by global sections. □

Let B be a prestable curve of genus 0, and let E be a locally free sheaf of positive rank on B. A smoothing of the pair (B, E) over a discrete valuation ring R is a pair (B, E) consisting of a family B → Spec R of prestable curves of genus 0 and a locally free sheaf E such that the generic fiber of B is a smooth curve, such that the closed fiber of B is isomorphic to B, and such that the restriction of E to the closed fiber is isomorphic to E. What conditions on (B, E) guarantee that for every smoothing (B, E), the restriction of E to the generic fiber is an ample locally free sheaf? Certainly if E is ample, this is true. But E need not be ample for this condition to hold: e.g. if E is an invertible sheaf such that the total degree of E is positive, then for every smoothing the restriction of E to the generic fiber is ample. Although it is not the most general criterion, the following criterion is used in the rest of the paper.

Definition 2.4. Let B be a connected, proper, at-worst-nodal curve of arithmetic genus 0. A locally free sheaf E on B with positive rank is deformation ample if

(i) E is generated by global sections, and

(ii) h^{1}(B, E(K_{B})) = 0, where O_{B}(K_{B}) is the dualizing sheaf of B. Remark 2.5.

RATIONAL CURVES ON HYPERSURFACES II 7

(i) Conditions (i) and (ii) above are independent.

(ii) If E is invertible, then E is deformation ample iff the restriction of E to every irreducible component has nonnegative degree and the restriction to at least one irreducible component has positive degree, cf. Lemma 2.11.

(iii) One can determine whether E is deformation ample in terms of the splitting type of the restriction of E to each irreducible component together with the patching isomorphisms at the nodes of B.

Let T be a scheme, let π : B → T be a family of prestable curves of genus 0, and let E be a locally free sheaf of positive rank on B. Definition 2.6. The sheaf E is π-relatively deformation ample (or simply deformation ample if π is understood) if

(i) E is π-relatively generated by global sections, and (ii) R^{1}π_{∗}(E(K_{π})) = {0}, where O_{B}(K_{π}) is the relative dualizing sheaf of π.

Let π : B → T be a proper, flat family of connected, at-worst-nodal curves of arithmetic genus 0, let E be a locally free sheaf on B of positive rank, and let f : T′ → T be a morphism of schemes. Denote the fiber product as in the diagram,

B′ ⟶^{g} B
π′↓  ↓^{π}
T′ ⟶^{f} T

Denote by E′ the pullback g^{∗}E. Lemma 2.7. If E is π-relatively deformation ample, then E′ is π′-relatively deformation ample. If f is surjective, the converse also holds.

Proof. (⇒): For the main direction, by [10, Sec. 8.5.2, Prop. 8.9.1], it suffices to consider the case when T and T′ are Noetherian affine schemes.

There is a canonical map of O_{T′}-modules, ν : f^{∗}π_{∗}E → (π′)_{∗}g^{∗}E, which fits into a commutative diagram,

(π′)^{∗}f^{∗}π_{∗}E ⟶^{=} g^{∗}π^{∗}π_{∗}E
(π′)^{∗}ν↓  ↓^{g^{∗}μ_{E}}
(π′)^{∗}(π′)_{∗}E′ ⟶^{μ_{E′}} E′

Since μ_{E} is surjective, g^{∗}μ_{E} is surjective. Hence also μ_{E′} is surjective, i.e., E′ is π′-relatively generated by global sections.

Since π has relative dimension 1, R^{2}π_{∗}E(K_{π}) = {0}. By [13, Thm. III.12.11(b)], for every closed point t ∈ T, h^{1}(B_{t}, E(K_{π})|_{B_{t}}) = 0. By [13, Prop. III.9.3], for every closed point t′ ∈ T′, h^{1}(B′_{t′}, E′(K_{π′})|_{B′_{t′}}) = 0. So by [13, Thm. III.12.11(a)] and Nakayama’s lemma, R^{1}π′_{∗}(E′(K_{π′})) = {0}. Hence E′ is π′-relatively deformation ample.

(⇐): Now suppose that f is surjective and that E′ is π′-relatively deformation ample. As above, it suffices to consider the case when T and T′ are Noetherian affine schemes. As above, for every closed point t′ ∈ T′, h^{1}(B′_{t′}, E′(K_{π′})|_{B′_{t′}}) = 0. Since T′ → T is surjective, by [13, Prop. III.9.3] for every closed point t ∈ T, h^{1}(B_{t}, E(K_{π})|_{B_{t}}) = 0. So by [13, Thm III.12.11(a)] and Nakayama’s lemma, R^{1}π_{∗}(E(K_{π})) = {0}.

It remains to prove that E is π-relatively generated by global sections. For every closed point t ∈ T, there is a closed point t′ ∈ T′ mapping to t. Since E′|_{B′_{t′}} is generated by global sections, also E|_{B_{t}} is generated by global sections. By Lemma 2.3, h^{1} (B_{t}, E|_{B_{t}}) = 0. By [13, Thm. III.12.11(a)] and Nakayama’s lemma, R^{1}π_{∗}(E) = {0}.

The claim is that for any coherent O_{T} -module F, R^{1}π_{∗}(π^{∗}F ⊗ E) = {0}. This is local on T. Locally on T, F is the cokernel of O_{T}^{⊕N} for some N, so π^{∗}F ⊗ E is the cokernel of E^{⊕N}. Since π has relative dimension 1, R^{1}π_{∗} is right exact on the category of coherent O_{B}-modules. Since R^{1}π_{∗}(E^{⊕N}) = {0}, also R^{1}π_{∗}(π^{∗}F ⊗ E) = {0}, which proves the claim.

In particular, applying the long exact sequence of higher direct images to the short exact sequence,

0 ⟶ π^{∗}I_{t} ⊗ E ⟶ E ⟶ E|_{B_{t}} ⟶ 0,

π_{∗}(E) → H^{0} (B_{t}, E|_{B_{t}}) is surjective. Since E|_{B_{t}} is generated by global sections for every closed point t ∈ T, E is π-relatively generated by global sections. So E is π-relatively deformation ample. □

Lemma 2.8. Let π : B → T be a proper, flat family of connected, smooth curves of genus 0 and let E be a locally free sheaf of positive rank. Then E is π-relatively deformation ample iff E is π-relatively ample.

Proof. Both properties are local on T and can be checked after étale, surjective base-change of T. So it suffices to consider the case when π : B → T is isomorphic to π_{T} : T × ℙ^{1} → T.

(⇒): Denote F = (π_{T})_{∗}(E ⊗ π_{ℙ^{1}}^{∗}O_{ℙ^{1}}(−1)). Tensoring the map μ_{E⊗π_{ℙ^{1}}^{∗} O_{ℙ^{1}} (−1)} with the identity map on π^{∗}O_{ℙ^{1}}(−1) gives a map ν : π_{T}^{∗} F ⊗ π_{ℙ^{1}}^{∗}O_{ℙ^{1}}(1) → E. Assume that E is deformation ample. The claim is that ν is surjective. To prove this, it suffices to prove,

(i) For every geometric point t of T, h^{1}(ℙ_{κ(t)}^{1}, E|_{B_{t}} ⊗ O_{ℙ^{1}}(−1)) = 0, (ii) F ⊗_{O_{T}} κ(t) = H^{0}(ℙ_{κ(t)}^{1}, E|_{B_{t}} ⊗ O_{ℙ^{1}}(−1)), and (iii) the map H^{0}(ℙ_{κ(t)}^{1}, E|_{B_{t}} ⊗ O_{ℙ^{1}}(−1)) ⊗ O_{ℙ^{1}}(1) → E|_{B_{t}} is surjective.

By Grothendieck’s lemma [13, Exer. V.2.6], E|_{B_{t}} splits as a direct sum O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{r}) for some integers a_{1} ≤ ⋯ ≤ a_{r}. By Lemma 2.7, E|_{B_{t}} is deformation ample, and in particular h^{1}(ℙ^{1}, E|_{B_{t}}(−2)) = 0. Hence a_{1} ≥ 1, and h^{1}(ℙ^{1}, E|_{B_{t}} ⊗ O_{ℙ^{1}}(−1)) = 0, i.e., (i) holds. By [13, Thm. III.12.11(b)], also (ii) holds. Finally, for a_{i} ≥ 1, H^{0}(ℙ_{κ}^{1}, O_{ℙ^{1}}(a_{i} − 1)) ⊗ O_{ℙ^{1}}(1) → O_{ℙ^{1}}(a_{i}) is surjective. Thus (iii) holds and the claim is proved.

Now π_{T}^{∗} F ⊗ π_{ℙ^{1}}^{∗}O_{ℙ^{1}}(1) is π_{T} -relatively ample. Since E is a quotient of π_{T}^{∗} F ⊗ π_{ℙ^{1}}^{∗}O_{ℙ^{1}}(1), also E is π_{T} -relatively ample, cf. Lemma 2.10 (i).

(⇐): The converse direction follows in the same way. □ Lemma 2.9. There exists an open subscheme i : U → T with the following property: for every morphism f : T′ → T, f(T′) is contained in U iff E′ is π′-relatively deformation ample.

Proof. By [10, Sec. 8.5.2, Prop. 8.9.1], it suffices to consider the case that T and T′ are Noetherian affine schemes.

Let Z_{1} ⊂ T be the closed subset,

Z_{1} = f[Supp(coker(π^{∗}π_{∗}E → E))].

RATIONAL CURVES ON HYPERSURFACES II 9

Let Z_{2} ⊂ T be the closed subset,

Z_{2} = Supp(R^{1}π_{∗}(E(K_{π}))).

And let Z_{3} ⊂ T be the closed subset,

Z_{3} = Supp(R^{1}π_{∗}E).

Let i : U → T be the open complement of Z_{1} ∪ Z_{2} ∪ Z_{3}.

Let f : T′ → T be a morphism of Noetherian affine schemes. By [13, Thm. III.12.11, Prop. III.9.3] and Nakayama’s lemma, R^{1}π′_{∗}(E′(K_{π′})) = {0} iff for each closed point t′ ∈ T′, h^{1}(B′_{t′}, E′(K_{π′})|_{B′_{t′}}) = 0. Denoting t = f(t′), h^{1}(B′_{t′}, E′(K_{π′})|_{B′_{t′}}) = 0 iff h^{1}(B_{t}, E(K_{π})|_{B_{t}}) = 0, i.e., iff t is contained in the complement of Z_{2}. Hence R^{1}π′_{∗}(E′(K_{π′})) = {0} iff f(T) is contained in the complement of Z_{2}.

By the same argument as in the converse direction of the proof of Lemma 2.7, E′ is π′-relatively generated by global sections iff for every closed point t′ ∈ T′, E′|_{B′_{t′}} is generated by global sections. Denoting t = f(t′), E′|_{B′_{t′}} is generated by global sections iff E|_{B_{t}} is generated by global sections. If E|_{B_{t}} is generated by global sections, then h^{1}(B_{t}, E|_{B_{t}}) = 0. By [13, Thm. III.12.11], t is not in Z_{3} and t is not in Z_{1}. Conversely, if t is not in Z_{3}, then E|_{B_{t}} is generated by global sections iff t is not in Z_{1}. Thus E′ is π′-relatively generated by global sections iff f(T′) is contained in the complement of Z_{1} ∪ Z_{3}. So E′ is π′-relatively deformation ample iff f(T′) is contained in U. □ Lemma 2.10.

(i) If χ : E → E′′ is a morphism of locally free sheaves on B whose cokernel is torsion in every fiber (in particular, if χ is surjective), if E′′ is nonzero, and if E is π-relatively deformation ample, then also E′′ is π-relatively deformation ample.

(ii) If E′ and E′′ are π-relatively deformation ample, then for every short exact sequence of coherent O_{B}-modules,

0 ⟶ E′ ⟶^{υ} E ⟶^{χ} E′′ ⟶ 0,

E is π-relatively deformation ample. (iii) If E is π-relatively deformation ample, then for every integer n ≥ 1, also E^{⊗n} is π-relatively

deformation ample.

Proof. (i): Let Q denote the cokernel of χ and let I ⊂ E′′ denote the image of χ. There is a short exact sequence of coherent O_{B}-modules:

0 ⟶ I ⟶ E′′ ⟶ Q ⟶ 0.

When we “twist” this exact sequence by O_{B}(K_{π}), it remains exact. Because π has relative dimension 1, R^{1}π_{∗} is right exact on the category of coherent O_{B}-modules. In particular, since R^{1}π_{∗}E(K_{π}) = {0}, also R^{1}π_{∗}I(K_{π}) = {0}. Since Q is torsion in every fiber, R^{1}π_{∗}Q(K_{π}) = {0}. Thus, by the long exact sequence of higher direct images associated to the twisted exact sequence above, R^{1}π_{∗}E′′(K_{π}) = {0}.

The surjective composition map

π^{∗}π_{∗}E_{1} ⟶^{μ_{E}} E_{1} ⟶ I

factors through the natural map μ_{I} : π^{∗}π_{∗}I → I. Hence μ_{I} is surjective, i.e., I is π-relatively generated by global sections. Since Q is torsion in every fiber, the support of Q is finite over T and it follows that Q is π-relatively generated by global sections. By Lemma 2.3 (iii), E′′ is π-relatively generated by global sections. So E′′ is π-relatively deformation ample.

(ii): By hypothesis, R^{1}π_{∗}E′(K_{π}) = R^{1}π_{∗}E′′(K_{π}) = {0}. By the long exact sequence of higher direct images, also R^{1}π_{∗}E(K_{π}) = {0}. By Lemma 2.3 (iii), E is π-relatively generated by global sections. So E is π-relatively deformation ample.

(iii): This is proved by induction on n, the case n = 1 being tautological. It suffices to consider the case when T is affine. Suppose n > 1 and suppose the result is known for n − 1. In particular, E^{⊗(n−1)} is generated by global sections. There is a natural surjection

π^{∗}π_{∗} (E^{⊗(n−1)}) ⊗_{O_{B}} E → E^{⊗n}.

And there is a surjective map O_{T}^{⊕N} → π_{∗}(E^{⊗(n−1)}). Hence there is a surjection E^{⊕N} → E^{⊗n}. By (ii) and induction, E^{⊕N} is π-relatively deformation ample. By (i), the quotient E^{⊗n} is π-relatively deformation ample. Thus (iii) is proved by induction. □ Lemma 2.11. Let B be a proper, connected, at-worst-nodal curve of arithmetic genus 0 over an algebraically closed field k. Let E be a locally free sheaf of positive rank such that,

(i) for every irreducible component B_{i} ⊂ B, E|_{B_{i}} is generated by global sections, and (ii) there exists a nonempty, connected, closed subcurve B′ ⊂ B such that E|_{B′} is deformation

ample. Then E is deformation ample.

Proof. Let δ be the number of irreducible components of B which are not contained in B′. The result is proved by induction on δ. The base case δ = 0 is tautological, for then B = B′. Assume that δ > 0 and that the result is true for all smaller values of δ.

Let B_{1} ⊂ B be an irreducible component of B. Let B_{2} ⊂ B denote the union of all irreducible components other than B_{1}. There exists an irreducible component B_{1} not contained in B′ such that B_{2} is connected: if the dual graph of B′ contains every leaf (= vertex of valence 1) of the dual graph of B, then the two graphs are equal. The intersection B_{1} ∩ B_{2} is a single node, denoted b. By the induction hypothesis, E|_{B_{2}} is deformation ample.

The claim is that E is generated by global sections. Denote by F ⊂ E the image of H^{0}(B, E) ⊗_{k} O_{B} → E. There is a short exact sequence of coherent sheaves:

0 ⟶ E ⊗_{O_{B}} O_{B_{1}}(−b) ⟶ E ⟶ E ⊗_{O_{B}} O_{B_{2}} ⟶ 0.

Since E|_{B_{1}} is a locally free sheaf on ℙ^{1} generated by global sections, Grothendieck’s lemma and the cohomology of line bundles on ℙ^{1} imply that h^{1}(B, E ⊗_{O_{B}} O_{B_{1}}(−b)) = 0 (h^{1}(ℙ^{1}, O_{ℙ^{1}}(a)) = 0 for a ≥ −1). Hence all the global sections of E|_{B_{2}} lift to global sections of E, i.e., F → E|_{B_{2}} is surjective. So E/F is supported on B_{1}. Hence E/F is a quotient of E|_{B_{1}}. Since E|_{B_{1}} is generated by global sections, also E/F is generated by global sections. There is a short exact sequence

0 ⟶ F ⟶ E ⟶ E/F ⟶ 0.

By Lemma 2.3 (iii), E is generated by global sections.

There is a short exact sequence of coherent sheaves,

0 → E(K_{B}) ⊗_{O_{B}} O_{B_{2}}(−b) → E(K_{B}) → E(K_{B}) ⊗_{O_{B}} O_{B_{1}} → 0.

This gives a long exact sequence in cohomology, part of which is,

H^{1}(B, E(K_{B}) ⊗_{O_{B}} O_{B_{2}}(−b)) → H^{1}(B, E(K_{B})) → H^{1}(B, E(K_{B}) ⊗_{O_{B}} O_{B_{1}}) → 0.

RATIONAL CURVES ON HYPERSURFACES II 11

The inclusion map B_{2} → B is finite, therefore there is a canonical isomorphism of O_{B}-modules (cf. [12, Sec. III.6], [17, Cor. 5.68]),

O_{B_{2}}(K_{B_{2}}) ≅ Hom_{O_{B}}(O_{B_{2}}, O_{B}(K_{B})) ≅ O_{B}(K_{B}) ⊗_{O_{B}} O_{B_{2}}(−b).

Hence h^{1}(B, E(K_{B}) ⊗_{O_{B}} O_{B_{2}}(−b)) = h^{1}(B_{2}, E ⊗_{O_{B}} O_{B_{2}}(K_{B_{2}})), which is zero by the induction assumption. Similarly, E(K_{B}) ⊗_{O_{B}} O_{B_{1}} is isomorphic to E ⊗_{O_{B}} O_{B_{1}}(−1) (identifying B_{1} with ℙ^{1}). Since E|_{B_{1}} is generated by global sections, it follows by Grothendieck’s lemma and the cohomology of line bundles on ℙ^{1} that h^{1}(B_{1}, E|_{B_{1}}(−1)) = 0. Hence h^{1}(B, E(K_{B})) = 0, and E is deformation ample. Therefore the lemma is proved by induction on δ. □

Remark 2.12. A particular case of Lemma 2.11 is when B′ is one irreducible component of B, in which case the lemma says that a locally free sheaf on B which is generically ample in the sense of Lazarsfeld [8] is deformation ample.

3. Deformation theory of stable maps

The Kontsevich moduli space of genus 0 stable maps, \overline{M}_{0,r}(X, e), and the Behrend-Manin moduli spaces, \overline{M}(X, τ), are described in Subsection 1.1. The deformation theory of stable maps has been worked out in [4] and [3]. Many specific deformation-theoretic results follow easily from these papers and are known to the experts, but have not been written down. Some of these specific results are proved in this section. Although these results will only be applied to genus 0 stable maps in this paper, the same arguments work for stable maps of arbitrary genus; in this section only, stable maps and stable A-graphs are not necessarily assumed to be of genus 0.

Let X and T be schemes.

Definition 3.1. A family of r-pointed, genus g prestable maps to X over B is a triple

ζ = ((p : Σ → B, σ_{1}, …, σ_{r}), g : Σ → X)

consisting of a family p : Σ → B of prestable curves of genus g over B, a sequence of r disjoint sections σ_{i} : B → Σ with image contained in the smooth locus of p (if r = 0, the sections are omitted), and a morphism g : Σ → X.

Let ((Σ, σ_{1}, …, σ_{r}), g) be a prestable map over an algebraically closed field. An irreducible component Σ_{i} ⊂ Σ is stable if the restriction of the log-dualizing sheaf, O_{Σ}(K_{Σ} + σ_{1} + ⋯ + σ_{r}) ⊗_{O_{Σ}} O_{Σ_{i}}, is g-relatively ample, i.e., one of the following hold,

(i) g : Σ_{i} → X is nonconstant, (ii) p_{a}(Σ_{i}) > 1, (iii) p_{a}(Σ_{i}) = 1 and Σ_{i} contains at least one marked point or external node of Σ, or (iv) p_{a}(Σ_{i}) = 0 and Σ_{i} contains at least three marked points and nodes of B.

The curve Σ is stable if O_{Σ}(K_{Σ}+σ_{1}+⋯+σ_{r}) is g-relatively ample, i.e., every irreducible component of Σ is stable. The family ζ is stable if for each geometric point t ∈ T, the curve Σ_{t} is stable.

Now assume that X is smooth.

Notation 3.2. Denote by L_{ζ} the complex of coherent sheaves on Σ

\begin{matrix}−1 & & 0 \\ g^{∗}Ω_{X} & ⟶^{(dg)^{†}} & Ω_{p}(σ_{1}(T) + ⋯ + σ_{r}(T)) \end{matrix}  (1)

For a scheme T and a bounded above complex of coherent sheaves C on T, C^{∨} denotes the object in the derived category of coherent sheaves on T,

C^{∨} := ℝHom_{O_{T}} (C, O_{T}).

12 J. HARRIS AND J. STARR

In particular, L_{ζ}^{∨} is the object

L_{ζ}^{∨} = ℝHom_{O_{Σ}}(L_{ζ}, O_{Σ})

in the derived category of coherent sheaves on Σ.

The relevance of the complex L_{ζ}^{∨} is the following.

Lemma 3.3. Let X be a smooth quasi-projective scheme. Let \overline{M}_{g,r}(X, β) denote the Deligne-Mumford stack of r-pointed stable maps to X of arithmetic genus g and degree β. Let p : Σ → \overline{M}_{g,r}(X, β) denote the universal curve, let σ_{i} : \overline{M}_{g,r}(X, β) → Σ denote the universal sections, and let g : Σ → X denote the universal map, i.e.

ζ = ((p : Σ → \overline{M}_{g,r}(X, β), σ_{1}, …, σ_{r}) , g : Σ → X)

is the universal family of stable maps. There is an obstruction theory for \overline{M}_{g,r}(X, β) in the sense of [4, Def. 4.4] of the form

(ℝp_{∗}(L_{ζ}^{∨})[1])^{∨} → L_{\overline{M}_{g,r}(X,β)}.

A similar result holds for prestable maps, cf. Remark 3.4.

Proof. Essentially this follows from [4] and [3]. □

Remark 3.4. Explicitly, if ζ = ((Σ, σ_{1}, …, σ_{r}), g : Σ → X) is a stable map or a prestable map, the space of first order deformations of ζ is 𝔼xt_{O_{Σ}}^{1}(L_{ζ}, O_{Σ}) and the obstruction group is a subgroup of 𝔼xt_{O_{Σ}}^{2}(L_{ζ}, O_{Σ}). In the case of a prestable map, the space of infinitesimal automorphisms of the map is 𝔼xt_{O_{Σ}}^{0}(L_{ζ}, O_{Σ}) (for stable maps this group is zero). In particular, if 𝔼xt_{O_{Σ}}^{2}(L_{ζ}, O_{Σ}) vanishes, then \overline{M}_{g,r}(X, β) is smooth at the point [ζ].

Lemma 3.5. Let ((Σ, σ_{1}, …, σ_{r}), g) be a prestable map. Let Θ_{Σ} denote the tangent sheaf of Σ, i.e. the dual of Ω_{Σ}. The space of infinitesimal automorphisms of ζ, 𝔼xt_{O_{Σ}}^{0}(L_{ζ}, O_{Σ}), is canonically isomorphic to a subspace of the space of infinitesimal automorphisms of (Σ, σ_{1}, …, σ_{r}), H^{0}(Σ, Θ_{Σ}(−(σ_{1} + ⋯ + σ_{r}))).

An irreducible component Σ_{i} ⊂ Σ is stable iff the restriction of every infinitesimal automorphism of ζ to Θ_{Σ}(−(σ_{1} + ⋯ + σ_{r})) ⊗_{O_{Σ}} O_{Σ_{i}} is zero. Moreover, if Σ_{i} is unstable, then every infinitesimal automorphism of the nodal curve Σ_{i} that fixes all marked points and nodes of Σ is the image of an infinitesimal automorphism of ζ.

Proof. Analyzing the spectral sequence for hypercohomology, 𝔼xt_{O_{Σ}}^{0}(L_{ζ}, O_{Σ}) is canonically isomorphic to the kernel of

Ext_{O_{Σ}}^{0}(Ω_{Σ}(σ_{1} + ⋯ + σ_{r}), O_{Σ}) → Ext_{O_{Σ}}^{0}(g^{∗}Ω_{X}, O_{Σ}).

And Ext_{O_{Σ}}^{0}(Ω_{Σ}(σ_{1} + ⋯ + σ_{r}), O_{Σ}) equals H^{0}(Σ, Θ_{Σ}(−(σ_{1} + ⋯ + σ_{r}))).

Assume that Σ_{i} is stable, i.e., Σ_{i} satisfies one of the Cases (i)–(iv) in Definition 3.1. In Cases (ii)–(iv), the image of

H^{0}(Σ, Θ_{Σ}(−(σ_{1} + ⋯ + σ_{r}))) → H^{0}(Σ_{i}, Θ_{Σ}(−(σ_{1} + ⋯ + σ_{r})) ⊗_{O_{Σ}} O_{Σ_{i}})

is zero, hence the restriction of every infinitesimal automorphism of ζ is zero. In Case (i), chasing through diagrams, the image of H^{0}(Σ, Θ_{Σ}(−(σ_{1} + ⋯ + σ_{r}))) is contained in the subsheaf which is the kernel of d(g|_{Σ_{i}}) : Θ_{Σ_{i}} → g^{∗}T_{X} ⊗_{O_{Σ}} O_{Σ_{i}}. Since g is non-constant, this map is nonzero. And Θ_{Σ_{i}} is a torsion-free sheaf, so the kernel of d(g|_{Σ_{i}}) is zero. Hence, also in Case (i), the restriction of every infinitesimal automorphism of ζ is zero.

RATIONAL CURVES ON HYPERSURFACES II 13

Assume that Σ_{i} is not stable. Then g contracts Σ_{i} to a point, and either Σ = Σ_{i} is a curve of arithmetic genus 1 and there are no marked points, or Σ_{i} is a smooth rational curve which contains at most two marked points and nodes of Σ. In both cases there is a positive dimensional group of automorphisms of Σ_{i} which fix all marked points and nodes of Σ: P ic^{τ} (Σ_{i}) acting by translation if Σ_{i} has arithmetic genus 1, and the group of automorphisms of ℙ^{1} fixing (at most) two points if Σ_{i} is a smooth rational curve. The Lie algebra of this positive dimensional group is the Lie algebra of infinitesimal automorphisms of the marked curve Σ_{i}. This group of automorphisms of Σ_{i} extends to a group of automorphisms of the map ζ. Therefore every infinitesimal automorphism of the marked curve Σ_{i} extends to an infinitesimal automorphism of ζ. In particular there exists an infinitesimal automorphism of ζ whose restriction to Σ_{i} is nonzero. □

Let f : X → Y be a smooth morphism of smooth quasi-projective varieties. Let ζ = ((Σ, σ_{1}, …, σ_{r}), g : Σ → X) be a prestable map. Denote by h : Σ → Y the composition h = f ∘ g, and denote by ξ the prestable map ((Σ, σ_{1}, …, σ_{r}), h : Σ → Y). There is a short exact sequence of complexes

0 ⟶ L_{ξ} ⟶ L_{ζ} ⟶ g^{∗}Ω_{f}[1] ⟶ 0,

defined by the following commutative diagram,

−1  0
L_{ξ} : g^{∗}f^{∗}Ω_{Y} ⟶^{dh^{†}} Ω_{Σ}(σ_{1} + ⋯ + σ_{r})
↓ ^{g^{∗}(df)^{†}}↓  ↓^{=}
L_{ζ} : g^{∗}Ω_{X} ⟶^{df^{†}} Ω_{Σ}(σ_{1} + ⋯ + σ_{r})
↓  ↓  ↓
g^{∗}Ω_{f}[1] : g^{∗}Ω_{f} ⟶  0

Lemma 3.6. (i) If ξ is stable, then ζ is stable.

(ii) If the dimension of the obstruction group of ζ is 0, then the dimension of the obstruction

group of ξ is 0. (iii) If h^{1}(Σ, g^{∗}Ω_{f}^{∨}) = 0 and if the dimension of the obstruction group of ξ is 0, then the dimension

of the obstruction group of ζ is 0, and the map from the space of first order deformations of ζ to the space of first order deformations of ξ is surjective.

Proof. (i): If the log dualizing sheaf of (Σ, σ_{1}, …, σ_{r}) is h-ample, then it is g-ample.

(ii): Associated to the short exact sequence of complexes above, there is a long exact sequence of hyperExt, part of which is

𝔼xt_{O_{Σ}}^{2}(L_{ζ}, O_{Σ}) ⟶ 𝔼xt_{O_{Σ}}^{2}(L_{ξ}, O_{Σ}) ⟶ 𝔼xt_{O_{Σ}}^{3}(g ∗ Ω_{f}[1], O_{Σ}).

Of course 𝔼xt_{O_{Σ}}^{k}(g^{∗}Ω_{f}[1], O_{Σ}) = H^{k−1}(Σ, g^{∗}Ω_{f}^{∨}). In particular dim𝔼xt_{O_{Σ}}^{3}(g^{∗}Ω_{f}[1], O_{Σ}) = h^{2}(Σ, g^{∗}Ω_{f}^{∨}) = 0. Therefore if dim𝔼xt_{O_{Σ}}^{2}(L_{ζ}, O_{Σ}) = 0, then dim𝔼xt_{O_{Σ}}^{2}(L_{ξ}, O_{Σ}) = 0.

(iii): If h^{1}(Σ, g^{∗}Ω_{f}^{∨}) = 0, then 𝔼xt_{O_{Σ}}^{2}(L_{ζ}, O_{Σ}) → 𝔼xt_{O_{Σ}}^{2}(L_{ξ}, O_{Σ}) is an isomorphism. Hence if the dimension of the obstruction group of ζ is 0, then the dimension of the obstruction group of ξ is 0. Moreover the preceding two terms in the long exact sequence of hyperExts give a surjection 𝔼xt_{O_{Σ}}^{1}(L_{ζ}, O_{Σ}) → 𝔼xt_{O_{Σ}}^{1}(L_{ξ}, O_{Σ}). □

Let q : S → B be a smooth morphism and let i : Σ → S be an unramified morphism of Bschemes. Then (g, i) : Σ → X × S is an unramified morphism of B-schemes, hence the map of

14 J. HARRIS AND J. STARR

coherent sheaves

d(g, i)^{†} : (g, i)^{∗}Ω_{(}X × S/B) → Ω_{p}

is surjective. Because p is flat of relative dimension 1 and the geometric fibers are reduced, local complete intersection schemes, the kernel of d(g, i)^{†} is a locally free sheaf N_{(g,i)}^{∨}. Similarly the kernel of di^{†} : i^{∗}Ω_{q} → Ω_{p} is a locally free sheaf N_{i}^{∨}.

There is a short exact sequence of coherent sheaves

0 ⟶ N_{i}^{∨} ⟶ N_{(g,i)}^{∨} ⟶ g^{∗}Ω_{X} ⟶ 0.

Denote by N_{(ζ,i)}^{∨} the subsheaf of N_{(g,i)}^{∨}(σ_{1}(B) + ⋯ + σ_{r}(B)) that contains N_{i}^{∨}(σ_{1}(B) + ⋯ + σ_{r}(B)) and such that N_{(ζ,i)}^{∨}/N_{i}^{∨}(σ_{1}(B)+⋯+σ_{r}(B)) is identified with g^{∗}Ω_{X} ⊂ g^{∗}Ω_{X}(σ_{1}(B)+⋯+σ_{r}(B)). Observe that N_{(ζ,i)}^{∨} is a locally free sheaf.

Notation 3.7. There is a canonical map N_{(g,i)}^{∨}(σ_{1}(B) + ⋯ + σ_{r}(B)) → (g, i)^{∗}Ω_{X×S/B}. The two projections give a canonical isomorphism of Ω_{X×S/B} with π_{X}^{∗}Ω_{X} ⊕π_{S}^{∗}Ω_{q}. Denote by γ_{(ζ,i)} : N_{(ζ,i)}^{∨} → g^{∗}Ω_{X}(σ_{1}(B) + ⋯ + σ_{r}(B)) ⊕ i^{∗}Ω_{q}(σ_{1}(B) + ⋯ + σ_{r}(B)) the induced morphism. Observe that the composition of γ_{(ζ,i)} with projection on the first summand factors through g^{∗}Ω_{X} ⊂ g^{∗}Ω_{X}(σ_{1}(B) + ⋯ + σ_{r}(B)). Denote by α_{(ζ,i)} : N_{(ζ,i)}^{∨} → g^{∗}Ω_{X} and β_{(ζ,i)} : N_{(ζ,i)}^{∨} → i^{∗}Ω_{q}(σ_{1}(B) + ⋯ + σ_{r}(B)) the composition of γ_{(ζ,i)} with the two projections. Denote by L_{(ζ,i)} the complex of locally free O_{Σ}-modules concentrated in degrees [−1, 0],

\begin{matrix}−1 & & 0 \\ N_{(ζ,i)}^{∨} & ⟶^{β_{(ζ,i)}} & i^{∗}Ω_{q}(σ_{1}(B) + ⋯ + σ_{r}(B)). \end{matrix}

Denote by λ_{(ζ,i)} : L_{(ζ,i)} → L_{ζ} the quasi-isomorphism of complexes of coherent O_{Σ}-modules,

\begin{matrix}N_{(ζ,i)}^{∨} & ⟶^{β_{(ζ,i)}} & i^{∗}Ω_{q}(σ_{1}(B) + ⋯ + σ_{r}(B)) \\ ^{α_{(ζ,i)}}↓ & ↓^{(di)^{†}} & . \\ g^{∗}Ω_{X} & ⟶^{(dg)^{†}} & Ω_{p}(σ_{1}(B) + ⋯ + σ_{r}(B)) \end{matrix}

The relevance of λ_{(ζ,i)} : L_{(ζ,i)} → L_{ζ} is that the complex L_{(ζ,i)}^{∨} is easy to compute since L_{(ζ,i)} is a complex of locally free sheaves; it is simply

\begin{matrix}0 & & 1 \\ i^{∗}T_{q}(−(σ_{1}(B) + ⋯ + σ_{r}(B))) & ⟶^{β_{(ζ,i)}^{†}} & N_{(ζ,i)}. \end{matrix}

In most applications, B will be the spectrum of a field and S will be a surface.

3.1. Contracting unstable components. In this subsection, the base B will always be the spectrum of an algebraically closed field. The changes necessary to get relative versions of the lemmas over a more general base are straightforward.

Let

ζ = ((Σ, σ_{1}, …, σ_{r}), g : Σ → X)

be a prestable map, let

(Σ′, σ′_{1}, …, σ′_{r}, σ′′_{1}, …, σ′′_{s})

be a proper, connected, at-worst-nodal curve, and let

u : (Σ′, σ′_{1}, …, σ′_{r}) → (Σ, σ_{1}, …, σ_{r})

RATIONAL CURVES ON HYPERSURFACES II 15

be a map which contracts some of the unstable components of (Σ′, σ′_{1}, …, σ′_{r}) (i.e., u contracts some of the irreducible components of Σ′ which have arithmetic genus 0 and contain fewer than three nodes and marked points). Denote g′ = g ∘ u and denote by ζ′ the prestable map

ζ′ = ((Σ′, σ′_{1}, …, σ′_{r}, σ′′_{1}, …, σ′′_{s}), g′ : Σ′ → X).

What is the relationship of 𝔼xt_{O_{Σ}}(L_{ζ}, O_{Σ}) and 𝔼xt_{O_{Σ′}} (L_{ζ′}, O_{Σ′})?

Any morphism u : Σ′ → Σ as above can be factored as a sequence of elementary morphisms, defined below.

Definition 3.8. (i) The morphism u is a Type I elementary morphism if ζ = (Σ, g : Σ → X) is a prestable map without marked points, u : Σ′ → Σ contracts a single unstable component to a smooth point of Σ, and ζ′ = (Σ′, g′ = g ∘ u : Σ′ → X).

(ii) The morphism u is a Type II elementary morphism if ζ = (Σ, g : Σ → X) is a prestable map without marked points, u : Σ′ → Σ contracts a single unstable component to a node of Σ, and ζ′ = (Σ′, g′ = g ∘ u : Σ′ → X).

(iii) The morphism u is a Type III elementary morphism if ζ = (Σ, (σ_{1}, …, σ_{r}), g : Σ → X) is a marked prestable map, ζ′ is the same prestable map but with one extra marked point, and u : Σ′ → Σ is the identity map.

Let S be a smooth surface and let i : Σ → S be a closed immersion. Let s ∈ Σ ⊂ S be a closed point.

Notation 3.9. Denote by v : S′ → S the blowing up of S at s. Denote by E ⊂ S the exceptional divisor. Denote by i′ : Σ′ → S′ the reduced total transform of B; i.e., the reduced scheme of v^{−1}(Σ). Denote by v also the morphism of pairs v : (S′, Σ′) → (S, Σ). Denote by u : Σ′ → Σ the restriction of v to Σ′. Denote by g′ : Σ′ → X the composition g′ = g ∘ u. Denote by ζ′ the prestable map (Σ′, g′ : Σ′ → X). Denote by Γ ⊂ Σ′ the closed (not necessarily connected) subcurve which is the union of all irreducible components other than E, and denote D = E ∩ Γ.

Definition 3.10. (i) The morphism of pairs is Type Ia if s ∈ Σ is a smooth point that lies on

a stable component. (ii) The morphism of pairs is Type Ib if s ∈ Σ is a smooth point that lies on an unstable

component. (iii) The morphism of pairs is Type I if it is Type Ia or Type Ib. (iv) The morphism of pairs is Type IIa if s ∈ Σ is a node, and there exists a first order defor-

mation of ζ that smoothes the node s (to first order).

(v) The morphism of pairs is Type IIb if s ∈ Σ is a node, and there is no first order deformation of ζ that smoothes the node s (to first order).

(vi) The morphism of pairs is Type II if it is Type IIa or Type IIb.

If v is Type I, then v^{∗}Σ = Σ′ as Cartier divisors. If v is Type II, then v^{∗}Σ = Σ′ + E as Cartier divisors.

Lemma 3.11. For every integer k the pullback morphism of sheaf hypercohomology groups, ℍ^{k}(Σ, L_{ζ}^{∨}) → ℍ^{k}(Σ′, 𝕃u^{∗}L_{ζ}^{∨}), is an isomorphism.

Proof. Of course v_{∗}O_{S′} = O_{S} and R^{k}v_{∗}O_{S′} = {0} if k > 0. Also v_{∗}(O_{S′}(E)) = O_{S} and R^{k}v_{∗}(O_{S′}(E)) = {0} for k > 0. If v is Type I then O_{S′}(−Σ′) = v^{∗}O_{S}(−Σ), and if v is Type II then O_{S′}(−Σ′) = v^{∗}O_{S}(−Σ) ⊗_{O_{S′}} O_{S′}(E). For both types, the projection formula implies that v_{∗}(O_{S′}(−Σ′)) = O_{S}(−Σ) and R^{k}v_{∗}(O_{S′}(−Σ′)) = {0} for k > 0.

16 J. HARRIS AND J. STARR

Associated to the short exact sequence of coherent O_{S′}-modules,

0 ⟶ O_{S′}(−Σ′) ⟶ O_{S′} ⟶ O_{Σ′} ⟶ 0,

there is a long exact sequence of higher direct images R^{k}v_{∗}, and the higher direct images of O_{S′} and O_{S′}(−Σ′) have just been computed. The conclusion is that u_{∗}O_{Σ′} = O_{Σ} and R^{k}u_{∗}O_{Σ′} = {0} for k > 0. In other words, the canonical morphism of complexes of coherent O_{Σ}-modules, O_{Σ}[0] → ℝu_{∗}O_{Σ′}, is a quasi-isomorphism. From this and the projection formula, it follows that the canonical morphism

L_{(ζ,i)}^{∨} → ℝu_{∗}𝕃u^{∗}(L_{(ζ,i)}^{∨})

is a quasi-isomorphism. Therefore the pullback morphisms

ℍ^{k}(Σ, L_{ζ}^{∨}) → ℍ^{k}(Σ′, 𝕃u^{∗}L_{ζ}^{∨})

are isomorphisms. □

There is a canonical map of coherent sheaves O_{S′}(Σ′) → v^{∗}O_{S}(Σ). It is an isomorphism if v is Type I, and is injective with cokernel v^{∗}O_{S}(Σ) ⊗_{O_{S′}} O_{E} if v is Type II. If v is Type II, then v^{∗}O_{X}(Σ) ⊗_{O_{S′}} O_{E} ≅ M ⊗_{ℂ} O_{E} where M = O_{S}(Σ)|_{s} is a one-dimensional vector space. If v is Type I, the canonical morphism N_{(ζ′,i′)} → u^{∗}N_{(ζ,i)} is an isomorphism. If v is Type II, there is an exact sequence:

0 → M ⊗_{ℂ} Tor_{1}^{O_{S′}} (O_{Σ′}, O_{E}) → N_{(ζ′,i′)} → u^{∗}N_{(ζ,i)} → M ⊗_{ℂ} O_{E} → 0

Lemma 3.12. Let N_{E/S′}^{∨} denote the conormal sheaf of E ⊂ S′. There is a canonical isomorphism

Tor_{1}^{O_{S′}} (O_{Σ′}, O_{E}) ≅ N_{E/S′}^{∨} ⊗_{O_{E}} O_{E}(−D).

Proof. There is an O_{S′}-flat resolution of O_{E},

0 ⟶ O_{S′}(−E) ⟶ O_{S} ⟶ O_{E} ⟶ 0.

Tensoring this resolution with O_{Σ′} over O_{S′} gives a canonical isomorphism,

Tor_{1}^{O_{S′}} (O_{Σ′}, O_{E}) ≅ I_{Γ} ⊗_{O_{S′}} O_{S′}(−E) ≅ N_{E/S′}^{∨} ⊗_{O_{E}} O_{E}(−D).

where I_{Γ} ⊂ O_{Σ′} is the ideal sheaf of Γ ⊂ Σ′. □

In particular, if v is Type II there is an exact sequence:

0 → M ⊗_{ℂ} N_{E/S′}^{∨} ⊗_{O_{E}} O_{E}(−D) → N_{(ζ′,i′)} → u^{∗}N_{(ζ,i)} → M ⊗_{ℂ} O_{E} → 0.

Lemma 3.13. There is a long exact sequence,

0 → T_{E}(−D) → (i′)^{∗}T_{S′} ⟶^{dv} u^{∗}i^{∗}T_{S} → T_{E} ⊗_{O_{E}} N_{E/S′} → 0.

(Both if v is Type I and if v is Type II.)

Proof. For both types, there is a short exact sequence of coherent O_{S′}-modules

0 ⟶ v^{∗}Ω_{S} ⟶^{(dv)^{†}} Ω_{S′} ⟶ Ω_{E} ⟶ 0.

There is an associated long exact sequence of higher derived functors of Hom_{O_{S′}} (·, O_{S′}), part of which is the short exact sequence,

0 ⟶ T_{S′} ⟶ v^{∗}T_{S} ⟶ Ext_{O_{S′}}^{1} (Ω_{E}, O_{S′}) ⟶ 0.

The resolution of O_{E} from the proof of Lemma 3.12 gives a canonical isomorphism,

Ext_{O_{S′}}^{1} (O_{E}, O_{S′}) ≅ N_{E/S′},

where N_{E/S′} is the dual of N_{E/S′}^{∨}. So the previous exact sequence is,

0 ⟶ T_{S′} ⟶ v^{∗}T_{S} ⟶ T_{E} ⊗_{O_{E}} N_{E/S′} ⟶ 0

Of course T_{E} ⊗_{O_{E}} N_{E/S′} is a locally free O_{E}-module; in particular, we have a canonical isomorphism,

Tor_{1}^{O_{S′}} (O_{Σ′}, T_{E} ⊗_{O_{E}} N_{E/S′}) ≅ Tor_{1}^{O_{S′}} (O_{Σ′}, O_{E}) ⊗_{O_{E}} T_{E} ⊗_{O_{E}} N_{E/S′}.

Tensoring the short exact sequence above with O_{Σ′} over O_{S′} and using Lemma 3.12, produces the exact sequence,

0 → T_{E}(−D) → (i′)^{∗}T_{S′} ⟶^{dv} u^{∗}i^{∗}T_{S} → T_{E} ⊗_{O_{E}} N_{E/S′} → 0.

□

The maps N_{(ζ′,i′)} → u^{∗}N_{(ζ,i)} and (i′)^{∗}T_{S′} → u^{∗}i^{∗}T_{S} are compatible with α_{(ζ′,i′)}^{†} and u^{∗}α_{(ζ,i)}^{†}. So there is an induced map of complexes L_{(ζ′,i′)}^{∨} → u^{∗}L_{(ζ,i)}^{∨}.

Notation 3.14. (i) Denote by du : L_{(ζ′,i′)}^{∨} → h^{∗}L_{(ζ,i)}^{∨} the induced map of complexes.

(ii) Denote by Image(du) ↪ u^{∗}L_{(ζ,i)}^{∨} the image of du in the Abelian category of complexes of

coherent O_{Σ′}-modules. (iii) If v is Type I, denote K_{I} = T_{E}(−D)[0] and Q_{I} = (T_{E} ⊗_{O_{E}} N_{E/S′}) [0]. (iv) If v is Type II, denote

\begin{aligned}K_{II} & = & T_{E}(−D)[0] ⊕ (M ⊗_{ℂ} N_{E/S′}^{∨}(−D)) [−1] \\ Q_{II} & = & (T_{E} ⊗_{O_{E}} N_{E/S′}) [0] ⊕ (M ⊗_{ℂ} O_{E}) [−1] \end{aligned}

Lemma 3.15. Both if v is Type I and if v is Type II, there are short exact sequences of complexes of coherent O_{Σ′}-modules,

0 ⟶  K  ⟶ L_{(ζ′,i′)}^{∨} ⟶^{du} Image(du) ⟶ 0
0 ⟶ Image(du) ⟶ u^{∗}L_{ζ}^{∨} ⟶  Q  ⟶ 0

Proof. If v is Type I, there is a commutative diagram with exact rows,

L_{(ζ′,i′)}^{∨}  ⟶^{du} u^{∗}L_{(ζ,i)}^{∨}

0 → T_{E}(−D) → (i′)^{∗}T_{S′}  ⟶^{(i′)^{∗}dv} u^{∗}i^{∗}T_{S} → T_{E} ⊗_{O_{E}} N_{E/S′} → 0
↓  ↓
N_{(ζ′,i′)}  ≅  h^{∗}N_{(ζ,i)}

The middle two columns of this diagram give du : L_{(ζ′,i′)}^{∨} ⟶ u^{∗}L_{(ζ,i)}^{∨}. By inspection the kernel of du is K_{I} and the cokernel of du is Q_{I}.

If v is Type II, there is a commutative diagram with exact rows,

L_{(ζ′,i′)}^{∨}  ⟶^{du} u^{∗}L_{(ζ,i)}^{∨}

0 →  T_{E}(−D)  → (i′)^{∗}T_{S′}  ⟶^{(i′)^{∗}dv} u^{∗}i^{∗}T_{S} → T_{E} ⊗_{O_{E}} N_{E/S′} → 0
↓  ↓
0 → M ⊗_{ℂ} N_{E/S′}^{∨}(−D) → N_{(ζ′,i′)} ⟶ u^{∗}N_{(ζ,i)} →  M ⊗_{ℂ} O_{E}  → 0

The middle two columns of this diagram give du : L_{(ζ′,i′)}^{∨} → u^{∗}L_{(ζ,i)}^{∨}.

The claim is that the induced map of kernels, T_{E}(−D) → M ⊗_{ℂ} N_{E/S′}^{∨}(−D), is zero, and the induced map of cokernels, T_{E} ⊗_{O_{E}} N_{E/S′} → M ⊗_{ℂ} O_{E} is zero. To see this, observe that in each

18 J. HARRIS AND J. STARR

map, the domain and target are locally free sheaves on E. So each map is really a section of a Hom_{O_{E}}(·, ·) sheaf. Up to canonical isomorphisms, the sheaf in both cases is simply

Hom_{O_{E}}(T_{E}, M ⊗_{ℂ} N_{E/S′}^{∨}) ≅ M ⊗_{ℂ} Ω_{S′}^{2} ⊗_{O_{S}} O_{E}.

Identifying E with ℙ^{1}, there are isomorphisms N_{E/S′} ≅ O_{E}(−1) and O_{E}(K_{E}) ≅ O_{E}(−2). By the adjunction formula for divisors on surfaces Ω_{S′}^{2} ⊗_{O_{S}} O_{E} = O_{S′}(K_{S′}) ⊗_{O_{S}} O_{E} is isomorphic to O_{E}(K_{E}) ⊗_{O_{E}} N_{E/S′}^{∨}, i.e., Ω_{S′}^{2} ⊗_{O_{S}} O_{E} ≅ O_{E}(−2 + 1) = O_{E}(−1). Since this sheaf has no nonzero global sections, both the induced maps are zero. Therefore the kernel of du is K_{II} and cokernel of du is Q_{II}. □

Assume first that v is Type I. Since T_{E}(−D) ≅ O_{E}(1), ℍ^{0}(Σ′, K_{I}) = H^{0}(E, T_{E}(−D)) is 2dimensional, and dimℍ^{i}(Σ′, K_{I}) = 0 for i ≠ 0. Similarly ℍ^{0}(Σ′, Q_{I}) = H^{0}(E, T_{E}(E)) is 2dimensional and dimℍ^{i}(Σ′, Q_{I}) = 0 for i > 0. Therefore there is a long exact sequence of hypercohomology groups:

0 → H^{0}(E, T_{E}(−D)) → ℍ^{0}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{0}(Σ, L_{(ζ,i)}^{∨}) → …
⋯ → H^{0}(E, T_{E} ⊗_{O_{E}} N_{E/S′}) → ℍ^{1}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{1}(Σ, L_{(ζ,i)}^{∨}) → 0
0 → ℍ^{2}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{2}(Σ, L_{(ζ,i)}^{∨}) → 0

Lemma 3.16. If v is Type Ia then there are exact sequences,

0 → H^{0}(E, T_{E}(−D)) → ℍ^{0}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{0}(Σ, L_{(ζ,i)}^{∨}) → 0,
0 → H^{0}(E, T_{E} ⊗_{O_{E}} N_{E/S′}) → ℍ^{1}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{1}(Σ, L_{(ζ,i)}^{∨}) → 0,
0 → ℍ^{2}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{2}(Σ, L_{(ζ,i)}^{∨}) → 0.

In other words, the canonical map from the space of infinitesimal automorphisms of ζ′ to the space of infinitesimal automorphisms of ζ is surjective with 2-dimensional kernel, the canonical map from the space of first order deformations of ζ′ to the space of first order deformations of ζ is surjective with 2-dimensional kernel, and the obstruction space of ζ′ equals the obstruction space of ζ.

Proof. The only claim that doesn’t follow from the long exact sequence of cohomology is that ℍ^{0}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{0}(Σ, L_{(ζ,i)}^{∨}) is surjective. Since the irreducible component Σ_{j} ⊂ Σ containing s is stable, Lemma 3.5 states that every infinitesimal automorphism of ζ vanishes on Σ_{j}. The infinitesimal automorphisms of ζ that vanish at s are the same as the infinitesimal automorphisms of ζ′ that vanish on E. Therefore every infinitesimal automorphism of ζ is the image of an infinitesimal automorphism of ζ′ that vanishes on E. □

Let v be a morphism of Type Ib. Let Σ_{j} ⊂ Σ denote the unstable component containing s. Let N denote the 1-dimensional vector space Θ_{Σ_{j}} ⊗_{O_{Σ_{j}}} O_{s}.

Lemma 3.17. If v is Type Ib then there are exact sequences,

0 → H^{0}(E, T_{E}(−D)) → ℍ^{0}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{0}(Σ, L_{(ζ,i)}^{∨}) → N → 0
0 → H^{0}(E, T_{E} ⊗_{O_{E}} N_{E/S′})/N → ℍ^{1}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{1}(Σ, L_{(ζ,i)}^{∨}) → 0
0 → ℍ^{2}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{2}(Σ, L_{(ζ,i)}^{∨}) → 0

In other words, the canonical map from the space of infinitesimal automorphisms of ζ′ to the space of infinitesimal automorphisms of ζ has a 2-dimensional kernel and a 1-dimensional cokernel, the canonical map from the space of first order deformations of ζ′ to the space of first order deformations of ζ is surjective with 1-dimensional kernel, and the obstruction space of ζ′ equals the obstruction space of ζ.

RATIONAL CURVES ON HYPERSURFACES II 19

Proof. As in Lemma 3.16, the only claim that doesn’t follow from the long exact sequence of cohomology is that the cokernel of du : ℍ^{0}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{0}(Σ, L_{(ζ,i)}^{∨}) is N. Let Σ′_{j} ⊂ Σ′ denote the strict transform of Σ_{j}; Σ′_{j} is canonically isomorphic to Σ_{j}. Composing du with the restriction map ℍ^{0}(Σ, L_{(ζ,i)}^{∨}) → H^{0}(Σ_{j}, Θ_{Σ_{j}}) gives a map which is canonically isomorphic to the restriction map ℍ^{0}(Σ′, L_{(ζ′,i′)}^{∨}) → H^{0}(Σ′_{j}, Θ_{Σ′_{j}}). By Lemma 3.5, every infinitesimal automorphism of ζ′ restricts to an infinitesimal automorphism of Σ′_{j} that vanishes at s. On the other hand, the infinitesimal automorphisms of ζ that vanish at s are the same as the infinitesimal automorphisms of ζ′ that vanish on E. Hence the image of du is precisely the subspace of infinitesimal automorphisms of ζ that vanish at s.

Since Σ_{j} is unstable, either Σ = Σ_{j} is a curve of arithmetic genus 1 with no markings and g is constant, or Σ_{j} is a smooth curve of genus 0 containing at most 2 marked points and nodes of Σ and g|_{Σ_{j}} is constant. In each case, it is easy to see that there is an infinitesimal automorphism of Σ_{j} that vanishes at all marked points and nodes of Σ and that does not vanish at s. By Lemma 3.5, this infinitesimal automorphism of Σ_{j} is the image of an infinitesimal automorphism of ζ, i.e., there exists an infinitesimal automorphism of ζ that does not vanish at s. Therefore the cokernel of du is N. □

Assume next that v is Type II. Because the divisor D ⊂ E has degree 2, the O_{E}-module N_{E/S′}^{∨}(−D) is isomorphic to O_{E}(−1). Since h^{0}(E, O_{E}(−1)) = h^{1}(E, O_{E}(−1)) = 0, the term M ⊗_{ℂ} N_{E/S′}^{∨}(−D)[−1] in K_{II} does not contribute to the hypercohomology; i.e., the hypercohomology of K_{II} is the sheaf cohomology of T_{E}(−D). The O_{E}-module T_{E}(−D) is isomorphic to O_{E}; hence h^{0}(E, T_{E}(−D)) = 1 and h^{1}(E, T_{E}(−D)) = 0. Therefore dimℍ^{0}(Σ′, K_{II}) = H^{0}(E, T_{E}(−D)) is 1-dimensional, and dimℍ^{i}(Σ′, K_{II}) = 0 for k ≠ 1.

For Q_{II} both T_{E} ⊗_{O_{E}} N_{E/S′}[0] and M ⊗_{ℂ} O_{E}[−1] contribute to the hypercohomology. The O_{E}- module T_{E}⊗_{O_{E}} N_{E/S′} is isomorphic to O_{E}(1). Hence ℍ^{0}(Σ′, Q_{II}) = H^{0}(E, T_{E}(E)) is 2-dimensional, ℍ^{1}(Σ′, Q_{II}) = M is 1-dimensional (recall M = O_{S}(Σ)|_{s}), and dimℍ^{k}(Σ′, Q_{II}) = 0 for k ≠ 0, 1. Therefore there is a long exact sequence in hypercohomology:

0 → H^{0}(E, T_{E}(−D)) → ℍ^{0}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{0}(Σ, L_{(ζ,i)}^{∨}) → …
⋯ → H^{0}(E, T_{E} ⊗_{O_{E}} N_{E/S′}) → ℍ^{1}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{1}(Σ, L_{(ζ,i)}^{∨}) → …
⋯ → M → ℍ^{2}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{2}(Σ, L_{(ζ,i)}^{∨}) → 0

Every infinitesimal automorphism of ζ vanishes at s. So the infinitesimal automorphisms of ζ are the same as the infinitesimal automorphisms of ζ′ that vanish on E. In particular, ℍ^{0}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{0}(Σ, L_{(ζ,i)}^{∨}) is surjective. The map ℍ^{1}(Σ, L_{(ζ,i)}^{∨}) → O_{S}(Σ)|_{s} is nonzero iff there are deformations of ζ that smooth the node s to first order. This proves the following two lemmas, which are stated separately for notational convenience.

Lemma 3.18. If v is Type IIa then there are exact sequences,

0 → H^{0}(E, T_{E}(−D)) → ℍ^{0}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{0}(Σ, L_{(ζ,i)}^{∨}) → 0  (2)
0 → H^{0}(E, T_{E} ⊗_{O_{E}} N_{E/S′}) → ℍ^{1}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{1}(Σ, L_{(ζ,i)}^{∨}) → O_{S}(Σ)|_{s} → 0  (3)
0 → ℍ^{2}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{2}(Σ, L_{(ζ,i)}^{∨}) → 0  (4)

In other words, the canonical map from the space of infinitesimal automorphisms of ζ′ to the space of infinitesimal automorphisms of ζ is surjective with a 1-dimensional kernel, the canonical map

20 J. HARRIS AND J. STARR

from the space of first order deformations of ζ′ to the space of first order deformations of ζ has a 2-dimensional kernel and a 1-dimensional cokernel, and the obstruction space to ζ′ equals the obstruction space to ζ.

Lemma 3.19. If v is Type IIb then there are exact sequences,

0 → H^{0}(E, T_{E}(−D)) → ℍ^{0}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{0}(Σ, L_{(ζ,i)}^{∨}) → 0  (5)
0 → H^{0}(E, T_{E} ⊗_{O_{E}} N_{E/S′}) → ℍ^{1}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{1}(Σ, L_{(ζ,i)}^{∨}) → 0  (6)
0 → O_{S}(Σ)|_{s} → ℍ^{2}(Σ′, L_{(ζ′,i′)}^{∨}) → ℍ^{2}(Σ, L_{(ζ,i)}^{∨}) → 0.  (7)

In other words, the canonical map from the space of infinitesimal automorphisms of ζ′ to the space of infinitesimal automorphisms of ζ is surjective with a 1-dimensional kernel, the canonical map from the space of first order deformations of ζ′ to the space of first order deformations of ζ is surjective with 1-dimensional kernel, and the map from the obstruction space of ζ′ to the obstruction space of ζ is surjective and has a 1-dimensional kernel.

Finally, assume that u : Σ′ → Σ is a Type III elementary morphism, i.e., u is the identity map, but there is one marked point σ′ ∈ Σ′ that is not in Σ.

Definition 3.20. Let u : Σ′ → Σ be a Type III elementary morphism.

(i) The morphism u is Type IIIa if σ′ ∈ Σ lies on an unstable component. (ii) The morphism u is Type IIIb if σ′ ∈ Σ lies on a stable component.

In both cases, there is a canonical short exact sequence of complexes:

0 ⟶ L_{ζ} ⟶ L_{ζ′} ⟶ Ω_{Σ}(σ′) ⊗_{O_{Σ}} O_{σ′}[0] ⟶ 0

Of course 𝔼xt_{O_{Σ}}^{1}(Ω_{Σ}(σ′)⊗_{O_{Σ}}O_{σ′}, O_{Σ}) is canonically isomorphic to T_{Σ}⊗_{O_{Σ}}O_{σ′}, and dim𝔼xt_{O_{Σ}}^{k}(Ω_{Σ}(σ′)⊗_{O_{Σ}} O_{σ′}, O_{Σ}) = 0 for k ≠ 1. In particular, there is an induced map

𝔼xt_{O_{Σ}}^{0}(L_{ζ}, O_{Σ}) → T_{Σ} ⊗_{O_{Σ}} O_{σ′}.

This map is zero iff σ′ lies on a stable component of ζ. Combined with the long exact sequence of hypercohomology associated to the short exact sequences, this proves the following two lemmas, which are stated separately for notational convenience.

Lemma 3.21. If u : Σ′ → Σ is Type IIIa then there are exact sequences,

0 → 𝔼xt_{O_{Σ′}}^{0} (L_{ζ′}, O_{Σ′}) → 𝔼xt_{O_{Σ}}^{0}(L_{ζ}, O_{Σ}) → T_{Σ} ⊗_{O_{Σ}} O_{σ′} → 0
0 → 𝔼xt_{O_{Σ′}}^{1} (L_{ζ′}, O_{Σ′}) → 𝔼xt_{O_{Σ}}^{1}(L_{ζ}, O_{Σ}) → 0
0 → 𝔼xt_{O_{Σ′}}^{2} (L_{ζ′}, O_{Σ′}) → 𝔼xt_{O_{Σ}}^{2}(L_{ζ}, O_{Σ}) → 0

In other words, the space of infinitesimal automorphisms of ζ′ maps isomorphically to a codimension 1 linear subspace of the space of infinitesimal automorphisms of ζ, the space of first order deformations of ζ′ equals the space of first order deformations of ζ, and the obstruction space of ζ′ equals the obstruction space of ζ.

Lemma 3.22. If u : Σ′ → Σ is Type IIIb then there are exact sequences,

0 → 𝔼xt_{O_{Σ′}}^{0} (L_{ζ′}, O_{Σ′}) → 𝔼xt_{O_{Σ}}^{0}(L_{ζ}, O_{Σ}) → 0
0 → T_{Σ} ⊗_{O_{Σ}} O_{σ′} → 𝔼xt_{O_{Σ′}}^{1} (L_{ζ′}, O_{Σ′}) → 𝔼xt_{O_{Σ}}^{1}(L_{ζ}, O_{Σ}) → 0
0 → 𝔼xt_{O_{Σ′}}^{2} (L_{ζ′}, O_{Σ′}) → 𝔼xt_{O_{Σ}}^{2}(L_{ζ}, O_{Σ}) → 0

RATIONAL CURVES ON HYPERSURFACES II 21

In other words, the space of infinitesimal automorphisms of ζ′ equals the space of infinitesimal automorphisms of ζ, the canonical map from the space of first order deformations of ζ′ to the space of first order deformations of ζ is surjective with 1-dimensional kernel, and the obstruction space of ζ′ equals the obstruction space of ζ′.

Taken together, Lemma 3.16 through Lemma 3.22, describe the canonical maps 𝔼xt_{O_{Σ′}}^{k} (L_{ζ′}, O_{Σ′}) → 𝔼xt_{O_{Σ}}^{k}(L_{ζ}, O_{Σ}) for any morphism u : Σ′ → Σ that removes a subset of marked points from Σ′ and then contracts a subset of the unstable components.

3.2. Gluing stable maps. Let τ be a stable A-graph, not necessarily of genus 0. To describe the analogue of Lemma 3.3, a more precise description of the stack \overline{M}(X, τ) is needed. Let B be a scheme. A 1-morphism ζ̃ : B → \overline{M}(X, τ) is equivalent to a pair (ζ, ϕ) consisting of a family of r-pointed stable maps,

ζ = ((p : Σ → B, σ_{1}, …, σ_{r}), g : Σ → X),

along with a natural assignment to each geometric point b ∈ B of a contraction of stable A-graphs ϕ_{b} : τ(ζ_{b}) → τ. The graph τ(ζ_{b}) is the stable A-graph of ((Σ_{b}, σ_{1}(b), …, σ_{r}(b)), g_{b}). A contraction of stable A-graphs, ϕ : τ′ → τ is a map that contracts subgraphs of τ′ to vertices of τ, cf. [5, Def. 1.8]. Geometrically the vertices of τ give a decomposition of Σ_{b} into connected subcurves. The main example of a contraction of stable A-graphs comes from a family of stable maps over a discrete valuation ring, ζ : Spec R → \overline{M}_{g,r}(X, e). Let ζ_{0} denote the fiber over the geometric closed point of R and let ζ_{η} denote the fiber over the geometric generic point of R. There is a canonical contraction ϕ_{(ζ,R)} : τ(ζ_{0}) → τ(ζ_{η}); a vertex of τ(ζ_{0}) corresponding to an irreducible component Σ_{0,j} ⊂ Σ_{0} maps to a vertex of τ(ζ_{η}) corresponding to an irreducible component of Σ_{η,k} ⊂ Σ_{η} iff Σ_{0,j} is in the closure of Σ_{η,k}. The assignment ϕ from above is called natural if it is compatible with the action of the Galois group of κ(b) and for each map from a discrete valuation ring to B, ρ : Spec R → B, the contractions ϕ_{ρ(0)} and ϕ_{ρ(η)} commute with the contraction ϕ_{ρ^{∗}ζ,R}.

Notation 3.23. Associated to each edge ϵ = {f_{1}, f_{2}} of τ, there is a section σ_{ϵ} : B → Σ such that for each geometric point b ∈ B, σ_{ϵ}(b) ∈ Σ_{b} is a node. Denote by N_{ϵ} the pullback σ_{ϵ}^{∗}Ext_{O_{Σ}}^{1}(Ω_{p}(σ_{1}(B) + ⋯ + σ_{r}(B)), O_{Σ}). Denote by N_{τ} the direct sum ⊕_{ϵ}N_{ϵ} where ϵ ranges over all edges of τ.

Definition 3.24. Let T be a scheme. Let C be a complex of coherent sheaves on T and let n be an integer. The good (≤ r)-truncation, C^{≤r}, is the complex of coherent sheaves on T,

(C^{≤r})^{k} = \{ \begin{matrix}C^{k} & k < r, \\ Ker(d^{r} : C^{r} → C^{r+1}) & k = r, \\ \{0\} & k > r \end{matrix}

The differentials on C^{≤r} are the obvious ones. The association C ↦ C^{≤r} defines a functor on the category of complexes of coherent sheaves on T. This functor takes quasi-isomorphisms to quasiisomorphisms and sends null-homotopic maps to null-homotopic maps, thus it induces a well-defined functor on the derived category of coherent sheaves on T. There is a natural transformation to the identity functor, C^{≤r} → C.

The O_{B}-module N_{ϵ} is invertible and the localization of Ext_{O_{Σ}}^{1}(Ω_{p}(σ_{1}(B)+⋯+σ_{r}(B)), O_{Σ}) along σ_{ϵ}(B) is canonically isomorphic to (σ_{ϵ})_{∗}N_{ϵ}. By construction there is a map of complexes of coherent O_{Σ}-modules, L_{ζ}^{∨} → ℝHom_{O_{Σ}}(Ω_{p}(σ_{1}(B)+⋯+σ_{r}(B)), O_{Σ}). For all k > 1, Ext_{O_{B}}^{k}(Ω_{B}(σ_{1}(B)+⋯+ σ_{r}(B)), O_{B}) = {0}. So the complex is quasi-isomorphic to its (≤ 1)-good truncation,

ℝ^{≤1}Hom_{O_{Σ}}(Ω_{p}(σ_{1}(B) + ⋯ + σ_{r}(B)), O_{Σ}) ⟶^{qism} ℝHom_{O_{Σ}}(Ω_{p}(σ_{1}(B) + ⋯ + σ_{r}(B)), O_{Σ}).

22 J. HARRIS AND J. STARR

Notation 3.25. After replacing L_{ζ}^{∨} by a quasi-isomorphic complex, there is a morphism of complexes denoted a_{(ζ,τ)} : L_{ζ}^{∨} → ℝ^{≤1}Hom_{O_{Σ}}(Ω_{p}(σ_{1}(B) + ⋯ + σ_{r}(B)), O_{Σ}) factoring the original morphism. There is a map of complexes of O_{Σ}-modules denoted

b_{(ζ,τ)} : ℝ^{≤1}Hom_{O_{Σ}}(Ω_{p}(σ_{1}(B) + ⋯ + σ_{r}(B)), O_{Σ}) → Ext_{O_{Σ}}^{1}(Ω_{p}(σ_{1}(B) + ⋯ + σ_{r}(B)), O_{Σ})[−1].

Denote by,

c_{(ζ,τ)} : Ext_{O_{Σ}}^{1}(Ω_{p}(σ_{1}(B) + ⋯ + σ_{r}(B), O_{Σ})_{τ} → Ext_{O_{Σ}}^{1}(Ω_{p}(σ_{1}(B) + ⋯ + σ_{r}(B)), O_{Σ}),

the kernel of,

Ext_{O_{Σ}}^{1}(Ω_{p}(σ_{1}(B) + ⋯ + σ_{r}(B)), O_{Σ}) → ⊕_{ϵ}(σ_{ϵ})_{∗}N_{ϵ}.

Denote by

d_{(ζ,τ)} : ℝ^{≤1}Hom_{O_{Σ}}(Ω_{p}(σ_{1}(B) + ⋯ + σ_{r}(B)), O_{Σ})_{τ} → ℝ^{≤1}Hom_{O_{Σ}}(Ω_{p}(σ_{1}(B) + ⋯ + σ_{r}(B)), O_{Σ}),

the fiber product of b_{(ζ,τ)} and c_{(ζ,τ)}. Denote by,

e_{(ζ,τ)} : L_{(ζ,τ)}^{∨} → L_{ζ}^{∨},

the fiber product of a_{(ζ,τ)} and d_{(ζ,τ)}.

Let |τ| denote the underlying modular graph of τ, i.e., the graph of τ along with the genus function on vertices, but without the degree functon on vertices, cf. [5]. There is a (highly nonseparated) Artin stack of |τ|-prestable curves, 𝔐(|τ|), and a 1-morphism \overline{M}(X, τ) → 𝔐(|τ|). The relative obstruction theory for this 1-morphism is described in [3]. From this the absolute obstruction theory of \overline{M}(X, τ) readily follows.

Lemma 3.26. Let X be a smooth quasi-projective scheme. Let τ be a stable A-graph. Let

ζ = ((p : Σ → \overline{M}(X, τ), σ_{1}, …, σ_{r}), g : Σ → X)

denote the universal family of stable maps over \overline{M}(X, τ). There exists an obstruction theory for \overline{M}(X, τ) in the sense of [4, Def. 4.4] of the form

(ℝp_{∗}(L_{(ζ,τ)}^{∨})[1])^{∨} → L_{\overline{M}(X,τ)}.

This obstruction theory is perfect, and there is a distinguished triangle in the derived category of complexes of coherent O_{\overline{M}(X,τ)}-modules,

ℝp_{∗}(L_{(ζ,τ)}^{∨})[1] ⟶ ℝp_{∗}(L_{ζ}^{∨})[1] ⟶ N_{τ} ⟶ ℝp_{∗}(L_{ζ,τ}^{∨})[2].

Remark 3.27. The definition of L_{(ζ,τ)}^{∨} seems very complicated, but in fact it is quite simple. Let ((Σ, σ_{1}, …, σ_{r}), g) be a stable τ-map. Let i : Σ → S be a closed immersion from Σ to a smooth surface S. Let N_{i} denote the normal sheaf of i. The complex L_{ζ}^{∨} is quasi-isomorphic to the complex L_{(ζ,i)}^{∨}. There is a surjective map N_{(ζ,i)} → N_{i}(σ_{1}(B) + ⋯ + σ_{r}(B)). Define N_{(ζ,i,τ)} ⊂ N_{(ζ,i)} to be the subsheaf which is the inverse image of I · N_{i}(σ_{1}(B) + ⋯ + σ_{r}(B)) ⊂ N_{i}(σ_{1}(B) + ⋯ + σ_{r}(B)), where I is the ideal sheaf of ∪_{ϵ}σ_{ϵ}. Then e_{(ζ,τ)} : L_{(ζ,τ)}^{∨} → L_{ζ}^{∨} is quasi-isomorphic to the following map of complexes.

0  1
L_{(ζ,i,τ)}^{∨} :i^{∗}T_{S}(σ_{1}(B) + ⋯ + σ_{r}(B)) ⟶ N_{(ζ,i,τ)}
e_{(ζ,i,τ)}↓  ^{=}↓  ↓
L_{(ζ,i)}^{∨} : i^{∗}T_{S}(σ_{1}(B) + ⋯ + σ_{r}(B)) ⟶ N_{(ζ,i)}

RATIONAL CURVES ON HYPERSURFACES II 23

Let ϵ = {f_{1}, f_{2}} is a disconnecting edge of τ. Let τ_{1} ⊂ τ be the maximal connected subgraph which contains f_{1} and not f_{2}, and let τ_{2} ⊂ τ be the maximal connected subgraph which contains f_{2} and not f_{1}. There are “forgetful” 1-morphisms F_{i} : \overline{M}(X, τ) → \overline{M}(X, τ_{i}) for i = 1, 2. Let ζ̃ = (ζ, ϕ) be a map in \overline{M}(X, τ_{i}), where ζ = ((Σ, σ_{1}, …, σ_{r}), g). Let σ_{ϵ} ∈ Σ be the node corresponding to ϵ. Let ζ̃_{i} = (ζ_{i}, ϕ_{i}) ∈ \overline{M}(X, τ_{i}) be the image F_{i}(ζ̃). Denote ζ_{i} = ((Σ_{i}, σ_{i,0}, …, σ_{i,r_{i}}), g_{i}) for i = 1, 2 where σ_{i,0} ∈ Σ_{i} is the point corresponding to the flag f_{i} of τ_{i}, i.e., σ_{i,0} = σ_{ϵ}.

Lemma 3.28. If

(i) the dimension of the obstruction group of \overline{M}(X, τ_{i}) at ζ̃_{i} is 0, i = 1, 2, and
(ii) the evaluation morphism ev_{f_{1}} : \overline{M}(X, τ_{1}) → X is smooth at ζ̃_{1},

then the dimension of the obstruction group of \overline{M}(X, τ) at ζ̃ is 0, and there is a short exact sequence,

0 ⟶ ζ_{1}^{∗}T_{ev_{f_{1}}} ⟶ ζ^{∗}T_{\overline{M}(X,τ)} ⟶ ζ_{2}^{∗}T_{\overline{M}(X,τ_{2})} ⟶ 0,

where T_{ev_{f_{1}}} is the dual of the sheaf of relative differentials of ev_{f_{1}}.

Proof. The proof follows from the fact that \overline{M}(X, τ) is an open substack of the 2-fibered product:

\overline{M}(X, τ_{1}) ×_{ev_{f_{1}} ,X,ev_{f_{2}}} \overline{M}(X, τ_{2}).

□

Let ϕ : τ → τ′ be the minimal contraction of stable A-graphs that contracts the edge {f_{1}, f_{2}}. The induced 1-morphism,

\overline{M}(X, ϕ) : \overline{M}(X, τ) → \overline{M}(X, τ′),

is unramified and the image has codimension at most 1. In some circumstances, this morphism is the normalization of a Cartier divisor.

Lemma 3.29. If

(i) τ has genus 0,

(ii) the dimension of the obstruction group of \overline{M}(X, τ_{2}) at ζ̃_{2} is 0, and (iii) the O_{Σ_{1}}-module g_{1}^{∗}T_{X} is generated by global sections,

then the dimension of the obstruction group of \overline{M}(X, τ′) at \overline{M}(X, ϕ)(ζ̃) is 0, the irreducible component of \overline{M}(X, τ) containing ζ̃ maps to a Cartier divisor under \overline{M}(X, ϕ), and there is a short exact sequence,

0 ⟶ T_{\overline{M}(X,τ)}|ζ̃ ⟶ T_{\overline{M}(X,τ′)}|ζ̃ ⟶ Θ_{Σ_{1}}|_{σ_{1,0}} ⊗ Θ_{Σ_{2}}|_{σ_{2,0}} ⟶ 0.

Proof. There is a short exact sequence of complexes,

0 ⟶ L_{Σ}(σ_{1} + ⋯ + σ_{r}) ⟶ L_{ζ} ⟶ g^{∗}Ω_{X}[1] ⟶ 0,

that gives rise to a long exact sequence in hypercohomology, part of which is,

ℍ^{1}(Σ, L_{ζ}^{∨}) → ℍ^{1}(Σ, L_{Σ}^{∨}(−(σ_{1} + ⋯ + σ_{r})) → H^{1}(Σ, g^{∗}T_{X}) → ℍ^{2}(Σ, L_{ζ}^{∨}) → 0.

The goal is to prove that dimℍ^{2}(Σ, L_{ζ}^{∨}) = 0, i.e., that ℍ^{1}(Σ, L_{Σ}^{∨}(−(σ_{1} + ⋯ + σ_{r}))) → H^{1}(Σ, g^{∗}T_{X}) is surjective. After replacing L_{Σ} and L_{ζ} by quasi-isomorphic complexes of locally free O_{Σ}-modules,

24 J. HARRIS AND J. STARR

there is a diagram of distinguished triangles of coherent O_{B}-modules,

L_{Σ}(σ_{1} + ⋯ + σ_{r})  →  L_{ζ}  → …
↓  ↓
L_{Σ}(σ_{1} + ⋯ + σ_{r}) ⊗_{O_{Σ}} O_{Σ_{2}} → L_{ζ} ⊗_{O_{Σ}} O_{Σ_{2}} → …

… →  g^{∗}Ω_{X}[1]  →  L_{Σ}(σ_{1} + ⋯ + σ_{r})[−1] ↓  ↓ … → g^{∗}Ω_{X} ⊗_{O_{Σ}} O_{Σ_{2}}[1] → L_{Σ}(σ_{1} + ⋯ + σ_{r}) ⊗_{O_{Σ}} O_{Σ_{2}}[−1] ,

that gives rise to a commutative diagram of exact sequences in hypercohomology,

ℍ^{1}(Σ, L_{Σ}^{∨}(−(σ_{1} + ⋯ + σ_{r})))  →  ℍ^{1}(Σ, g^{∗}T_{X}) ↓  ↓ ℍ^{1}(Σ, L_{Σ}^{∨}(−(σ_{1} + ⋯ + σ_{r})) ⊗_{O_{Σ}} O_{Σ_{2}}) → ℍ^{1}(Σ, g^{∗}T_{X} ⊗_{O_{Σ}} O_{Σ_{2}}) .

To prove the top horizontal arrow is surjective, it suffices to prove (i) the left vertical arrow is surjective, (ii) the right vertical arrow is an isomorphism, and (iii) the bottom vertical arrow is surjective.

Let S be a smooth surface, and let i : Σ → S be an unramified morphism. Then L_{Σ}^{∨}(−(σ_{1} + ⋯ + σ_{r})) is represented in the derived category of coherent sheaves by the complex

\begin{matrix}0 & & 1 \\ L_{i}^{∨}(−(σ_{1} + ⋯ + σ_{r})) : & i^{∗}T_{S}(−(σ_{1} + ⋯ + σ_{r})) & ⟶^{β_{i}^{†}} & N_{i}(−(σ_{1} + ⋯ + σ_{r})). \end{matrix}

Let K denote the kernel of β_{i}^{†} and let Q denote the cokernel of β_{i}^{†}. The sheaf K is torsion-free and is locally free of rank 1 on a dense open subset of Σ. The sheaf Q is torsion. There is a short exact sequence of complexes

0 ⟶ K[0] ⟶ L_{i}^{∨}(−(σ_{1} + ⋯ + σ_{r})) ⟶ Q[−1] ⟶ 0.

This gives rise to a long exact sequence in hypercohomology, part of which is

H^{1}(Σ, K) ⟶ ℍ^{1}(Σ, L_{i}^{∨}(−(σ_{1} + ⋯ + σ_{r})) ⟶ H^{0}(Σ, Q) ⟶ 0.

Let K′ denote the kernel of β_{i}^{†} ⊗_{O_{Σ}} O_{Σ_{2}} and let Q′ denote the cokernel. There are induced maps of coherent sheaves on Σ, K → K′ and Q → Q′. These maps give rise to a commutative diagram with exact rows,

H^{1}(Σ, K) ⟶  ℍ^{1}(Σ, L_{i}^{∨}(−(σ_{1} + ⋯ + σ_{r}))  ⟶ H^{0}(Σ, Q) ⟶ 0
↓  ↓  ↓
H^{1}(Σ, K′) ⟶ ℍ^{1}(Σ, L_{i}^{∨}(−(σ_{1} + ⋯ + σ_{r}) ⊗_{O_{Σ}} O_{Σ_{2}}) ⟶ H^{0}(Σ, Q′) ⟶ 0.

Now Q → Q′ is a surjective map of torsion sheaves, so the map on the right is surjective. And the cokernel of K → K′ is torsion, and hence the map on the left is surjective. Therefore the map in the middle is surjective, i.e., (i) holds.

There is a short exact sequence of O_{Σ}-modules,

0 ⟶ (g_{1})^{∗}T_{X}(−σ_{1,0}) ⟶ g^{∗}T_{X} ⟶ g^{∗}T_{X} ⊗_{O_{Σ}} O_{Σ_{2}} ⟶ 0

By hypothesis (g_{1})^{∗}T_{X} is generated by global sections, hence h^{1}(Σ_{1}, (g_{1})^{∗}T_{X}(−σ_{1,0})) = 0 by Lemma 2.3. Therefore H^{1}(Σ, g^{∗}T_{X}) → H^{1}(Σ, g^{∗}T_{X} ⊗_{O_{Σ}} O_{Σ_{2}}) is an isomorphism, i.e., (ii) is true.

Denote by j : Σ_{2} → Σ the canonical closed immersion. There is a morphism of complexes of O_{Σ}-modules,

−1  0
L_{ζ} : g^{∗}Ω_{X}  ⟶^{(dg)^{†}}  Ω_{Σ}(σ_{1} + ⋯ + σ_{r})
↓  ↓  ↓^{(dj)^{†}}
j_{∗}L_{ζ_{2}} :j_{∗}(g_{2})^{∗}Ω_{X} ⟶^{j_{∗}(dg_{2})^{†}} j_{∗}Ω_{Σ}(σ_{2,0} + ⋯ + σ_{2,r_{2}})

By adjunction, there is a morphism of complexes of O_{Σ_{2}}-modules, L_{ζ} ⊗_{O_{Σ}} O_{Σ_{2}} → L_{ζ_{2}}. So there is a commutative diagram in hypercohomology

ℍ^{1}(Σ_{2}, L_{Σ_{2}}^{∨}(−(σ_{1} + ⋯ + σ_{r}))  ⟶ H^{1}(Σ_{2}, (g_{2})^{∗}T_{X})
↓  ↓
ℍ^{1}(Σ_{2}, L_{Σ}^{∨}(−(σ_{2,0} + ⋯ + σ_{2,r_{2}})) ⊗_{O_{Σ}} O_{Σ_{2}}) ⟶ H^{1}(Σ_{2}, j^{∗}g^{∗}T_{X})

Of course (g_{2})^{∗}T_{X} = j^{∗}g^{∗}T_{X}, so H^{1}(Σ_{2}, (g_{2})^{∗}T_{X}) → H^{1}(Σ_{2}, j^{∗}g^{∗}T_{X}) is an isomorphism. By hypothesis, the obstruction group of ζ_{2} vanishes, hence ℍ^{1}(Σ_{2}, L_{Σ_{2}}^{∨}(−(σ_{2,0} + ⋯ + σ_{2,r_{2}}))) → H^{1}(Σ_{2}, (g_{2})^{∗}T_{X}) is surjective. Therefore also ℍ^{1}(Σ_{2}, L_{Σ}^{∨}(−(σ_{1}+⋯+σ_{r}))⊗_{O_{Σ}}O_{Σ_{2}}) → H^{1}(Σ_{2}, j^{∗}g^{∗}T_{X}) is surjective. This proves (iii). Therefore the dimension of the obstruction group of \overline{M}(X, τ′) at \overline{M}(X, ϕ)(ζ̃) is 0.

Because the obstruction groups of \overline{M}(X, τ) and \overline{M}(X, τ′) both vanish, each stack is smooth of the expected dimension at ζ̃. The expected dimension of \overline{M}(X, τ) is 1 less than the expected dimension of \overline{M}(X, τ′), because τ has one extra edge (before stabilization). Therefore the image of the irreducible component of \overline{M}(X, τ) containing ζ̃ is a Cartier divisor in \overline{M}(X, τ′).

Finally, the short exact sequence follows from the 2-Cartesian diagram of Artin stacks,

\overline{M}(X, τ) ⟶^{\overline{M}(X,ϕ)} \overline{M}(X, τ′)
↓  ↓
𝔐(|τ|) ⟶^{𝔐(|ϕ|)} 𝔐(|τ′|).

The image of 𝔐(|ϕ|) is a Cartier divisor. Because the image of \overline{M}(X, ϕ) is a Cartier divisor, the normal bundle of \overline{M}(X, ϕ) is simply the pullback of the normal bundle of 𝔐(|ϕ|), which is the bundle whose fiber at ζ̃ is the space of first-order deformations of the node σ_{ϵ} ∈ Σ. It is well-known that this space is canonically isomorphic to Θ_{Σ_{1}}|_{σ_{1,0}} ⊗ Θ_{Σ_{2}}|_{σ_{2,0}}. □

Remark 3.30. There is a relative version of Lemma 3.29. Let B be a scheme and let ζ̃ = (ζ, ϕ) be a family of τ-maps over B. If for every geometric point b ∈ B, the map ζ̃_{b} satisfies the hypotheses of Lemma 3.29, then the obstruction group of \overline{M}(X, σ) at ζ̃_{b} vanishes, the image of \overline{M}(X, τ) in \overline{M}(X, τ′) is a Cartier divisor, and there exists a short exact sequence of locally free O_{B}-modules,

0 ⟶ ζ̃^{∗}T_{\overline{M}(X,τ)} ⟶ ζ̃^{∗}T_{\overline{M}(X,τ′)} ⟶ σ_{1,0}^{∗}Θ_{Σ_{1}/B} ⊗_{O_{B}} σ_{2,0}^{∗}Θ_{Σ_{2}/B} ⟶ 0.

26 J. HARRIS AND J. STARR

4. Conditions on families of stable maps

This section defines and proves basic results about the many conditions discussed in Subsection 1.2. Because of the process of modification given in the next section, it is necessary to work with families of stable maps that are parametrized by reducible curves of genus 0. This is the setting in which all definitions are made.

Let ζ = ((Σ, σ_{1}, …, σ_{r}), g : Σ → X) be an r-pointed genus 0 stable map to X.

Definition 4.1. The stable map ζ is very stable if the prestable map (Σ, g : B → X) is stable.

Let X be a quasi-projective scheme. Let B be a scheme. Let ζ : B → \overline{M}_{g,r}(X, e) be a 1morphism.

Notation 4.2. The 1-morphism ζ : B → \overline{M}_{g,r}(X, e) consists of a datum,

ζ = ((p_{ζ} : Σ_{ζ} → B, σ_{(ζ,1)}, …, σ_{(ζ,r)}) , g_{ζ} : Σ → X) ,

where (i) p_{ζ} : Σ_{ζ} → B is a proper, flat family of connected prestable curves of arithmetic genus g, (ii) σ_{(ζ,i)} : B → Σ_{ζ}, i = 1, …, r is a collection of everywhere disjoint sections with image in the

smooth locus of p_{ζ}, and (iii) g_{ζ} : Σ_{ζ} → X is a morphism of schemes,

that satisfies the stability condition in Definition 3.1. Denote by h_{ζ,i} : B → X the composition g_{ζ} ∘ σ_{(ζ,i)}. When there is no risk of confusion, the subscript ζ will be omitted.

Let X be a smooth, quasi-projective variety. Let π : B → T be a proper, flat family of connected prestable curves of arithmetic genus 0. Let ζ : B → \overline{M}_{0,1}(X, 1) be a 1-morphism ((p : Σ → B, σ), f : Σ → X). Definition 4.3. The 1-morphism ζ : B → \overline{M}_{0,1}(X, 1) is twisting (resp. very twisting) if,

(i) (π : B → T, h : B → X) is a family of stable maps to X, i.e., a 1-morphism ξ : T → \overline{M}_{0,0}(X, e) for some e ≥ 0,

(ii) the dimension of the obstruction group of \overline{M}_{0,0}(X, e) at each point of ξ(T) is 0, (iii) the dimension of the relative obstruction group of the evaluation morphism ev : \overline{M}_{0,1}(X, 1) →

X at each point of ζ(T) is 0, (iv) denoting by T_{ev} the dual of the sheaf of relative differentials Ω_{ev}, the pullback ζ^{∗}T_{ev} is

π-relatively generated by global sections (resp. π-relatively deformation ample), and

(v) denoting by pr : \overline{M}_{0,1}(X, 1) → \overline{M}_{0,0}(X, 1) the projection map, and denoting by T_{pr} the dual of the sheaf of relative differentials Ω_{pr}, the pullback ζ^{∗}T_{pr} is π-relatively generated by global sections, i.e., the line bundle σ^{∗}O_{Σ}(σ(B)) is π-relatively generated by global sections.

Remark 4.4.

(i) The sheaf ζ^{∗}T_{pr} is canonically isomorphic to σ^{∗}O_{Σ}(σ(B)).

(ii) The product morphism (p, g) : Σ → B × X is a closed immersion whose ideal sheaf is everywhere locally defined by a regular sequence, i.e., it is a regular embedding. Denote by N the normal bundle of this regular embedding. Then (iii) of Definition 4.3 is equivalent to the condition that R^{1}p_{∗} (N (−σ(B))) = {0}. Under this hypothesis, ζ^{∗}T_{ev} is canonically isomorphic to p_{∗} (N (−σ(B))).

(iii) Condition (ii) is superfluous. Since the prestable family of maps (π : B → T, ξ : B → X) is

stable, also (π : B → T, ζ : B → \overline{M}_{0,1}(X, 1)) is stable by Lemma 3.6 (i). (iv) The conditions in Definition 4.3 impose some restrictions on the degrees of the locally free

sheaves involved. By [19, Lem. 2.2.2], the total degree of σ^{∗}O_{Σ}(σ(B)) is simply δ = 2e − e′

RATIONAL CURVES ON HYPERSURFACES II 27

where e is the degree of h : B → ℙ^{N} and e′ is the degree of f : Σ → ℙ^{N}. If ζ is twisting then 2e ≥ e′.

Let B be a prestable, connected curve of arithmetic genus 0. Let B_{1}, B_{2} ⊂ B be connected subcurves of B such that B_{1} ∩ B_{2} = \{q\} is a single node, and B = B_{1} ∪ B_{2}. Let ζ : B → \overline{M}_{0,1}(X, 1) be a 1-morphism.

Lemma 4.5. If ζ|_{B_{i}} : B_{i} → \overline{M}_{0,1}(X, 1) is twisting for i = 1, 2, then ζ is twisting. If both ζ_{1}, ζ_{2} are twisting, and if at least one of them is very twisting, then ζ is very twisting.

Proof. This follows from Lemma 2.11. □

Let π : B → T be a proper, flat family of connected, prestable curves of arithmetic genus 0. Let ζ : B → \overline{M}_{0,1}(X, 1) be a 1-morphism. For every morphism of schemes g : T′ → T, denote by π_{T′} : B_{T′} → T′ the base-change of π by g, and denote by ζ_{T′} : B_{T′} → \overline{M}_{0,1}(X, 1) the composition of B_{T′} → B with ζ.

Lemma 4.6. There is an open subscheme U_{twist} ⊂ T (resp. U_{vtwist} ⊂ T) with the following property: for every morphism of schemes g : T′ → T, the pullback family π_{T′} : B_{T′} → T′ and ζ_{T′} : B_{T′} → \overline{M}_{0,1}(X, 1) is twisting (resp. very twisting) iff g(T′) ⊂ U_{twist} (resp. f(T′) ⊂ U_{vtwist}).

Proof. By [3, Lemma 1] there is an open subscheme U_{1} ⊂ T with the property that for every morphism g, (π_{T′} : B_{T′} → T′, h_{T′} : B_{T′} → X) is a family of stable maps iff g(T′) ⊂ U_{1}. For every morphism g such that ζ_{T′} is twisting, g(T′) ⊂ U_{1}. Hence the lemma for ζ_{U_{1}} implies the lemma for ζ. After replacing T by U_{1}, the morphism h : B → X is a family of stable maps over T, i.e., a 1-morphism ξ : T → \overline{M}_{0,0}(X, e) for some integer e.

Because π has relative dimension 1, the cohomology sheaf H^{k}(ℝπ_{∗}(L_{h}^{∨})) = {0} for k ≥ 2. By cohomology and base change, for every geometric point t of the support of the sheaf H^{1}(ℝπ_{∗}(L_{h}^{∨})), dimℍ^{1}(B_{t}, L_{h_{t}}^{∨}) > 0, i.e. the obstruction group of ξ_{t} does not vanish. Therefore the dimension of the obstruction group of \overline{M}_{0,0}(X, e) at each point of ξ_{T′}(T′) is 0 iff g(T′) is contained in the complement of the support of H^{1}(ℝπ_{∗}(L_{h}^{∨})). The complement of the support of this sheaf is an open subset of T. After replacing T by this open set, the dimension of the obstruction group of \overline{M}_{0,0}(X, e) at each point of ξ(T) is 0.

For similar reasons, T can be replaced by the complement of

π[Supp (R^{1}p_{∗}(N (−σ(B))))],

where N is as in (ii) of Remark 4.4. After replacing T by this open subset, for every g, ζ_{T′} satisfies (i), (ii), and (iii) of Definition 4.3. In order that (v) is satisfied, it is necessary and sufficient that g(T′) is contained in the complement of the support of the sheaf

Coker (π^{∗}π_{∗}σ^{∗}O_{Σ}(σ) → σ^{∗}O_{Σ}(σ)) .

After replacing T by the complement of the support of this sheaf, for every g, ζ_{T′} satisfies (i), (ii), (iii) and (v) of Definition 4.3.

Define U_{twist} to be the complement in T of the image under π of the support of

Coker (π^{∗}π_{∗}ζ^{∗}T_{ev} → ζ^{∗}T_{ev}) .

For every morphism g, (ζ^{∗}T_{ev})_{T′} is π_{T′}-relatively generated by global sections iff g(T′) ⊂ U_{twist}. Therefore, for every morphism g, ζ_{T′} is twisting iff g(T′) ⊂ U_{twist}. By Lemma 2.9, there exists an open subset U_{vtwist} ⊂ T such that for every g, (ζ^{∗}T_{ev})_{T′} is π_{T′}-relatively deformation ample iff g(T′) ⊂ U_{vtwist}. Therefore ζ_{T′} is very twisting iff g(T′) ⊂ U_{vtwist}. □

28 J. HARRIS AND J. STARR

Let (π : B → T, h : B → X) be a family of genus 0 stable maps, i.e., a 1-morphism ξ : T → \overline{M}_{0,0}(X, e) for some e ≥ 0. For every morphism of schemes g : T′ → T, denote by π_{T′} : B_{T′} → T′ the base-change of π by g, and denote by ξ_{T′} : B_{T′} → \overline{M}_{0,1}(X, 1) the composition of B_{T′} → B with ξ.

Definition 4.7. The family of stable maps ξ : T → \overline{M}_{0,0}(X, e) is twistable (resp. very twistable) if there exists a surjective étale morphism u : T′ → T and a morphism ζ : B_{T′} → \overline{M}_{0,1}(X, 1) such that ξ_{ζ} = ξ_{T′} and such that ζ is twisting (resp. very twisting).

Proposition 4.8. There is an open subscheme U_{t-able} ⊂ T (resp U_{vt-able} ⊂ T) such that for every morphism of schemes g : T′ → T, ξ_{T′} is twistable (resp. very twistable) iff g(T′) ⊂ U_{t-able} (resp. g(T′) ⊂ U_{vt-able}).

Proof. It suffices to check that if t ∈ T is a geometric point such that h_{t} : B_{t} → X is twistable (resp. very twistable), then there is an étale neighborhood of t ∈ T over which ξ is twistable (resp. very twistable). Denote by ζ_{t} : B_{t} → \overline{M}_{0,1}(X, 1) the twisting morphism. Consider \overline{M}_{0,1}(X, 1) as a

quasi-projective scheme via the Plücker and Segre embeddings of 𝔾(1, n) × ℙ^{n} ↪ ℙ^{\frac{n(n+1)^{2}}{2} −1}. Let β denote the degree of the stable map ζ_{t}.

Define M = T × \overline{M}_{0,0}(\overline{M}_{0,1}(X, 1), β), i.e., M parametrizes pairs (s, ζ) consisting of a point s ∈ T together with a genus 0 stable map ζ : B → \overline{M}_{0,1}(X, 1) of degree β. Denote the universal stable map over \overline{M}_{0,0}(\overline{M}_{0,1}(X, 1), β) by

\begin{matrix}ρ : B → \overline{M}_{0,0}(\overline{M}_{0,1}(X, 1), β), \\ ζ : B → \overline{M}_{0,1}(X, 1). \end{matrix}

As in Notation 4.2, let p : Σ → B be the pullback by ζ of the universal curve over \overline{M}_{0,1}(X, 1), let σ : B → Σ be the pullback of the universal section, let g : Σ → X be the pullback of the universal map, and let h = g ∘ σ. This gives a family of prestable maps,

ξ̃ = (ρ : B → \overline{M}_{0,0}(\overline{M}_{0,1}(X, 1), β), h : B → X) .  (8)

By [3, Lemma 1] there is a maximal open substack U_{e} ⊂ \overline{M}_{0,0}(\overline{M}_{0,1}(X, 1), β) over which ξ̃ is stable of degree e. By hypothesis, (t, ζ_{t}) is in T × U_{e}.

Because ζ_{t} is twisting, the hypotheses of Lemma 3.6 (iii) are satisfied where f : X → Y corresponds to ev : \overline{M}_{0,1}(X, 1) → X and where (Σ, g : Σ → X) corresponds to (B_{t}, ζ_{t} : B_{t} → \overline{M}_{0,1}(X, 1)). Therefore at the point (B_{t}, ζ_{t} : B_{t} → \overline{M}_{0,1}(X, 1)), the stack U_{e} is smooth and the morphism ξ̃: U_{e} → \overline{M}_{0,0}(X, e) is smooth by the Jacobian criterion.

Consider the 1-morphism (1_{T} , ξ) : T → T × \overline{M}_{0,0}(X, e). Denote by M the 2-fibered product of the 1-morphism (1_{T} , ξ) and the 1-morphism (1_{T} , ξ̃) : T ×U_{e} → T ×\overline{M}_{0,0}(X, e). The 2-fibered product M is the stack whose objects are triples (t, ζ, θ) consisting of a point t ∈ T, an object ζ : B → \overline{M}_{0,1}(X, 1) of U_{e}, and an equivalence θ : ξ_{t} → ζ̃ of objects in the groupoid \overline{M}_{0,0}(X, e)(Spec κ(t)). Because (1_{T} , ξ̃) is smooth at (t, (B_{t}, ζ_{t})), the projection pr_{1} : M → T is smooth at (t, (B_{t}, ζ_{t})). Hence there exists an étale 1-morphism f : M → M from a scheme M to M such that (t, (B_{t}, ζ_{t})) is in the image of f and such that the composite morphism pr_{1} ∘ f : M → T is smooth. By [6, Prop. 2.2.14], there exists an étale morphism u : (T′, t′) → (T, t) and a section z : T′ → M such that f(z(t′)) is (t, (B_{t}, ζ_{t})). Denote by ζ_{pre} : T′ → U_{e} the composition pr_{2} ∘ f ∘ z.

RATIONAL CURVES ON HYPERSURFACES II 29

Denote by ζ : B′ → \overline{M}_{0,1}(X, 1) the pullback by ζ_{pre} : T′ → U_{e} of the universal stable map. By construction, ξ̃(ζ) : B′ → \overline{M}_{0,0}(X, e) is equivalent to u^{∗}ξ : u^{∗}B → \overline{M}_{0,0}(X, e). Hence, after replacing T′ by a surjective, étale cover, B′ = u^{∗}B as T′-schemes, and ξ̃(ζ) = u^{∗}ξ. The fiber of ζ : u^{∗}B → \overline{M}_{0,1}(X, 1) over every preimage of (t, (B_{t}, ζ_{t})) is twisting. So by Lemma 4.6, after replacing T′ by a Zariski open subscheme whose image contains t, ζ : u^{∗}B → \overline{M}_{0,1}(X, 1) is twisting. Similarly, if (t, (B_{t}, ζ_{t})) is very twisting, after replacing T′ by a Zariski open subscheme whose image contains t, ζ is very twisting. On the Zariski open subscheme of T that is the image of u : T′ → T, the family ξ : B → \overline{M}_{0,0}(X, e) is twistable (resp. very twistable). □

Let

ξ_{i} = ((π_{i} : B_{i} → T, σ_{i} : T → B_{i}) , h_{i} : B_{i} → X) , i = 1, 2,

be two families of 1-pointed stable maps such that h_{1} ∘ σ_{1} = h_{2} ∘ σ_{2}. Denote by

ξ = (π : B → T, h : B → X)

the family obtained by taking B to be the union of B_{1} and B_{2} where the section σ_{1} is identified with the section σ_{2}.

Lemma 4.9. If for i = 1, 2 the family of unpointed stable maps (π_{i} : B_{i} → T, h_{i} : B_{i} → X) is twistable, and if for every geometric point t ∈ T, the variety parametrizing free lines that contain h(σ(t)) is irreducible, then ξ is twistable. If also one of ξ_{1}, ξ_{2} is very twistable, then ξ is very twistable.

Proof. This follows from Lemma 4.5. First of all, using Proposition 4.8, it suffices to prove the result when T = Spec k for some algebraically closed field k.

For each of i = 1, 2, let M_{i} denote the 2-fibered product constructed in the proof of Proposition 4.8, i.e., the objectst of M_{i} are pairs (ζ_{i}, θ_{i}) consisting of a twisting family (resp. very twisting family), ζ_{i} : B_{i} → \overline{M}_{0,1}(X, 1), such that the induced map

ζ̃_{i} = ((B_{i}, σ_{i}), g_{i} ∘ ρ_{i} : B_{i} → X)

is stable, and an equivalence of objects, θ_{i} : ξ_{i} → ζ̃_{i}. Since each of ξ_{i} is twistable, each of M_{i} is nonempty.

By the proof of Proposition 4.8, each of M_{i} is smooth. By the definition of twisting families, for each i = 1, 2 the morphism

e_{i} : M_{i} → \overline{M}_{0,1}(X, 1), ζ_{i} ↦ ζ_{i}(σ_{i})

has image contained in the unobstructed locus of ev : \overline{M}_{0,1}(X, 1) → X. Let P ⊂ \overline{M}_{0,1}(X, 1) denote the preimage under ev of the point p = h_{1}(σ_{1}) = h_{2}(σ_{2}). The image of e_{i} is contained in the smooth locus of P. The claim is that e_{i} : M_{i} → P is smooth. The obstruction space at a point ζ_{i} is a quotient of the cohomology group H^{1} (B_{i}, ζ_{i}^{∗}T_{ev}(−σ_{i})). By Definition 4.3 (iv), ζ_{i}^{∗}T_{ev} is generated by global sections. Hence, by Lemma 2.3 (ii), this cohomology group is zero, and e_{i} is smooth.

Since both e_{1} : M_{1} → P and e_{2} : M_{2} → P are smooth, both have nonempty, open image contained in the locus of free lines. By hypothesis, the open subset of P parametrizing free lines is irreducible. Therefore the image of e_{1} and the image of e_{2} intersect. Choose a family ζ_{1} ∈ M_{1} and ζ_{2} ∈ M_{2} such that e_{1}(ζ_{1}) = e_{2}(ζ_{2}). Then ζ_{1} and ζ_{2} can be glued to obtain a morphism ζ : B → \overline{M}_{0,1}(X, 1) such that ζ|_{B_{1}} = ζ_{1} and ζ|_{B_{2}} = ζ_{2}. By Lemma 4.5, ζ is twisting. Moreover, if one of ζ_{1}, ζ_{2} is very twisting, then ζ is very twisting. And ζ̃ = ξ. Therefore ξ is twistable, and it is very twistable if one of ξ_{1}, ξ_{2} is very twistable. □

30 J. HARRIS AND J. STARR

Hypothesis 4.10. Let U ⊂ \overline{M}_{0,1}(X, 1) denote the preimage of U_{t-able} ⊂ \overline{M}_{0,0}(X, 1) under pr. The evaluation morphism ev : U → X has Zariski dense image, i.e., a general point of X is contained in a twistable line.

Let π : B → T be a family of prestable curves of arithmetic genus 0.

Definition 4.11. A 1-morphism ζ : B → \overline{M}_{0,1}(X, e) is positive (resp. very positive) if

(i) (π : B → T, h : B → X) is a family of stable maps, i.e., a 1-morphism ξ : T → \overline{M}_{0,0}(X, ϵ) for some ϵ ≥ 0,

(ii) the dimension of the obstruction group of \overline{M}_{0,0}(X, ϵ) is 0 at each point of ξ(T), (iii) the dimension of the obstruction group of \overline{M}_{0,0}(X, e) is 0 at each point of pr(ζ(T)), (iv) the pullback (pr ∘ ζ)^{∗}T_{\overline{M}_{0,0}(X,e)} is π-relatively deformation ample, and

(v) the pullback σ^{∗}O_{Σ}(σ) is π-relatively generated by global sections (resp. π-relatively ample).

Remark 4.12. This definition is similar to Definition 4.3. It differs in that e need not equal 1 and that pr ∘ ζ has image in the unobstructed locus, instead of requiring ζ to have image in the unobstructed locus of the morphism ev.

Lemma 4.13. Let π : B → T be a family of prestable curves of arithmetic genus 0 and let ζ : B → \overline{M}_{0,1}(X, 1) be a 1-morphism. There is an open subscheme U_{pos} ⊂ T (resp. U_{v-pos} ⊂ T) with the following property: for every morphism of schemes f : T′ → T, the pullback family f^{∗}π : f^{∗}B → T′ and f^{∗}ζ : f^{∗}B → \overline{M}_{0,1}(X, 1) is positive (resp. very positive) iff f(T′) ⊂ U_{pos} (resp. f(T′) ⊂ U_{v-pos}).

Proof. The proof is almost identical to the proof of lemma 4.6. □

Lemma 4.14. Let B be a prestable curve of arithmetic genus 0, and let ζ : B → \overline{M}_{0,1}(X, e) be a positive 1-morphism whose image is contained in the locus of very stable maps.

(i) If B is smooth, then ζ : B → \overline{M}_{0,1}(X, e) is free, i.e., ζ^{∗}T_{\overline{M}_{0,1}(X,e)} is generated by global sections. If ζ is very positive, then ζ is very free, i.e., ζ^{∗}T_{\overline{M}_{0,1}(X,e)} is ample.

(ii) The dimension of the obstruction group of the 1-morphism ζ : B → \overline{M}_{0,1}(X, e) is 0. In particular there is a discrete valuation ring, R, a family of prestable curves, π : B → Spec R, and a positive 1-morphism ζ_{R} : B → \overline{M}_{0,1}(X, e) such that the geometric closed fiber is ζ, such that the geometric generic fiber B_{η} is smooth, and such that ζ_{R}(B) is contained in the locus of very stable maps.

Proof. (i): By hypothesis, ζ is positive (resp. very positive). By the relative version of Lemma 3.21, the image of ζ is in the smooth locus of \overline{M}_{0,1}(X, e) and there is a short exact sequence,

0 ⟶ σ^{∗}O_{Σ}(σ) ⟶ ζ^{∗}T_{\overline{M}_{0,1}(X,e)} ⟶ (pr ∘ ζ)^{∗}T_{\overline{M}_{0,0}(X,e)} ⟶ 0.

By Definition 4.11 (iv) and (v), (pr ∘ ζ)^{∗}T_{\overline{M}_{0,0}(X,e)} is deformation ample and σ^{∗}O_{Σ}(σ) is generated by global sections (resp. deformation ample). Hence ζ^{∗}T_{\overline{M}_{0,1}(X,e)} is generated by global sections (resp. deformation ample by Lemma 2.10 (ii)). Since B is smooth ζ is free (resp. very free).

(ii): Let R′ be a discrete valuation ring and let π : B′ → Spec R′ be a smoothing of B, i.e., there is an isomorphism i : B → B′_{0} and the generic fiber B′_{η} is smooth. For technical reasons it is necessary to compactify \overline{M}_{0,1}(X, e). Let j : X ↪ \overline{X} be an open immersion of X into a projective scheme, and let j_{∗} : \overline{M}_{0,1}(X, e) ↪ \overline{M}_{0,1}(\overline{X}, e) be the corresponding open immersion of moduli stacks. By [9], \overline{M}_{0,1}(\overline{X}, e) is a proper Deligne-Mumford stack with projective coarse moduli space. Form the R′-stack, B′ ×\overline{M}_{0,1}(\overline{X}, e). With respect to an ample invertible sheaf on the coarse moduli

RATIONAL CURVES ON HYPERSURFACES II 31

space, let ϵ denote the degree of the map (i, ζ) : B → B′ × \overline{M}_{0,1}(\overline{X}, e). Then (i, ζ) is an object of the Abramovich-Vistoli stack,

K = K_{0,0}(B′ × \overline{M}_{0,1}(\overline{X}, e), ϵ) → Spec R′,

that parametrizes twisted stable maps to B′ × \overline{M}_{0,1}(\overline{X}, e) over Spec R′, cf. [1]. The stack K → Spec R′ is a Deligne-Mumford stack that is proper over Spec R′, and the coarse moduli space is projective over Spec R′.

The deformation theory of twisted stable maps is developed in [1, Section 5], and is exactly analogous to the deformation theory of stable maps. In particular, the obstruction space to the morphism o : K → Spec R′ at [(i, ζ)] is a quotient of H^{1}(B, ζ^{∗}T_{\overline{M}_{0,1}(X,e)}). Since ζ^{∗}T_{\overline{M}_{0,1}(X,e)} is generated by global sections, by Lemma 2.3 (i), h^{1}(B, ζ^{∗}T_{\overline{M}_{0,1}(X,e)}) = 0. Therefore o is smooth at [(i, ζ)]. By [6, Prop. 2.2.14], there exists an étale morphism of discrete valuation ring’s, Spec R → Spec R′ and a section s : Spec R → K of o. In other words, there exists a family of twisted stable maps over Spec R whose closed fiber is isomorphic to the pullback of (i, ζ), i.e.,

(π : B → Spec R′, λ : B → Spec R ×_{Spec R′} (B′ × \overline{M}_{0,1}(\overline{X}, e))).

Because B is a scheme, the family of curves over Spec R is a scheme, i.e., π : B → Spec R is a morphism of schemes. Because the composition of (i, ζ) with projection onto the closed fiber of B′ is an isomorphism, the composition

B ⟶^{λ} Spec R ×_{Spec R′} (B′ × \overline{M}_{0,1}(\overline{X}, e)) ⟶^{pr_{B′}} Spec R ×_{Spec R′} B′,

is an isomorphism. Since the image of (i, ζ) is contained in the open substack B′ × \overline{M}_{0,1}(X, e), the image of λ is contained in the the open substack Spec R ×_{Spec R′} (B′ × \overline{M}_{0,1}(X, e). Putting the pieces together, π : B → Spec R is a smoothing of B, and λ induces a 1-morphism of R-stacks, ζ_{R} : B → \overline{M}_{0,1}(X, e) such that the closed fiber is ζ.

By Lemma 4.13, ζ_{R} is positive. Since the locus of very stable maps in \overline{M}_{0,1}(X, e) is open, the image of ζ_{R} is contained in this locus. □

Remark 4.15. The use of the Abramovich-Vistoli stack above is a bit contrived. The most natural stack to use is the Hom stack parametrizing 1-morphisms between two flat, proper, tame Deligne-Mumford stacks. Martin Olsson has proved existence and some foundational properties of the Hom stack in the generality needed above, but the details have not yet been published.

Following is the main definition of this section.

Definition 4.16. Let π : B → T be a family of prestable curves of arithmetic genus 0. An inducting pair of degree e is a pair of 1-morphisms

(ζ_{1} : B → \overline{M}_{0,1}(X, 1), \overline{ζ}_{e} : B → \overline{M}_{0,1}(X, e)) ,

such that (i) ζ_{1} is very twisting, (ii) \overline{ζ}_{e} is very positive and the image of \overline{ζ}_{e} is contained in the locus of very stable maps, and (iii) the morphisms h_{ζ_{1}} : B → X and h_{\overline{ζ}_{e}} : B → X are equal.

Lemma 4.17. Let π : B → T be a family of prestable curves of arithmetic genus 0, and let

(ζ_{1} : B → \overline{M}_{0,1}(X, 1), \overline{ζ}_{e} : B → \overline{M}_{0,1}(X, e)) ,

32 J. HARRIS AND J. STARR

be a pair of 1-morphisms such that h_{ζ_{1}} = h_{\overline{ζ}_{e}}. Then there is an open subscheme U_{induct} ⊂ T with the following property: for every morphism of schemes f : T′ → T, the pullback of (ζ_{1}, \overline{ζ}_{e}) is inducting iff f(T′) ⊂ U.

Proof. Define U_{induct} to be the intersection of the open subset U_{vtwist} ⊂ T from Lemma 4.6 for ζ_{1} and the open subset U_{v-pos} ⊂ T from Lemma 4.13 for \overline{ζ}_{e}. □

The final definition of this section is the following.

Definition 4.18. Let (π : B → T, \overline{ζ}_{e} : B → \overline{M}_{0,1}(X, e)) be a very positive family whose image is contained in the locus of very stable maps. The 1-morphism \overline{ζ}_{e} is inductable if there exists a surjective étale morphism u : T′ → T and a morphism ζ_{1} : u^{∗}B → \overline{M}_{0,1}(X, 1) such that

(i) h_{ζ_{1}} = u^{∗}h_{\overline{ζ}_{e}}, and

(ii) (ζ_{1}, \overline{ζ}_{e}) is an inducting pair.

Lemma 4.19. Let (π : B → T, \overline{ζ}_{e} : B → \overline{M}_{0,1}(X, e)) be a very free family. There is an open subscheme U_{i−able} ⊂ T with the following property: for every morphism of schemes f : T′ → T, the pullback (f^{∗}π : f^{∗}B → T′, f^{∗}\overline{ζ}_{e} : f^{∗}B → \overline{M}_{0,1}(X, e)) is inductable iff f(T′) ⊂ U_{i−able}.

Proof. Apply Proposition 4.8 to

ξ := (π : B → T, h_{\overline{ζ}_{e}} : B → X).

□

5. The induction argument

In this section it is proved that if X satisfies Hypotheses 1.5, 1.6, 1.7, and 4.10, and if there exists an inductable 1-morphism \overline{ζ}_{e} : B → \overline{M}_{0,1}(X, e), then there exists an inductable 1-morphism \overline{ζ}_{e+1} : B → \overline{M}_{0,1}(X, e + 1). The basic idea is, given an inducting pair (ζ_{1}, \overline{ζ}_{e+1}), to form the family of “connected sums”, i.e., the family of reducible curves obtained by gluing the families ζ_{1} and \overline{ζ}_{e+1} along the two sections. However, this is a family of unpointed curves, and an inducting family is a family of pointed curves. By hypothesis, the section of \overline{ζ}_{e}, \overline{σ} : B → \overline{Σ}, is such that the Cartier divisor \overline{σ}(B) ⊂ \overline{Σ} moves. A general member of the linear system |\overline{σ}(B)| is the image of a section, σ′ : B → \overline{Σ}. Except at finitely many points of B, the sections \overline{σ} and σ′ are disjoint. Away from these points, the section of the family of reducible curves is taken to be σ′. The stable limit over the finitely many points of B is obtained by blowing up \overline{Σ} along the zero-dimensional scheme \overline{σ}(B) ∩ σ′(B). Unfortunately, the resulting family of pointed, reducible curves is no longer very positive.

To make the resulting family very positive, the family is altered by a modification.

5.1. Modification.

Definition 5.1. An input triple is a triple I = (ζ, L, σ_{i}) where (i) ζ = ((Σ, σ_{1}, …, σ_{r}), g : Σ → X) is an r-pointed stable map to X of arithmetic genus g, (ii) L ⊂ Σ is an irreducible component that is smooth, and (iii) σ_{i} ∈ L is a marked point.

Let I = (ζ, L, σ_{i}) be an input triple. Denote by M ⊂ Σ the union of all irreducible components other than L. Denote by R = (ρ_{1}, …, ρ_{c}) the intersection L ∩ M, denote by S = (σ_{j_{1}}, …, σ_{j_{d}}) the marked points that are contained in L other than σ_{i}, and denote by S′ = (σ_{k_{1}}, …, σ_{k_{e}}) the marked points that are contained in M.

Denote by Δ : L → L × L the diagonal, and denote by u : Λ → L × L the blowing up of L × L along Δ(R ∪ S). For each closed point λ ∈ R ∪ S, denote by F_{λ} ⊂ Λ the proper transform of L × {λ} ⊂ L × L. Denote by F_{Δ} ⊂ Λ the proper transform of Δ(L) ⊂ L × L. For each λ ∈ R ∪ S, there exists a unique section of pr_{1} ∘ u : Λ → L, σ_{I,λ} : L → Λ, such that F_{λ} = σ_{I,λ}(L). Also there is a section of pr_{1} ∘ u, σ_{I,Δ}, such that σ_{I,Δ}(L) = F_{Δ}.

Consider the projection pr_{1} : L × M → L. For each closed point λ ∈ R ∪ S′ there is a unique section of pr_{1}, σ′_{I,λ}, such that σ′_{I,λ}(L) = L × {λ}. Let Σ_{I} be the simple normal crossing surface containing Λ and L×M that is obtained by identifying the divisor σ_{I,λ}(L) ⊂ Λ with σ′_{I,λ}(L) ⊂ L×M for each λ ∈ R. The divisors are identified by σ_{I,λ}(t) ↔ σ′_{I,λ}(t) for each t ∈ L. There is a unique morphism p_{I} : Σ_{I} → L whose restriction to Λ is pr_{1} ∘ u and whose restriction to L × M is pr_{1}. There is a unique morphism g_{I} : Σ_{I} → X whose restriction to Λ is g|_{L} ∘pr_{2} ∘u and whose restriction to L × M is g|_{M} ∘ pr_{2}. For each integer j = 1, …, r such that j ≠ i, there exists a section of p_{I}, σ_{I,j} : L → Σ_{I} given by

σ_{I,j} = \{ \begin{matrix}σ_{I,λ}, & λ = σ_{j} ∈ S, \\ σ′_{I,λ}, & λ = σ_{j} ∈ S′ \end{matrix}

And there exists a section σ_{I,i} : L → Σ_{I} given by σ_{I,i} = σ_{I,Δ}. By construction, the sections σ_{I,j} are pairwise disjoint. Hence the datum

ζ_{I} = ((p_{I} : Σ_{I} → L, σ_{I,1}, …, σ_{I,r}), g_{I} : Σ_{I} → X)

is a family of r-pointed stable maps to X of arithmetic genus g. And the fiber over σ_{i} ∈ L is canonically isomorphic to ζ.

Definition 5.2. The family of stable maps, ζ_{I}, is the modification associated to I.

The relevance of this construction is the following. Let B be a prestable curve and let

ζ = ((p : Σ → B, σ_{1}, …, σ_{r}), g : Σ → X)

be a family of r-pointed stable maps to X of arithmetic genus g. Let b ∈ B be a smooth point, let L ⊂ Σ_{b} be an irreducible component that is smooth, and let σ_{i}(b) ∈ L be a marked point. Then I = (ζ_{b}, L, σ_{i}(b)) is an input triple. Let ζ_{I} be the modification associated to I. The fiber ζ_{I,σ_{i}(b)} is canonically isomorphic to the fiber ζ_{b}. Let B̃ be the prestable curve containing B and L obtained by identifying b ∈ B with σ_{i}(b) ∈ L. There is a unique family of r-pointed stable maps to X of arithmetic genus g,

ζ̃ = ((p̃: Σ̃ → B̃, σ̃_{1}, …, σ̃_{r}), g̃: Σ̃ → X),

whose restriction to B is ζ and whose restriction to L is ζ_{I}.

Definition 5.3. The family of stable maps, ζ̃, is the modification of ζ associated to (b, L, σ_{i}(b)).

Let I = (ζ, L, σ_{i}) be an input triple. Denote by ζσ̂_{i} the (r − 1)-pointed stable map obtained by forgetting the marked point σ_{i}.

Lemma 5.4. Let I = (ζ, L, σ_{i}) be an input triple.

(i) The dimension of the obstruction space of ζσ̂_{i} is 0 and for each closed point λ ∈ R, there exists a first-order deformation of ζσ̂_{i} that smoothes the node λ iff the same is true for ζ. (ii) For every closed point λ ∈ L, the dimension of the obstruction space of ζ_{I,λ} is 0 iff both the dimension of the obstruction space of ζσ̂_{i} is 0 and for each closed point λ ∈ R there exists a first-order deformation of ζσ̂_{i} that smoothes the node λ.

(iii) If the equivalent conditions of (ii) are satisfied then there is an exact sequence,

\begin{aligned}0 & → & T_{L} & → & ζ_{I}^{∗}T_{\overline{M}_{g,r}(X,e)} & → & … \\ … & → & T_{\overline{M}_{g,r−1}(X,e),ζσ̂_{i}} ⊗_{ℂ} O_{L} & → & ⨁_{λ∈R} T_{L,λ} ⊗_{O_{λ}} T_{M,λ} & → & 0. \end{aligned}

(iv) If the equivalent conditions of (iii) are satisfied, if L has genus 0, and if the map,

T_{\overline{M}_{g,r−1}(X,e),ζσ̂_{i}} → ⊕_{λ∈R}T_{L,λ} ⊗ T_{M,λ},

is surjective, then h^{1}(L, ζ_{I}^{∗}T_{\overline{M}_{g,r}(X,e)}) = 0.

Proof. The item (i) follows by Lemma 3.22. The items (ii) and (iii) follow by Lemmas 3.18, 3.19, 3.21 and 3.22. If L has genus 0, then T_{L} is generated by global sections. So (iv) follows from the long exact sequences in cohomology associated to the exact sequence in (ii) and by Lemma 2.3 (i). □

Let ζ be a stable map of arithmetic genus 0 and let I = (ζ, L, σ_{i}) be an input pair.

Lemma 5.5. If the dimension of the obstruction group of g|_{L} : L → X is 0, and if the pullback of T_{X} by g|_{M} : M → X is generated by global sections, then for every closed point λ ∈ L, the dimension of the obstruction space of ζ_{I,λ} is 0, and h^{1}(L, ζ_{I}^{∗}T_{\overline{M}_{0,r}(X,e)}) = 0.

Proof. By Lemma 3.17, Lemma 3.21 and Lemma 3.22, the family of pointed stable curves,

((pr_{1} ∘ u : Λ → L, σ_{λ}, σ_{Δ}), g|_{L} ∘ pr_{2} ∘ u : Λ → X),

is such that for each point of L the dimension of the obstruction space is 0. Consider the families obtained by successively gluing on the connected components of L × M. Applying Lemma 3.29 to each of these families, the hypotheses of Lemma 5.4 (iv) are satisfied for ζ_{I}. □

Let B be a prestable curve of arithmetic genus 0, and let

ζ = ((p : Σ → B, σ_{1}, …, σ_{r}), g : Σ → X),

be a family of r-pointed stable maps of arithmetic genus 0. Let b ∈ B be a smooth point, let L ⊂ Σ_{b} be an irreducible component that is not contracted by g, and let σ_{i}(b) ∈ L be a marked point. Let M be the union of all irreducible components of Σ_{b} other than L.

Lemma 5.6. If for every point b′ ∈ B the dimension of the obstruction group of ζ_{b′} is 0, if ζ^{∗}T_{\overline{M}_{0,r}(X,e)} is generated by global sections, if the dimension of the obstruction group of g|_{L} : L → X is 0, and if the pullback of T_{X} by g|_{M} : M → X is generated by global sections, then for every point b′ ∈ B̃ the dimension of the obstruction group of ζ̃_{b′} is 0 and h^{1}(B̃, ζ̃^{∗}T_{\overline{M}_{0,r}(X,e)}) = 0.

Therefore the dimension of the obstruction group of ζ̃: B̃ → \overline{M}_{0,r}(X, e) is 0. In particular there is a discrete valuation ring, R, a family of prestable curves, π : B → Spec R, and a 1-morphism ζ_{R} : B → \overline{M}_{0,r}(X, e) such that the geometric closed fiber is ζ̃ and such that the geometric generic fiber B_{η} is smooth.

Proof. The hypotheses of Lemma 5.5 are satisfied for the modification associated to I = (ζ_{b}, L, σ_{i}(b)), ζ̃_{L} : L → \overline{M}_{0,r}(X, e). Hence the dimension of the obstruction group of ζ̃ at each closed point of L is 0 and h^{1}(L, ζ̃_{L}^{∗}T_{\overline{M}_{0,r}(X,e)}) = 0. There is a short exact sequence of coherent sheaves on B̃,

0 ⟶ ζ^{∗}T_{\overline{M}_{0,r}(X,e)}(−b) ⟶ ζ̃^{∗}T_{\overline{M}_{0,r}(X,e)} ⟶ ζ̃_{L}^{∗}T_{\overline{M}_{0,r}(X,e)} ⟶ 0.

By Lemma 2.3 (ii), h^{1}(B, ζ^{∗}T_{\overline{M}_{0,r}(X,e)}(−b)) = 0. Hence by the long exact sequence of cohomology

associated to the short exact sequence above, h^{1}(B̃, ζ̃^{∗}T_{\overline{M}_{0,r}(X,e)}) = 0.

The second part of the lemma follows by the same proof as in Lemma 4.14 (ii). □

5.2. The induction argument.

Notation 5.7. The notation for the generators of the group of ℚ-Cartier divisor classes on \overline{M}_{0,1}(ℙ^{n}, e) is taken from [19].

(i) For each integer 0 ≤ i ≤ ⌊e⌋ denote by Δ_{i}, or sometimes Δ_{i,e−i}, the class of the ℚ-Cartier divisor that is the closure of the locus parametrizing embedded curves that have one irreducible component of degree i, one irreducible component of degree e − i and where the marked point is on the irreducible component of degree i.

(ii) Denote by L the first Chern class of ev^{∗}O_{ℙ^{n}}(1).

(iii) Denote by H the class of the ℚ-Cartier divisor parametrizing stable maps g : Σ → ℙ^{n} such that g(Σ) ∩ Λ ≠ ∅, where Λ ⊂ ℙ^{n} is a fixed linear ℙ^{n−2}.

Let B be a prestable curve of genus 0. Let ζ = ((p : Σ → B, σ), g : Σ → X) be a family of stable maps of genus 0 and degree e > 1. Denote by h : B → X the composition g ∘ σ.

Definition 5.8. The 1-morphism ζ : B → \overline{M}_{0,1}(X, e) is typical if, (i) the curve B is smooth, (ii) for every closed point b ∈ B, the obstruction group of ζ_{b} has dimension 0, (iii) for every point b ∈ B, the stable map ζ_{b} has only the trivial automorphism, (iv) for every integer j = 1, …, e − 1, the preimage ζ^{−1}(Δ_{j}) consists of finitely many points,

and for each b ∈ ζ^{−1}(Δ_{j}), the curve Σ_{b} has only two irreducible components,

(v) for every point b ∈ ζ^{−1}(Δ_{1}), Σ_{b} is a union of 2 irreducible components L ∪ M with σ(b) ∈ L such that g|_{L} : L → X is an isomorphism to a twistable line.

Lemma 5.9. Let π : B → T be a family of prestable curves of arithmetic genus 0 and let ζ : B → \overline{M}_{0,1}(X, e) be a 1-morphism. There is an open subscheme U_{typ} ⊂ T with the following property: for every morphism of schemes f : T′ → T, the pullback family f^{∗}π : f^{∗}B → T′ and f^{∗}ζ : f^{∗}B → \overline{M}_{0,1}(X, e) is typical iff f(T′) ⊂ U_{typ}.

Proof. Each of the conditions in Definition 5.8 is clearly an open condition. □

Lemma 5.10. If X satisfies Hypotheses 1.5, 1.6, 1.7, and 4.10, if for every closed point b ∈ B the obstruction group of ζ_{b} has dimension 0, and if ζ^{∗}T_{\overline{M}_{0,1}(X,e)} is generated by global sections, then there exists a discrete valuation ring, R, a family of prestable curves, π : B → Spec R, and a 1-morphism ζ_{R} : B → \overline{M}_{0,1}(X, e) such that the geometric closed fiber is ζ, such that the geometric generic fiber B_{η} is smooth, and such that (ζ_{R})_{η} is typical.

Proof. By the same argument as in the proof of Lemma 4.13, there exists a family ζ_{R} : B → \overline{M}_{0,1}(X, e) such that B_{η} is smooth. The condition that for every point the obstruction group has dimension 0 is stable under generization, so it also holds for (ζ_{R})_{η}. Similarly, (ζ_{R})_{η}^{∗}T_{\overline{M}_{0,1}(X,e)} is generated by global sections. By [16, Prop. 3.7], the family ζ_{R} may be chosen so that (ζ_{R})(B_{η}) is disjoint from any given finite collection of closed substacks of codimension at least 2. By Hypothesis 1.5, the locus of stable maps such that the domain curve has 3 or more irreducible components has codimension at least 2. Let Z ⊂ Δ_{i} be a closed substack whose image in the coarse moduli space of Δ_{i} is an ample divisor. Again by Hypothesis 1.5, Z has codimension 2 in \overline{M}_{0,1}(X, e). So (ζ_{R})(B_{η}) does not intersect Z, and therefore it is not contained in Δ_{i}. By Hypothesis 1.7, the locus of stable maps that have a nontrivial automorphism has codimension at least 2. Finally, by Hypothesis 1.6 and Hypothesis 4.10, the locus in Δ_{1} parametrizing stable maps g : L ∪ M → X such that g|_{L} : L → X is an isomorphism to a twistable line is a dense open in Δ_{1}. In particular the complement in Δ_{1} is a closed substack that has codimension 2 in \overline{M}_{0,1}(X, e). Therefore (ζ_{R})_{η} is typical. □

By Hypothesis 4.10, a general line L ⊂ X is twistable. Denote by,

ζ_{L} = ((p_{L} : Σ_{L} → L, σ_{L}), g_{L} : Σ_{L} → X),

a twisting family such that h_{L} : L → X is the inclusion. By Remark 4.4 (iv), the degree of g_{L} is either 1 or 2, i.e., either g_{L} : Σ_{L} → X is a birational map to a linear ℙ^{2} in X (obtained by blowing up a point on the ℙ^{2}), or g_{L} is an isomorphism of Σ_{L} to a nonsingular quadric surface in X.

Definition 5.11. Let X be a variety that satisfies Hypotheses 1.6 and 4.10. The variety X is planar type if for a general line L there exists a twisting family such that the degree of g_{L} is 1. The variety is quadric type if it is not planar type.

Remark 5.12. It is not hard to prove that if X is planar type, then there exists an integer r ≥ 2 such that for a general point x ∈ X, the set of lines in X that contain x sweep out a linear ℙ^{r}. Let Q denote the irreducible component of the Hilbert scheme of ℙ^{r}’s in X that contains the ℙ^{r}’s constructed in this way. Then X is birationally a ℙ^{r}-bundle over Q and, with its natural Plücker embedding, Q is not uniruled by lines. If X satisfies Hypothesis 1.5, then Q is not uniruled by rational curves of any degree. If also X contains a very twistable curve, then Q is a point and X is isomorphic to ℙ^{r} (not merely birational to ℙ^{r}). Thus, in what follows, X will usually be of quadric type.

Let (ζ_{1}, \overline{ζ}_{e}) be an inducting pair denoted by,

ζ_{1} = ((p : Σ → B, σ), g : Σ → X),
\overline{ζ}_{e} = ((\overline{p} : \overline{Σ} → B, \overline{σ}), \overline{g} : \overline{Σ} → X).

Denote

\begin{aligned}δ & = & deg(ζ_{1}^{∗}(2L − H)), \\ \overline{δ} & = & deg(\overline{ζ}_{e}^{∗}(\frac{2}{e}L − \frac{1}{e^{2}} H − ∑_{i=1}^{e−1} \frac{(e−i)^{2}}{e^{2}} Δ_{i})). \end{aligned}

By [19, Lem. 2.2.2], deg(σ^{∗}O_{Σ}(σ(B))) = δ and deg(\overline{σ}^{∗}O\overline{Σ}(\overline{σ}(B))) = \overline{δ}. In particular, both δ and \overline{δ} are nonnegative integers, and \overline{δ} is positive by Definition 4.11.

Theorem 5.13. For each integer d = 1, …, \overline{δ}, there exists an inducting pair (ξ_{d,1}, \overline{ξ}_{d,e+1}) such that \overline{ξ}_{d,e+1} is typical and such that the following conditions are satisfied,

\{ \begin{matrix}deg(\overline{ξ}_{d,e+1}^{∗}H) & = & deg(\overline{ζ}_{e}^{∗}H) + deg(ζ_{1}^{∗}H), \\ deg(ξ_{d,1}^{∗}H) & = & deg(ζ_{1}^{∗}H) + d, & if X is planar type, \\ deg(ξ_{d,1}^{∗}H) & = & deg(ζ_{1}^{∗}H) + 2d, & if X is quadric type, \\ deg(ξ_{d,1}^{∗}L) & = & deg(ζ_{1}^{∗}L) + d, \\ deg(\overline{ξ}_{d,e+1}^{∗}L) & = & deg(\overline{ζ}_{e}^{∗}L) + d, \\ deg(\overline{ξ}_{d,e+1}^{∗}Δ_{i}) & = & deg(\overline{ζ}_{e}^{∗}Δ_{i−1}), & i = 2, …, e − 1, \\ deg(\overline{ξ}_{d,e+1}^{∗}Δ_{e}) & = & deg(\overline{ζ}_{e}^{∗}Δ_{e−1}) + δ + d, & if i > 1, \\ deg(\overline{ξ}_{d,e+1}^{∗}Δ_{1}) & = & \overline{δ} − d, & if i > 1, \\ deg(\overline{ξ}_{d,2}^{∗}Δ_{1}) & = & δ + \overline{δ}, & if i = 1. \end{matrix}  (9)

The families are denoted,

\overline{ξ}_{d,e+1} = ((\overline{p}_{d,e+1} : \overline{Σ}_{d,e+1} → B, \overline{σ}_{d,e+1}), \overline{g}_{d,e+1} : \overline{Σ}_{d,e+1} → X),
ξ_{d,1} = ((p_{d,1} : Σ_{d,1} → B, σ_{d,1}), g_{d,1} : Σ_{d,1} → X).

Also \overline{h}_{d,e+1} denotes \overline{g}_{d,e+1} ∘ \overline{σ}_{d,e+1} and h_{d,1} denotes g_{d,1} ∘ σ_{d,1}.

RATIONAL CURVES ON HYPERSURFACES II 37

Proof. By Lemma 4.14 (ii) and Lemma 4.19, it suffices to consider the case that B is smooth. By Lemma 4.14(i) \overline{ζ}_{e} : B → \overline{M}_{0,1}(X, e) is very free. Hence, the same argument as in the proof of Lemma 4.14 (ii) and [16, Prop. 3.7], it suffices to consider the case that \overline{ζ}_{e}(B_{e}) is in general position, i.e., for any finite collection of codimension 2 subvarieties (Z_{α}|α = 1, …, M) and any finite collection of divisors (D_{β}|β = 1, …, N), \overline{ζ}_{e}(B_{e}) is disjoint from each Z_{α} and has 0-dimensional intersection with each D_{β}.

The family ζ^{(1)}: Denote by Σ^{(1)} the surface containing Σ and \overline{Σ} obtained by identifying the divisor σ(B) ⊂ Σ and \overline{σ}(B) ⊂ \overline{Σ} via σ(b) ↔ \overline{σ}(b). Denote by p^{(1)} : Σ^{(1)} → B the unique morphism whose restriction to Σ is p and whose restriction to \overline{Σ} is \overline{p}. Denote by g^{(1)} : Σ^{(1)} → X the unique morphism whose restriction to Σ is g and whose restriction to \overline{Σ} is \overline{g}. Then ζ^{(1)} = (p^{(1)} : Σ^{(1)} → B, g^{(1)} : Σ^{(1)} → X) is a family of stable maps in the boundary divisor Δ_{1} ⊂ \overline{M}_{0,0}(X, e + 1). Denote by τ the stable A-graph with two vertices v and \overline{v} of degree 1 and e respectively, with no tails, and with one edge connecting v and \overline{v}. Then ζ^{(1)} factors through the canonical 1-morphism of Behrend-Manin stacks \overline{M}(X, τ) → \overline{M}_{0,0}(X, e + 1).

Claim 5.14. The pullback (ζ^{(1)})^{∗}T_{\overline{M}_{0,0}(X,e)} is ample.

By Lemma 3.28, there is a short exact sequence,

0 ⟶ ζ_{1}^{∗}T_{ev} ⟶ (ζ^{(1)})^{∗}T_{\overline{M}(X,τ)} ⟶ \overline{ζ}_{e}^{∗}T_{\overline{M}_{0,1}(X,e)} ⟶ 0.  (10)

Since ζ_{1} is very twisting, by Definition 4.3 ζ_{1}^{∗}T_{ev} is ample. Since \overline{ζ}_{e} is very positive, by Lemma 4.14, \overline{ζ}_{e}^{∗}T_{\overline{M}_{0,1}(X,e)} is ample. Hence (ζ^{(1)})^{∗}T_{\overline{M}(X,τ)} is ample. By Lemma 3.29, there is a short exact sequence,

0 → (ζ^{(1)})^{∗}T_{\overline{M}(X,τ)} → (ζ^{(1)})^{∗}T_{\overline{M}_{0,0}(X,e+1)} → σ^{∗}O_{Σ}(σ) ⊗ \overline{σ}^{∗}O\overline{Σ}(\overline{σ}) → 0.  (11)

By Definition 4.3 and Definition 4.11, both σ^{∗}O_{Σ}(σ) and \overline{σ}^{∗}O\overline{Σ}(\overline{σ}) are ample. Therefore the tensor product is ample, and (ζ^{(1)})^{∗}T_{\overline{M}_{0,0}(X,e+1)} is ample.

The family ζ^{(2)}: The family ζ^{(1)} cannot be an inductable family, because it is a family of unpointed curves rather than 1-pointed curves. The next approximation to ξ_{d,e+1} “adds” a marked section to ζ^{(1)}. The self-intersection of \overline{σ}(B) ⊂ \overline{Σ} is the degree of the invertible sheaf \overline{σ}^{∗}O\overline{Σ}(\overline{σ}(B)), which is also

\overline{δ} = deg (2\overline{ζ}_{e}^{∗}L − \overline{ζ}_{e}^{∗}H) .

Let ς : B → \overline{Σ} be a section such that ς(B) ⊂ \overline{Σ} is a general member of the linear system |\overline{σ}(B)|. Because \overline{σ}^{∗}O\overline{Σ}(\overline{σ}(B)) is generated by global sections, there exists ς such that ς(B) has only transverse intersections with \overline{σ}(B). Denote the points of intersection by q_{1}, …, q\overline{δ} ∈ \overline{Σ}. Denote by

w : \overline{Σ}^{(2)} → \overline{Σ} the blowing up of \overline{Σ} at the points q_{1}, …, q\overline{δ}. Let \overline{p}^{(2)} : \overline{Σ}^{(2)} → B denote the projection

\overline{p} ∘ w. Let \overline{g}^{(2)} denote \overline{g} ∘ w. Let \overline{σ}^{(2)} : B → \overline{Σ}^{(2)} and ς^{(2)} : B → \overline{Σ}^{(2)} denote the proper transforms of \overline{σ} and ς respectively. Notice that \overline{σ}^{(2)}(B) and ς^{(2)}(B) are disjoint by construction. So the data

\overline{ζ}_{e}^{(2)} = ((\overline{p}^{(2)} : \overline{Σ}^{(2)} → B, \overline{σ}^{(2)}, ς^{(2)}), \overline{g}^{(2)} : \overline{Σ}^{(2)} → X)

is a family of stable 2-pointed maps, i.e., a 1-morphism \overline{ζ}_{e}^{(2)} : B → \overline{M}_{0,2}(X, e).

Claim 5.15. The pullback (\overline{ζ}_{e}^{(2)})^{∗}T_{\overline{M}_{0,2}(X,e)} is generated by global sections.

By Lemmas 3.16, 3.21 and 3.22, there is a short exact sequence:

0 ⟶ (ς^{(2)})^{∗}O\overline{Σ}_{(2)}(ς^{(1)}(B)) ⟶ (\overline{ζ}_{e}^{(2)})^{∗}T_{\overline{M}_{0,2}(X,e)} ⟶ \overline{ζ}_{e}^{∗}T_{\overline{M}_{0,1}(X,e)} ⟶ 0.

38 J. HARRIS AND J. STARR

By construction, the self-intersection of ς^{(2)}(B) ⊂ \overline{Σ}^{(2)} is 0, i.e., (ς^{(2)})^{∗}O\overline{Σ}_{(2)}(ς^{(2)}(B)) is the trivial invertible sheaf O_{B}. By hypothesis \overline{ζ}_{e}^{∗}T_{\overline{M}_{0,1}(X,e)} is generated by global sections. Therefore by

Lemma 2.3 (iii), (\overline{ζ}_{e}^{(2)})^{∗}T_{\overline{M}_{0,2}(X,e)} is generated by global sections.

Denote by Σ^{(2)} the surface containing \overline{Σ}^{(2)} and Σ obtained by identifying the divisors \overline{σ}^{(2)}(B) ⊂ \overline{Σ}^{(2)} and σ(B) ⊂ Σ via \overline{σ}^{(2)}(b) ↔ σ(b). Denote by p^{(2)} : Σ^{(2)} → B the unique morphism whose restriction to \overline{Σ}^{(2)} is \overline{p}^{(2)} and whose restriction to Σ is p. Denote by g^{(2)} : Σ^{(2)} → X the unique morphism whose restriction to \overline{Σ}^{(2)} is \overline{g}^{(2)} and whose restriction to Σ is g. Denote by σ^{(2)} : B → Σ^{(2)} the morphism obtained by composing ς^{(2)} with the inclusion \overline{Σ}^{(2)} ⊂ Σ^{(2)}. Then

ζ^{(2)} = ((p^{(2)} : Σ^{(2)} → B, σ^{(2)}), g^{(2)} : Σ^{(2)} → X)

is a family of stable maps in the boundary Δ_{e} ⊂ \overline{M}_{0,1}(X, e + 1). Denote by τ^{(2)} the stable A-graph with two vertices v and \overline{v} of degree 1 and e respectively, a single tail attached to \overline{v}, and a single edge connecting v and \overline{v}. Then ζ^{(2)} factors through the canonical 1-morphism of Behrend-Manin stacks \overline{M}(X, τ^{(2)}) → \overline{M}_{0,1}(X, e + 1).

Claim 5.16. The pullback (ζ^{(2)})^{∗}T_{\overline{M}_{0,1}(X,e)} is generated by global sections.

By Lemma 3.28, there is a short exact sequence,

0 ⟶ ζ_{1}^{∗}T_{ev} ⟶ (ζ^{(2)})^{∗}T_{\overline{M}(X,τ^{(2)})} ⟶ (\overline{ζ}_{e}^{(2)})^{∗}T_{\overline{M}_{0,2}(X,e)} ⟶ 0.

By Claim 5.15, the third term is generated by global sections. Since ζ_{1} is very twisting, by Definition 4.3, ζ_{1}^{∗}T_{ev} is ample. In particular it is generated by global sections. By Lemma 2.3 (iii), (ζ^{(2)})^{∗}T_{\overline{M}(X,τ^{(2)})} is generated by global sections. By Lemma 3.29 there is a short exact sequence,

0 → (ζ^{(2)})^{∗}T_{\overline{M}(X,τ^{(2)})} → (ζ^{(2)})^{∗}T_{\overline{M}_{0,1}(X,e+1)} → σ^{∗}O_{Σ}(σ(B)) ⊗_{O_{B}} (\overline{σ}^{(2)})^{∗}O\overline{Σ}_{(2)}(\overline{σ}^{(2)}(B)) → 0.

By construction, O\overline{Σ}_{(2)}(\overline{σ}^{(2)}(B)) is isomorphic to O_{B}. Since ζ_{1} is very twisting, by Definition 4.3, σ^{∗}O_{Σ}(σ(B)) is generated by global sections. Hence the third term in the short exact sequence is generated by global sections. Since also the first term is generated by global sections, by Lemma 2.3 (iii), (ζ^{(2)})^{∗}T_{\overline{M}_{0,1}(X,e+1)} is generated by global sections.

If e > 1, the image of ζ^{(2)} intersects the divisor Δ_{1} transversely at the images of the points q_{1}, …, q\overline{δ}. In particular, the degree of the ℚ-Cartier divisor class (ζ^{(2)})^{∗}O_{\overline{M}_{0,1}(X,e+1)}(Δ_{1}) is \overline{δ}, which is positive. If e = 1, then Δ_{1} = Δ_{e}. In this case \overline{M}(X, τ^{(2)}) is the normalization of Δ_{1,1} in a neighborhood of ζ^{(2)}(B). So the degree of (ζ^{(2)})^{∗}O_{\overline{M}_{0,1}(X,2)}(Δ_{1}) is the sum of the degree of the pullback of the normal sheaf of \overline{M}(X, τ^{(2)}) → \overline{M}_{0,1}(X, 2) and the degree of the divisor q_{1} +⋯+q\overline{δ}, i.e. δ + \overline{δ}. So also in this case the degree of (ζ^{(2)})^{∗}O_{\overline{M}_{0,1}(X,e+1)}(Δ_{1}) is positive.

Similar computations give that deg((ζ^{(2)})^{∗}Δ_{i}) = \overline{ζ}_{e}^{∗}Δ_{i−1} for i = 2, …, e − 1. The curve ζ^{(2)}(B) is contained in the divisor Δ_{e}, which is the image of \overline{M}(X, τ^{(2)}). As noted above, by Lemma 3.29, the morphism \overline{M}(X, τ^{(2)}) → \overline{M}_{0,1}(X, e + 1) is unramified and the pullback by ζ^{(2)} of the normal sheaf has degree deg(σ^{∗}O_{Σ}(σ(B))) = δ. If e > 1, then the image of ζ^{(2)}(B) is contained in the smooth locus of Δ_{e} so that the degree of (ζ^{(2)})^{∗}O_{\overline{M}_{0,1}(X,e+1)}(Δ_{e}) is precisely δ. If e = 1, then each point q_{1}, …, q\overline{δ} maps to a point of Δ_{1} where Δ_{1} intersects itself transversely. So if e = 1, the total degree of (ζ^{(2)})^{∗}O_{\overline{M}_{0,1}(X,2)}(Δ_{1}) is δ + \overline{δ} (this is the same result from the last paragraph – it is included for the sake of consistency).

It is evident that,

deg((ζ^{(2)})^{∗}H) = deg(\overline{ζ}_{e}^{∗}H) + deg(ζ_{1}^{∗}H),

and deg((ζ^{(2)})^{∗}L) = deg(\overline{ζ}_{e}^{∗}L). As a “consistency check” for the computations of these degrees, observe that by [19, Lem. 2.2.2],

deg((σ^{(2)})^{∗}O_{Σ^{(2)}}(σ^{(2)}(B)) = deg ((ζ^{(2)})^{∗} (\frac{2}{e + 1}L − \frac{1}{(e + 1)^{2}} H − ∑_{i=1}^{e} \frac{(e + 1 − i)^{2}}{(e + 1)^{2}} Δ_{i})) .

Substituting the values above, using the formulas for δ and \overline{δ}, and using that ζ_{1}^{∗}L = \overline{ζ}_{e+1}^{∗}L, the degree of the right-hand-side is 0. Of course this is correct; σ^{(2)}(B) ⊂ Σ^{(2)} has self-intersection 0 by construction.

The family \overline{ξ}_{0,e+1}: Before proceeding to the construction of the families \overline{ξ}_{d,e+1}, the family ζ^{(2)} is deformed to a typical family \overline{ξ}_{0,e+1}, cf. Definition 5.8. The families \overline{ξ}_{d,e+1} will be constructed as suitable modifications of \overline{ξ}_{0,e+1}.

Since ζ^{(2)} : B → \overline{M}_{0,1}(X, e + 1) is free, i.e., (ζ^{(2)})^{∗}T_{\overline{M}_{0,1}(X,e+1)} is generated by global sections, it follows by the same argument as in the proof of Lemma 4.14 (ii) and [16, Prop. 3.7] that there exists a deformation \overline{xi}_{0,e+1} : B → \overline{M}_{0,1}(X, e + 1) of ζ̃ that is in general position. Here deformation means that both ζ^{(2)} and \overline{ξ}_{0,e+1} have a common generization. Moreover, for any finite collection of open conditions that are satisfied by ζ^{(2)}, the deformation \overline{ξ}_{0,e+1} may be chosen to also satisfy these conditions. Denote the family of stable maps by,

\overline{ξ}_{0,e+1} = ((\overline{p}_{0,e+1} : \overline{Σ}_{0,e+1} → B, \overline{σ}_{0,e+1}), \overline{g}_{0,e+1} : \overline{Σ}_{0,e+1} → X).

Denote by \overline{h}_{0,e+1} : B → X the composition \overline{g}_{0,e+1} ∘ \overline{σ}_{0,e+1}.

Claim 5.17. There exists a deformation \overline{ξ}_{0,e+1} of ζ^{(2)} such that

(i) \overline{ξ}_{0,e+1} is typical, (ii) (pr ∘ \overline{ξ}_{0,e+1})^{∗}T_{\overline{M}_{0,0}(X,e+1)} is ample, (iii) \overline{h}_{0,e+1} : B → X is very twistable, (iv) and (\overline{σ}_{0,e+1})^{∗}O_{\overline{Σ}_{0,e+1}}(\overline{σ}_{0,e+1}(B)) is isomorphic to O_{B}.

It suffices to prove that for a general deformation \overline{ξ}_{0,e+1} of ζ^{(2)}, \overline{ξ}_{0,e+1} satisfies the properties in Claim 5.17, i.e., for every irreducible component of the Hom stack Hom(B, \overline{M}_{0,1}(X, e)) that contains ζ^{(2)}, each of the conditions (i)–(iv) holds on a dense open substack.

(i): By Lemmas 5.9 and 5.10, a general deformation of ζ^{(2)} is typical.

(ii): The pullback by pr ∘ ζ^{(2)} : B → \overline{M}_{0,0}(X, e + 1) of T_{\overline{M}_{0,0}(X,e+1)} equals (ζ^{(1)})^{∗}T_{\overline{M}_{0,0}(X,e)}. By Claim 5.14, this is ample. The condition on deformations of ζ^{(2)} that the pullback of T_{\overline{M}_{0,0}(X,e)} is ample is an open condition by Lemma 2.8 and Lemma 2.9 (or more direct arguments). Therefore the pullback (pr ∘ \overline{ξ}_{0,e+1})^{∗}T_{\overline{M}_{0,0}(X,e)} is ample.

(iii): The morphism g^{(2)} ∘ σ^{(2)} : B → X equals \overline{g} ∘ ς, and this is a deformation of \overline{g} ∘ \overline{σ} = g ∘ σ. Because ζ_{1} is very twisting, g ∘ σ is very twistable. By Proposition 4.8, ς can be chosen so that g^{(2)} ∘ σ^{(2)} is very twistable. Since \overline{h}_{0,e+1} is a deformation of g^{(2)} ∘ σ^{(2)}, by Proposition 4.8, \overline{ξ}_{0,e+1} can be chosen so that \overline{h}_{0,e+1} is very twistable.

40 J. HARRIS AND J. STARR

(iv): Since (σ^{(2)})^{∗}O_{Σ^{(2)}}(σ^{(2)}(B)) is isomorphic to O_{B}, also (\overline{σ}_{0,e+1})^{∗}O_{\overline{Σ}_{0,e+1}}(\overline{σ}_{0,e+1}(B)) is isomorphic to O_{B}. This proves Claim 5.17. In particular, \overline{ξ}_{0,e+1} is positive and \overline{ξ}_{0,e+1}(B) is contained in the locus of very stable maps. Unfortunately it is not very positive!

To be a bit more precise in the proof of (iii), there exists a deformation of ζ_{1}, ξ_{0,1} : B → \overline{M}_{0,1}(X, 1) that is very twisting and such that h_{0,1} := g_{0,1} ∘ σ_{0,1} : B → X equals \overline{h}_{0,e+1}.

Because \overline{ξ}_{0,e+1} is a deformation of ζ^{(2)}, the intersection number of \overline{ξ}_{0,e+1}(B) with any divisor of \overline{M}_{0,1}(X, e + 1) is equal to the intersection number of ζ^{(2)}(B) with that divisor. These intersection numbers were computed above. Similarly, the intersection number of ξ_{0,1}(B) with any divisor in \overline{M}_{0,1}(X, 1) is equal to the intersection number of ζ_{1}(B) with that divisor.

The families \overline{ξ}_{d,e+1}: The families \overline{ξ}_{d,e+1} and ξ_{d,1} are constructed by induction on d. The base case d = 0 is the pair (ξ_{0,1}, \overline{ξ}_{0,e+1}) constructed above, and the induction step consists of performing a single modification and then deforming.

Let d be an integer 1 ≤ d ≤ \overline{δ}. By way of induction, suppose that a pair (ξ_{d−1,1}, \overline{ξ}_{d−1,e+1}) has been constructed such that ξ_{d−1,1} : B → \overline{M}_{0,1}(X, 1) is very twisting, \overline{ξ}_{d−1,e+1} : B → \overline{M}_{0,1}(X, e + 1) is very positive and typical (in particular \overline{ξ}_{d−1,e+1}(B) is contained in the locus of very stable maps), the map h_{d−1,1} : B → X equals the map \overline{h}_{d−1,e+1} : B → X, and the degree conditions of Equation (9) hold for d − 1. In particular, because d ≤ \overline{δ}, deg(\overline{ξ}_{d−1,e+1}^{∗}Δ_{1}) ≥ 1.

Let b ∈ B be a closed point such that the stable map (\overline{ξ}_{d−1,e+1})_{b} is in Δ_{1}. By Definition 5.8, this stable map is of the form,

(\overline{ξ}_{d−1,e+1})_{b} = ((L ∪ M, \overline{σ}_{d−1,e+1}(b)), (\overline{g}_{d−1,e+1})_{b} : L ∪ M → X),

where \overline{σ}_{d−1,e+1}(b) ∈ L and g|_{L} : L → X is an isomorphism to a twistable line.

Denote by B̃ the prestable curve of genus 0 containing B and L obtained by identifying the divisor b ∈ B with the divisor \overline{σ}_{d−1,e+1}(b) ∈ L. Denote by,

ξ̃_{d−1,e+1} = ((p̃_{d−1,e+1} : Σ̃_{d−1,e+1} → B̃, σ̃_{d−1,e+1}), g̃_{d−1,e+1} : B̃ → X),

the modification of \overline{ξ}_{d−1,e+1} associated to (b, L, \overline{σ}_{d−1,e+1}(b)), cf. Definition 5.3.

By construction, for each divisor in \overline{M}_{0,1}(X, e + 1), the difference of the intersection number with ξ̃_{d−1,e+1}(B̃) and the intersection number with \overline{ξ}_{d−1,e+1}(B) equals the intersection number with ξ̃_{d−1,e+1}(L), i.e.,

ξ̃_{d−1,e+1}^{∗}H = \overline{ξ}_{d−1,e+1}^{∗}H,
ξ̃_{d−1,e+1}^{∗}L = \overline{ξ}_{d−1,e+1}^{∗}L + 1,
ξ̃_{d−1,e+1}^{∗}Δ_{i} = \overline{ξ}_{d−1,e+1}^{∗}Δ_{i},  i = 2, …, e − 1,
ξ̃_{d−1,e+1}^{∗}Δ_{1} = \overline{ξ}_{d−1,e+1}^{∗}Δ_{1} − 1, i > 1,
ξ̃_{d−1,e+1}^{∗}Δ_{e} = \overline{ξ}_{d−1,e+1}^{∗}Δ_{e} + 1, i > 1,
ξ̃_{d−1,2}^{∗}Δ_{1} = \overline{ξ}_{d−1,e+1}^{∗}Δ_{1}, i = 1.

Just a few remarks about this list: The second line arises because h̃_{d−1,e+1} : L → X equals g|_{L} : L → X, which is an isomorphism to a line. The fourth line arises by applying Lemma 3.29 and using the fact that the pullback of O_{\overline{M}_{0,1}(X,e+1)}(Δ_{1}) equals

σ_{I,λ}^{∗}O_{Λ}(σ_{I,λ}(L)) ⊗ (σ′_{I,λ})^{∗}O_{L×M}(σ′_{I,λ}(L)),

in the notation of Subsection 5.1, where λ is the point in L ∩ M. Of course the second factor in the tensor product is isomorphic to O_{L} and the first factor is O_{L}(−λ) (because of the blowing up of L × L at (λ, λ)); hence the net degree change is −1. Similarly, the fifth line arises because the intersection of ξ̃_{d−1,e+1}(L) and Δ_{e} is exactly ξ̃_{d−1,e+1}(λ); hence the degree increases by 1. In case e = 1, the last two contributions exactly cancel each other, which gives the last line.

As a “consistency check”, substituting the computations above into the formula [19, Lem 2.2.2] for the self-intersection of σ̃_{d−1,e+1}(B̃) ⊂ Σ̃_{d−1,e+1} yields a net change of +1. Of course this is correct because the change equals the self-intersection of σ_{I,Δ}(L) ⊂ Λ, which is T_{L}(−λ) (because of the blowing up of L × L at (λ, λ)), and this has degree 2 − 1 = 1.

Claim 5.18. The 1-morphism ξ̃_{d−1,e+1} : B̃ → \overline{M}_{0,1}(X, e + 1) is very positive, the image is contained in the locus of stable maps with only the trivial automorphism, and the hypotheses of Lemma 5.6 are satisfied.

First of all, the restriction of h̃_{d−1,e+1} : B̃ → X to B is \overline{h}_{d−1,e+1} and the restriction to L is g|_{L} : L → X. By hypothesis, \overline{h}_{d−1,e+1} is stable, and g|_{L} is stable because it is a closed immersion. Therefore \overline{h}_{d−1,e+1} is stable. By hypothesis, the dimension of the obstruction group of \overline{h}_{d−1,e+1} is

0. Moreover, possibly after deforming \overline{ξ}_{d−1,e+1}, it may be assumed that the node of L ∩ M maps to a very general point of X. Therefore g|_{L} : L → X is free by [7, Prop. 4.14]. So the dimension of the obstruction group of g|_{L} is 0. Also, by Hypothesis 4.10 it may be assumed that ev : \overline{M}_{0,1}(X, 1) → X is smooth at ((L, \overline{σ}_{d−1,e+1}(b)), g|_{L} : L → X). So by Lemma 3.28, the dimension of the obstruction group of h̃_{d−1,e+1} is 0. Hence (i) and (ii) of Definition 4.11 are satisfied.

By hypothesis, the dimension of the obstruction group of \overline{M}_{0,0}(X, e + 1) at each point of pr(ξ̃_{d−1,e+1}(B)) = pr(\overline{ξ}_{d−1,e+1}(B)) is 0. And pr(ξ̃_{d−1,e+1}(L)) is the point pr((ξ̃_{d−1,e+1})_{b}), which is one of the points above. Thus (iii) of Definition 4.11 is satisfied. Similarly, the restriction to B of the pullback,

(pr ∘ ξ̃_{d−1,e+1})^{∗}T_{\overline{M}_{0,0}(X,e+1)},

equals the pullback associated to pr ∘ \overline{ξ}_{d−1,e+1}, which is deformation ample by hypothesis. And the restriction to L is the pullback of a vector bundle by a constant map, hence it is isomorphic to a direct sum of copies of O_{L}. In particular the restriction to L is generated by global sections. By Lemma 2.11, the pullback is deformation ample on all of B̃, hence (iv) of Definition 4.11 is satisfied.

Finally, the restriction to B of the pullback,

σ_{d−1,e+1}^{∗}OΣ̃_{d−1,e+1}(σ̃_{d−1,e+1}(B̃)),

is equal to the analogous sheaf for \overline{ξ}_{d−1,e+1}, and this is generated by global sections by construction. The restriction to L is the pullback by the diagonal of T_{L}, twisted down by \overline{σ}_{d−1,e+1}(b), i.e., O_{L}(2 − 1) = O_{L}(1). Thus by Lemma 2.11, the restriction to all of B̃ is deformation ample. Hence

(v) of Definition 4.11 is satisfied, therefore ξ̃_{d−1,e+1} is very positive.

By hypothesis ξ̃_{d−1,e+1}(B) = \overline{ξ}_{d−1,e+1}(B) is contained in the locus of stable maps that have only the trivial automorphism. In particular, (\overline{ξ}_{d−1,e+1})_{b} has only the trivial automorphism, from which it easily follows that ξ̃_{d−1,e+1}(L) is contained in the locus of stable maps with only the trivial automorphism. Therefore the image of all of B̃ is contained in the locus of stable maps with only the trivial automorphism.

Because \overline{ξ}_{d−1,e+1} : B → \overline{M}_{0,1}(X, e + 1) is positive and the image is contained in the locus of very stable curves, the hypotheses of Lemma 3.22 are satisfied and there is a short exact sequence of locally free O_{B}-modules,

\begin{aligned}0 & → & \overline{σ}_{d−1,e+1}^{∗}O_{\overline{Σ}_{d−1,e+1}}(\overline{σ}_{d−1,e+1}(B)) & → & … \\ & & \overline{ξ}_{d−1,e+1}^{∗}T_{\overline{M}_{0,1}(X,e+1)} & → & (pr ∘ \overline{ξ}_{d−1,e+1})^{∗}T_{\overline{M}_{0,0}(X,e+1)} & → & 0. \end{aligned}

By hypothesis, the third term is deformation ample, and the first term is generated by global sections. By Lemma 2.3 (iii), the pullback of T_{\overline{M}_{0,1}(X,e+1)} is generated by global sections. Because the node of L ∩ M is mapped to a very general point of X, by [7, Prop. 4.14], the pullback of T_{X} to L and the pullback to M are each generated by global sections; in particular the dimension of the obstruction group of g|_{L} : L → X is 0. Hence the hypotheses of Lemma 5.6 are satisfied; therefore Claim 5.18 is true.

Of course the next step will be to apply Lemma 5.6 to construct \overline{ξ}_{d,e+1}. But first the family ξ̃_{d−1,1} is constructed. By hypothesis, the line L is twistable. Denote by

ξ_{L,1} = ((p_{L,1} : Σ_{L,1} → L, σ_{L,1}), g_{L,1} : Σ_{L,1} → X),

a twisting family such that h_{L,1} = g|_{L}. If X is of planar type, assume that the degree of g_{L,1} is 1; otherwise the degree of g_{L,1} is 2. By Hypotheses 1.6 and 4.10, Lemma 4.9 applies to ξ_{d−1,1} and ξ_{L,1}, i.e., possibly after deforming the two families (without deforming h̃_{d−1,e+1}),

(ξ_{d−1,1})_{b} = (ξ_{L,1})_{\overline{σ}_{d−1,e+1}(b)}.

Define ξ̃_{d−1,1} : B̃ → \overline{M}_{0,1}(X, 1) to be the unique 1-morphism whose restriction to B is ξ_{d−1,1} and whose restriction to L is ξ_{L,1}. By Lemma 4.5, ξ̃_{d−1,1} is very twisting. By construction, h̃_{d−1,1} = h̃_{d−1,e+1}. Also,

deg(ξ̃_{d−1,1}^{∗}H) = deg(ξ_{d−1,1}^{∗}H) + deg(g_{L,1}).

By definition, deg(g_{L,1}) is 1 if X is planar type, and 2 if X is quadric type. Finally,

deg(ξ̃_{d−1,1}^{∗}L) = deg(ξ_{d−1,1}^{∗}L) + 1,

because h_{L,1} : L → X is an isomorphism to a line.

By Lemma 5.6, there exists a 1-morphism \overline{ξ}_{d,e+1} : B → \overline{M}_{0,1}(X, e + 1), i.e., both ξ̃_{d−1,e+1} and \overline{ξ}_{d,e+1} have a common generization. Since the image of ξ̃_{d−1,e+1} is contained in the locus of stable maps with only the trivial automorphism, \overline{ξ}_{d,e+1} can be chosen with the same property. Because the dimension of the obstruction group at every point of the image of ξ̃_{d−1,e+1} equals 0, the same is true for \overline{ξ}_{d,e+1}. Hence the image of \overline{ξ}_{d,e+1} is contained in the locus of \overline{M}_{0,1}(X, e + 1) that is a smooth scheme. Because ξ̃_{d−1,e+1} can be chosen to contain a very general point of \overline{M}_{0,1}(X, e + 1) (because \overline{ξ}_{d−1,e+1} is typical), the same is true of \overline{ξ}_{d,e+1}. By [7, Prop. 4.14], the pullback \overline{ξ}_{d,e+1}^{∗}T_{\overline{M}_{0,1}(X,e+1)} is generated by global sections. Therefore, after deforming further, \overline{ξ}_{d,e+1} can be chosen to be typical. Because ξ̃_{d−1,e+1} is very positive, by Lemma 4.13, \overline{ξ}_{d,e+1} can be chosen to be very positive. Also, by the proof of Proposition 4.8, \overline{ξ}_{d,e+1} can be chosen so that there exists a deformation ξ_{d,1} of ξ̃_{d−1,1} that is very twisting and such that h_{d,1} = \overline{h}_{d,e+1}.

RATIONAL CURVES ON HYPERSURFACES II 43

Of course for any divisor in \overline{M}_{0,1}(X, e + 1) (resp. \overline{M}_{0,1}(X, 1)), the intersection number with \overline{ξ}_{d,e+1} (resp. ξ_{d,1}) equals the intersection number with ξ̃_{d−1,e+1} (resp. ξ̃_{d−1,1}), and these are computed above. This finishes the proof of the induction step. Hence, by induction on d, for each d = 1, …, \overline{δ}, there exists an inducting pair (ξ_{d,1}, \overline{ξ}_{d,e+1}) as claimed. □

6. Twistable lines on hypersurfaces

In this section it is proved that if n + 1 ≥ d^{2}, if X ⊂ ℙ^{n} is a general hypersurface of degree d, and if L ⊂ X is a general line on X, then L is twistable.

Remark 6.1. There is one exceptional case, namely d = 1. For n = 1, 2, there is no twistable line on a hyperplane in ℙ^{n}, i.e., there is no twistable line on ℙ^{n−1}. For n ≥ 3 there is a twistable line on ℙ^{n−1}. In this section it is proved there is a twistable line on ℙ^{n−1} if n ≥ 4. Let ((π : Σ → B, σ), g) be a twisting family in ℙ^{n} such that h : B → ℙ^{n} is a line, and let p ∈ ℙ^{n} − g(Σ). Denote by proj_{p} : ℙ^{n} −−→ ℙ^{n−1} projection from p. Then ((π : Σ → B, σ), proj_{p} ∘ g) is a twisting family in ℙ^{n−1} and proj_{p} ∘ h : B → ℙ^{n−1} is a line. Therefore there is also a twistable line on ℙ^{2}.

Notation 6.2. Denote N_{d} = \binom{n+d}{n} − 1 and denote by ℙ^{N_{d}} the projective space parametrizing hypersurfaces X ⊂ ℙ^{n} of degree d. Denote by X ⊂ ℙ^{N_{d}} × ℙ^{n} the universal family of degree d hypersurfaces in ℙ^{n}. Denote by 𝔾(1, n) the Grassmannian variety of lines in ℙ^{n}. Denote by F (X) ⊂ ℙ^{N_{d}} × 𝔾(1, n) the parameter space of pairs ([X], [L]) consisting of a hypersurface of degree d, X ⊂ ℙ^{n}, and a line L ⊂ X. Denote by P (t) = (t + 1)^{2} the Hilbert polynomial of a quadric surface in ℙ^{3}. Denote by U ⊂ Hilb_{ℙ^{n}}^{P (t)} the open subscheme of the Hilbert scheme that parametrizes subschemes of ℙ^{n} projectively equivalent to a smooth quadric surface in ℙ^{3} ⊂ ℙ^{n}. Denote by V ⊂ U × 𝔾(1, n) the parameter space of pairs ([Σ], [L]) consisting of a smooth quadric surface, Σ, and a line L ⊂ Σ. Denote by V → Ũ → U the Stein factorization of the projection pr_{U} : V → U. Denote by W ⊂ ℙ^{N_{d}} × U × 𝔾(1, n) the parameter space of triples ([X], [Σ], [L]) consisting of a hypersurface of degree d, X ⊂ ℙ^{n}, a smooth quadric surface Σ ⊂ X, and a line L ⊂ Σ.

Observe that the projection F (X) → 𝔾(1, d) is a projective bundle of relative dimension N_{d} − (d + 1). Observe that Ũ → U is a finite, étale morphism of degree 2. Observe that V → Ũ is a ℙ^{1}-bundle. Observe that W → V is a projective space bundle of relative dimension N_{d} − (d + 1)^{2}.

Let ([X], [Σ], [L]) be a triple in W. There is a map (well-defined up to nonzero scalar) ∂_{X} : ℂ^{n+1} → H^{0}(ℙ^{n}, O_{ℙ^{n}}(d − 1)) that evaluates the partial derivatives of a defining equation of X. Compose this map with the restriction map H^{0}(ℙ^{n}, O_{ℙ^{n}}(d − 1)) → H^{0}(Σ, O_{Σ}(d − 1)), and denote the composition by,

∂_{X,Σ} : ℂ^{n+1} → H^{0}(Σ, O_{Σ}(d − 1)).

Denote E = O_{W}^{⊕(n+1)}. Denote by G the unique quotient of H^{0}(ℙ^{n}, O_{ℙ^{n}}(d−1))⊗_{ℂ} O_{U} that is locally free and whose fiber at each point Σ is the quotient,

H^{0}(ℙ^{n}, O_{ℙ^{n}}(d − 1)) → H^{0}(Σ, O_{Σ}(d − 1)).

Denote by F the locally free O_{W} -module,

F = pr_{ℙ^{N_{d}}}^{∗} O_{ℙ^{N_{d}}} (1) ⊗ pr_{U}^{∗}G.

There is a map of O_{W} -modules, ∂ : E → F whose fiber at each point ([X], [Σ], [L]) is the map ∂_{X,Σ}. Denote by W^{o} ⊂ W the open subscheme that is the complement of the support of Coker(∂), i.e., W^{o} is the maximal open subscheme on which ∂ is surjective.

Lemma 6.3. Let ([X], [Σ], [L]) be a point in W^{o}. Then, (i) X is smooth along Σ,

44 J. HARRIS AND J. STARR

(ii) h^{i}(Σ, N_{Σ/ℙ^{n}}) = h^{i}(Σ, N_{Σ/ℙ^{n}}(−L)) = h^{i}(Σ, N_{Σ/ℙ^{n}}(−1)) = 0, for i > 0, (iii) h^{i}(Σ, N_{Σ/X}) = h^{i}(Σ, N_{Σ/X}(−L)) = h^{i}(Σ, N_{Σ/X} ⊗ O_{Σ}(−1)) = 0, for i > 0, (iv) h^{1}(L, N_{L/X}(−1)) = h^{1}(L, N_{L/X}) = 0,

(v) the projection morphism pr_{ℙ^{N_{d}}} : W → ℙ^{N_{d}} is smooth at ([X], [Σ], [L]), (vi) the projection morphism pr_{ℙ^{N_{d}}} : F (X) → ℙ^{N_{d}} is smooth at ([X], [L]), and (vii) the projection morphism pr_{F (X)} : W → F (X) is smooth at ([X], [Σ], [L]).

Proof. (i): Since the partial derivatives of a defining equation of X generate H^{0}(Σ, O_{Σ}(d − 1)), the subscheme of X where the partial derivatives all vanish is disjoint from Σ. By the Jacobian criterion, X is smooth at each point of Σ.

(ii): Denote by ℙ^{3} ⊂ ℙ^{n} the span of Σ. There is a short exact sequence,

0 ⟶ N_{Σ/ℙ^{3}} ⟶ N_{Σ/ℙ^{n}} ⟶ N_{ℙ^{3}/ℙ^{n}}|_{Σ} ⟶ 0.

Since N_{Σ/ℙ^{3}} ≅ O_{Σ}(2) and since N_{ℙ^{3}/ℙ^{n}}|_{Σ} ≅ O_{Σ}(1)^{⊕(n−3)}, the short exact sequence above is,

0 ⟶ O_{Σ}(2) ⟶ N_{Σ/ℙ^{n}} ⟶ O_{Σ}(1)^{⊕(n−3)} ⟶ 0.

From this it is easy to compute that,

h^{i}(Σ, N_{Σ/ℙ^{n}}) = h^{i}(Σ, N_{Σ/ℙ^{n}} ⊗ O_{Σ}(−1)) = h^{i}(Σ, N_{Σ/ℙ^{n}} ⊗ O_{Σ}(−L)) = 0,

for i > 0.

(iii): There is a short exact sequence,

0 ⟶ N_{Σ/X} ⟶ N_{Σ/ℙ^{n}} ⟶ N_{X/ℙ^{n}}|_{Σ} ⟶ 0.

Of course N_{X/ℙ^{n}}|_{Σ} ≅ O_{Σ}(d). For the three cases L = O_{Σ}, L = O_{Σ}(−L), and L = O_{Σ}(−1), h^{i}(Σ, O_{Σ}(d) ⊗ L) = 0 for i > 0. By (ii), h^{i}(Σ, N_{Σ/ℙ^{n}} ⊗ L) = 0 for i > 0. By the long exact sequence in cohomology associated to the twist by L of the short exact sequence above, h^{i}(Σ, N_{Σ/X} ⊗ L) = 0 for i > 2. Also h^{1}(Σ, N_{Σ/X} ⊗ L) = 0 iff the map e_{L} : H^{0}(Σ, N_{Σ/ℙ^{n}} ⊗ L) → H^{0}(Σ, N_{X/ℙ^{n}}|_{Σ}) is surjective.

The map e_{O_{Σ}} factors the map ∂_{X,Σ}. Since ∂_{X,Σ} is surjective, also e_{O_{Σ}} is surjective and h^{1}(Σ, N_{Σ/X}) = 0.

For the case L = O_{Σ}(−L), observe that O_{Σ}(1) ≅ O_{Σ}(L + L′) where L′ ⊂ Σ is a line of the ruling opposite to L. There is a commutative diagram,

H^{0}(Σ, N_{Σ/ℙ^{n}}(−1)) ⊗_{ℂ} H^{0}(Σ, O_{Σ}(L′)) ⟶ H^{0}(Σ, N_{Σ/ℙ^{n}}(−L))
↓  ↓
H^{0}(Σ, N_{X/ℙ^{n}}|_{Σ}(−1)) ⊗_{ℂ} H^{0}(Σ, O_{Σ}(L′)) ⟶ H^{0}(Σ, N_{X/ℙ^{n}}|_{Σ}(−L))

The left vertical arrow is surjective by (ii). The bottom arrow is

H^{0}(Σ, O_{Σ}((d − 1)L + (d − 1)L′)) ⊗ H^{0}(Σ, O_{Σ}(L′)) → H^{0}(Σ, O_{Σ}((d − 1)L + dL′)),

which is surjective. Therefore also the right vertical arrow is also surjective, i.e., h^{1}(Σ, N_{Σ/X}(−L)) = 0. The proof that h^{1}(Σ, N_{Σ/X}) = 0 is almost identical to the proof that h^{1}(Σ, N_{Σ/X}(−L) = 0.

(iv) There is a short exact sequence,

0 ⟶ N_{Σ/X}(−1) ⟶ N_{Σ/X}(−L′) ⟶ N_{Σ/X}|_{L}(−1) ⟶ 0.

RATIONAL CURVES ON HYPERSURFACES II 45

By the associated long exact sequence in cohomology and the computations above, h^{1} (L, N_{Σ/X}|_{L}(−1)) = 0. There is a short exact sequence,

0 ⟶ N_{L/Σ}(−1) ⟶ N_{L/X}(−1) ⟶ N_{Σ/X}|_{L}(−1) ⟶ 0.

Of course N_{L/Σ} ≅ O_{L}(1), so h^{1}(L, N_{L/Σ}(−1)) = 0. As proved above, h^{1} (L, N_{Σ/X}|_{L}(−1)) = 0. Therefore by the long exact sequence in cohomology, h^{1}(L, N_{L/X}(−1)) = 0. Since N_{L/X} is a locally free O_{L}-module and h^{1}(L, N_{L/X}(−1)) = 0, by Grothendieck’s lemma N_{L/X} is generated by global sections. In particular, also h^{1}(L, N_{L/X}) = 0.

(v): By [16, Prop. I.2.14.2], the obstruction space for the relative Hilbert scheme, Hilb_{X /ℙ^{N_{d}}}^{P (t)} , at the point ([X], [Σ]) is contained in H^{1}(Σ, N_{Σ/X}), which has dimension 0 by (iv). Since the obstruction space vanishes, it follows by [16, Thm. 2.10] that Hilb_{X /ℙ^{N_{d}}}^{P (t)} → ℙ^{N_{d}} is smooth at

([X], [Σ]). Also W → Hilb_{X /ℙ^{N_{d}}}^{P (t)} is smooth. Therefore the composition pr_{ℙ^{N_{d}}} : W → ℙ^{N_{d}} is smooth at ([X], [Σ], [L]).

(vi): The proof is almost identical to the proof of (v).

(vii): By (v), W → ℙ^{N_{d}} is smooth at ([X], [Σ], [L]). By (vi), F (X) → ℙ^{N_{d}} is smooth at ([X], [L]). So the Jacobian criterion for the smoothness of pr_{F (X)} at ([X], [Σ], [L]) is that the map of vertical tangent bundles, dpr_{F (X)} : T_{W/ℙ^{N_{d}}} → π^{∗}T_{F (X)/ℙ^{N_{d}}} is surjective at ([X], [Σ], [L]). This reduces to the surjectivity of H^{0} (Σ, N_{Σ/X}) → H^{0} (L, N_{Σ/X}|_{L}). The cokernel is contained in H^{1}(Σ, N_{Σ/X}(−L)), which is zero by (iii). Therefore pr_{F (X)} is smooth at ([X], [Σ], [L]). □

We associate to each ([X], [Σ], [L]) ∈ W^{o} a morphism ζ : L → \overline{M}_{0,1}(X, 1) as follows. Let σ : L → Σ be the inclusion and let pr_{L} : Σ → L be the unique projection such that σ is a section of pr_{L}. Let g : Σ → X be the inclusion. Then,

ζ = ((pr_{L} : Σ → L, σ), g : Σ → X),

is a family of stable maps, i.e., a morphism ζ : L → \overline{M}_{0,1}(X, 1).

Lemma 6.4. For every point ([X], [Σ], [L]) ∈ W^{o}, the morphism ζ : L → \overline{M}_{0,1}(X, 1) is twisting.

Proof. Since g ∘ σ : L → X is an embedding, Axiom (i) of Definition 4.3 is satisfied. By Lemma 6.3 (vi), the dimension of the obstruction group of \overline{M}_{0,0}(X, 1) at [g ∘ σ : L → X] is 0, i.e., Axiom (ii) of Definition 4.3 is satisfied.

Denote by N the normal bundle of the regular embedding (pr_{L}, g) : Σ → L × X. There is a short exact sequence,

0 ⟶ pr_{L}^{∗}T_{L} ⟶ N ⟶ N_{Σ/X} ⟶ 0.

By Remark 4.4 (ii), Axiom (iii) holds if R^{1} (pr_{L})_{∗} N (−σ) = {0}. For each fiber L′ of pr_{L} : Σ → L, N (−σ)|_{L′} ≅ N_{L′/X}(−1). Since ([X], [Σ], [L′]) is also in W^{o}, by Lemma 6.3 (iv), h^{1}(L′, N_{L′/X}(−1)) = 0. Therefore R^{1}(pr_{L})_{∗}N (−σ) = {0}, i.e., Axiom (iii) holds.

By Remark 4.4 (ii), ζ^{∗}T_{ev} ≅ (pr_{L})_{∗}N (−σ). Part of the long exact sequence of higher direct images associated to the twist by O_{Σ}(−L) of the short exact sequence above is,

(pr_{L})_{∗}pr_{L}^{∗}T_{L}(−σ(L)) → (pr_{L})_{∗}N (−σ(L)) → (pr_{L})_{∗}N_{Σ/X}(−σ(L)) → R^{1}(pr_{L})_{∗}pr_{L}^{∗}T_{L}(−σ(L)) → 0.

For each fiber L′ of pr_{L}, T_{L}(−σ(L))|_{L′} ≅ O_{L′}(−1). Therefore (pr_{L})_{∗}pr_{L}^{∗}T_{L}(−σ(L)) = {0} and R^{1}(pr_{L})_{∗}pr_{L}^{∗}T_{L}(−σ(L)) = {0}, i.e., (pr_{L})_{∗}N (−σ(L)) ≅ (pr_{L})_{∗}N_{Σ/X}(−σ(L)).

Let p′ ∈ L be a closed point and denote L′ = pr_{L}^{−1} {p′}. Since R^{1} (pr_{L})_{∗} pr_{L}^{∗}T_{L}(−L − L′) = {0} and R^{1} (pr_{L})_{∗} N (−L − L′) = {0}, by the long exact sequence of higher direct images, also R^{1} (pr_{L})_{∗} N_{Σ/X}(−L − L′) = {0}. By the Leray spectral sequence, h^{1} (Σ, N_{Σ/X}(−L − L′)) = h^{1} (L, (pr_{L})_{∗} (N_{Σ/X}(−σ(L))) (−p′)). By Lemma 6.3 (iii), h^{1}(Σ, N_{Σ/X}(−L − L′)) = 0, so also h^{1}(L, (pr_{L})_{∗}(N_{Σ/X}(−σ(L)))(−p′)) = 0. By Grothendieck’s lemma, (pr_{L})_{∗}N_{Σ/X}(−σ(L)) is a direct sum of line bundles of degree ≥ 0, i.e., it is generated by global sections. Therefore Axiom (iv) is satisfied. Finally, σ^{∗}O_{Σ}(σ(L)) ≅ O_{L}, so Axiom (v) is satisfied. □

Proposition 6.5. If either, (i) d = 1 and n ≥ 4, or

(ii) d ≥ 2 and n + 1 ≥ d^{2},

then pr_{F (X)} : W^{o} → F (X) is dominant. Therefore, for a general pair ([X], [L]) ∈ F (X), L is a twistable line on X.

Proof. By Lemma 6.3 (v), it suffices to prove that W^{o} is nonempty.

(i): If d = 1 and n ≥ 4, then for any quadric surface Σ ⊂ ℙ^{n} and any hyperplane X ⊂ ℙ^{n} containing span(Σ), ∂ is an isomorphism so ([X], [Σ], [L]) is in W^{o} for any line L ⊂ Σ.

(ii): Next suppose that d ≥ 2. Denote by I_{d} the set,

I_{d} = {(i, j) ∈ ℤ × ℤ : 0 ≤ i, j ≤ d − 1, i + j ≥ 3},

which has d^{2} − 4 elements. Denote by,

(Y_{0}, Y_{1}, Y_{2}, Y_{3}) ∪ (X_{i,j})_{(i,j)∈I_{d}} ∪ (Z_{m} : m = 1, …, n + 1 − d^{2}),

a basis of H^{0}(ℙ^{n}, O_{ℙ^{n}}(1)), i.e., a basis of homogeneous coordinates on ℙ^{n}. Denote by Σ ⊂ ℙ^{n} the smooth quadric surface with ideal,

I_{Σ} = ⟨Y_{0}Y_{3} − Y_{1}Y_{2}⟩ + ⟨X_{i,j}|(i, j) ∈ I_{d}⟩ + ⟨Z_{m}|m = 1, …, n + 1 − d^{2}⟩.

This is the image of the closed immersion f : ℙ^{1} × ℙ^{1} → ℙ^{n},

([U_{0} : U_{1}], [V_{0} : V_{1}]) ↦ [U_{0}V_{0} : U_{0}V_{1} : U_{1}V_{0} : U_{1}V_{1} : 0 : ⋯: 0].

For each (i, j) ∈ I_{d}, denote k = min(i, j), denote i′ = i − k, and denote j′ = j − k. Denote by X ⊂ ℙ^{n} the hypersurface with defining equation,

F = (Y_{0}Y_{3} − Y_{1}Y_{2}) Y_{3}^{d−2} + ∑_{(i,j)∈I_{d}} Y_{0}^{k}Y_{1}^{i′}Y_{2}^{j′}Y_{3}^{d+k−i−j−1}X_{i,j}.

Clearly Σ ⊂ X. The claim is that ∂F : ℂ^{n+1} → H^{0} (Σ, O_{Σ}(d − 1)) is surjective. By construction,

\frac{∂F}{∂Y_{0}} ↦ U_{1}^{d−1}V_{1}^{d−1}, \frac{∂F}{∂Y_{1}} ↦ U_{1}^{d−1}V_{0}V_{1}^{d−2}, \frac{∂F}{∂Y_{2}} ↦ U_{0}U_{1}^{d−2}V_{1}^{d−1}, \frac{∂F}{∂Y_{3}} ↦ U_{0}U_{1}^{d−2}V_{0}V_{1}^{d−2}.

For each (i, j) ∈ I_{d},

\frac{∂F}{∂X_{i,j}} ↦ U_{0}^{i}U_{1}^{d−1−i}V_{0}^{j}V_{1}^{d−1−j}.  (12)

Since the partial derivatives of the form \frac{∂F}{∂Y_{i}} give the terms U_{0}^{i}U_{1}^{d−1−i}V_{0}^{j}V_{1}^{d−1−j} with (i, j) = (0, 0), (0, 1), (1, 0), and (1, 1), and since these are precisely the pairs (i, j) not contained in I_{d}, ∂F is surjective. Therefore, for every line L ⊂ Σ, ([X], [Σ], [L]) is in W^{o}. □

Together with Remark 6.1, Lemma 6.4 and Proposition 6.5 imply the following corollary.

RATIONAL CURVES ON HYPERSURFACES II 47

Corollary 6.6. If X ⊂ ℙ^{n} is a general hypersurface of degree d and either, (i) d = 1 and n ≥ 3, or

(ii) d ≥ 2 and n + 1 ≥ d^{2},

then Hypothesis 4.10 holds.

7. Base case of the induction for hypersurfaces

In this section it is proved that if n ≥ d^{2} + d + 1 and if X ⊂ ℙ^{n} is a general hypersurface of degree d, then there exists a morphism \overline{ζ}_{1} : ℙ^{1} → \overline{M}_{0,1}(X, 1) that is both very twisting and very positive. This provides the base case for the induction argument of Section 5.

Remark 7.1. There is one exceptional case, d = 1. It will be proved that for n ≥ 7, there is a morphism \overline{ζ}_{1} : ℙ^{1} → \overline{M}_{0,1}(ℙ^{n−1}, 1) that is both very twisting and very positive. As in Remark 6.1, repeatedly projecting from a point produces a morphism \overline{ζ}_{1} : ℙ^{1} → \overline{M}_{0,1}(ℙ^{n−1}, 1) that is very twisting and very positive for all n ≥ 3.

The techniques in this section are the same as those of Section 6. Proposition 6.5 is proved by finding a single degree d polynomial F on ℙ^{n}, vanishing on some quadric surface Σ, and such that

∂_{F,Σ} : ℂ^{n+1} → H^{0} (Σ, O_{Σ}(d − 1))

is surjective. In this section, the role of L ⊂ X is replaced by a rational normal curve C_{0} ⊂ X of degree k ≤ n (in the end, only the case k = 2d will be needed). The role of the quadric surface is replaced by a rational normal scroll Σ of degree 2k − 1 such that C_{0} ⊂ Σ ⊂ X. The cohomology vanishing results of the last section are replaced by the vanishing of h^{i}(Σ, N_{Σ/X}(−C_{0} − 2L)) for i > 0, where L is a line of the ruling of Σ. The computation in this section will be to find a single degree d polynomial F on ℙ^{n}, vanishing on Σ, and such that the image, W, of the map,

∂_{F,Σ} : ℂ^{n+1} → H^{0}(Σ, O_{Σ}(d − 1)),

has the property that the induced map,

W ⊗ H^{0}(Σ, O_{Σ}((k − 3)L)) → H^{0}(Σ, O_{Σ}(d − 1) ⊗ O_{Σ}((k − 3)L)),

is surjective. A similar polynomial F to that of the last section satisfies this condition.

7.1. Generating linear systems on 𝔽_{1}. In the last section, the relevant surface was the Hirzebruch surface 𝔽_{0} = ℙ^{1} × ℙ^{1} embedded as a quadric surface. In this section, the relevant surface is the Hirzebruch surface 𝔽_{1} embedded as a rational normal scroll of degree 2k − 1. The projective model of 𝔽_{1} used here is,

𝔽_{1} = {([T_{0} : T_{1}], [T_{0}U : T_{1}U : V]) ∈ ℙ^{1} × ℙ^{2}|T_{0}(T_{1}U) = T_{1}(T_{0}U)} .

In the equation above, “T_{0}U” and “T_{1}U” are just names of homogeneous coordinates on ℙ^{2} (although the term U does have a meaning described below). Denote by pr_{ℙ^{1}} : 𝔽_{1} → ℙ^{1} and pr_{ℙ^{2}} : 𝔽_{1} → ℙ^{2} the projection morphisms. Denote by O_{𝔽_{1}}(F) the invertible sheaf pr_{ℙ^{1}}^{∗}O_{ℙ^{1}} and by O_{𝔽_{1}}(E + F) the invertible sheaf pr_{ℙ^{2}}^{∗}O_{ℙ^{2}}. The invertible sheaf O_{𝔽_{1}}(E) is associated to the directrix E ⊂ 𝔽_{1}. (This explains the terminology T_{0}U and T_{1}U; U is a nonzero element of H^{0} (𝔽_{1}, O_{𝔽_{1}}(E)), and T_{0}U and T_{1}U are the products of U with the two global sections T_{0} and T_{1} of H^{0} (𝔽_{1}, O_{𝔽_{1}}(F)).)

The invertible sheaves O_{𝔽_{1}}(E + F) and O_{𝔽_{1}}(F) generate the Picard group of 𝔽_{1}; thus motivating the notation,

O(a, b) := O_{𝔽_{1}} (a(E + F) + bF) .

The divisors E + F and F are each nef, but not ample. Therefore they generate the nef cone; an invertible sheaf O(a, b) is nef iff 0 ≤ a, b, and it is ample iff 0 < a, b.

Let O(a, b) be a nef invertible sheaf, and let W ⊂ H^{0} (𝔽_{1}, O(a, b)) be a linear system. Let c ≥ 0 be an integer.

Definition 7.2. The linear system W is a c-generating linear system if the associated map,

μ_{W,c} : W ⊗ H^{0} (𝔽_{1}, O(0, c)) → H^{0} (𝔽_{1}, O(a, b + c)) ,

is surjective.

When is W a c-generating linear system?

Notation 7.3. For each integer i, denote r(i) = ⌊\frac{b+i−1}{c+1}⌋. Denote by β_{d}, β_{r}, α_{d} and α_{r} the unique integers such that 0 ≤ α_{r}, β_{r} < c + 1 and b − 1 = β_{d}(c + 1) + β_{r}, a + b − 1 = α_{d}(c + 1) + α_{r}. Denote by N(a, b, c) the integer,

N(a, b, c) = ∑_{i=0}^{a}(r(i)+2) = 2a+2+\{ \begin{matrix}β_{d}(c + 1 − β_{r}) + α_{d}(1 + α_{r}) + (α_{d} − β_{d} − 1)(c + 1) & α_{d} > β_{d}, \\ β_{d}(α_{r} − β_{r} + 1) & α_{d} = β_{d}. \end{matrix}

Denote by W_{0}(a, b, c) ⊂ H^{0}(𝔽_{1}, O(a, b)) the linear system,

W_{0}(a, b, c) = span{U^{i}V^{a−i}T_{0}^{(b+i)−j(c+1)}T_{1}^{j(c+1)}|i = 0, …, a, j = 0, …, r(i)}
+span {U^{i}V^{a−i}T_{1}^{b+i}| i = 0, …, a}.

Lemma 7.4. The linear system W_{0}(a, b, c) is a c-generating linear system of dimension N(a, b, c).

Proof. For each pair of nonnegative integers a′, b′ there is a decreasing filtration on H^{0}(𝔽_{1}, O(a′, b′)),

F^{i}H^{0}(𝔽_{1}, O(a′, b′)) = H^{0}(𝔽_{1}, O(a′, b′)(−iE)) ≅ H^{0}(𝔽_{1}, O(a′ − i, b′)).

For any linear system W ⊂ H^{0}(𝔽_{1}, O(a, b)), there is an induced filtration F^{i}W = F^{i} ∩ W. The multiplication map μ_{W,c} respects the filtrations on W and on H^{0}(𝔽_{1}, O(a, b + c)). Hence μ_{W,c} is surjective if every associated graded map,

gr^{i}μ_{W,c} : gr^{i}W ⊗ H^{0}(𝔽_{1}, O(0, c)) → gr^{i}H^{0}(𝔽_{1}, O(a, b + c)),

is surjective. Now dim(W) is the sum of all terms dim(gr^{i}W). For each i, what is the minimum possible dimension of a vector subspace W^{i} ⊂ gr^{i}H^{0}(𝔽_{1}, O(a, b)) such that the associated map,

gr^{i}μ_{W^{i},c} : W^{i} ⊗ H^{0}(𝔽_{1}, O(0, c)) → gr^{i}H^{0}(𝔽_{1}, O(a, b + c)),

is surjective?

The associated graded pieces of O(a′, b′) are,

gr^{i}H^{0}(𝔽_{1}, O(a′, b′)) ≅ \{ \begin{matrix}H^{0}(E, O_{E}(b′ + i)), & 0 ≤ i ≤ a′ \\ \{0\}, & i > a′ \end{matrix} .

Let W^{i} ⊂ H^{0} (E, O_{E}(b + i)) be a linear system such that the multiplication map,

gr^{i}μ_{W^{i},c} : W^{i} ⊗ H^{0}(E, O_{E}(c)) → H^{0}(E, O_{E}(b + c + i)),

is surjective. Counting dimensions on the left and right side of the equation, dim(W^{i}) · (c + 1) ≥ (b + c + i + 1), i.e.,

dim(W^{i}) ≥ ⌊\frac{b + i − 1}{c + 1}⌋ + 2 = r(i) + 2.

The linear system W^{i} = gr^{i}W_{0}(a, b, c), which is generated by the set of monomials

{U^{i}V^{a−i}T_{0}^{(b+i)−j(c+1)}T_{1}^{j(c+1)}|j = 0, …, r(i)} ∪ {U^{i}V^{a−i}T_{1}^{b+i}},

has the property that W^{i} ⊗ H^{0}(E, O_{E}(c)) → H^{0}(E, O_{E}(b + c + i)) is surjective. And dim(W^{i}) = r(i) + 2. So W_{0}(a, b, c) is a c-generating linear system of dimension,

2a + 2 + ∑_{i=0}^{a} ⌊\frac{b + i − 1}{c + 1}⌋ = N(a, b, c).

Moreover, this is the minimum dimension among c-generating linear systems for which each map gr^{i}μ_{W^{i},c} is surjective. □

Notation 7.5. Denote by S the Cox homogeneous coordinate ring of 𝔽_{1}, i.e.,

S := ⊕_{(a,b)∈ℤ^{2}}H^{0} (𝔽_{1}, O(a, b)) = ℂ[T_{0}, T_{1}, U, V].

This is a ℤ^{2}-graded ring, where deg(T_{0}) = deg(T_{1}) = (0, 1), deg(V) = (1, 0) and deg(U) = (1, −1). For every (a, b) ∈ ℤ^{2}, S_{(a,b)} = H^{0} (𝔽_{1}, O(a, b)). Denote by deg : S → ℤ ∪ {−∞} the total degree defined by deg(M) = a + b for all elements M ∈ S_{(a,b)}. Denote by ≺ the graded lexicographical monomial order on S that refines the grading by total degree by U > V > T_{0} > T_{1}. For every linear system W ⊂ H^{0}(𝔽_{1}, O(a, b)) denote by IN(W) the linear system generated by the initial terms of W.

Lemma 7.6. If the linear system IN(W) contains W_{0}(a, b, c), then W is a c-generating linear system.

Proof. The linear system of initial terms of Image(μ_{W,c}) satisfies

IN(W) · S_{(0,c)} ⊂ IN (Image(μ_{W,c})) .

Since IN(W) contains W_{0}(a, b, c),

W_{0}(a, b, c) · S_{(0,c)} ⊂ IN (Image(μ_{W,c})) .

By Lemma 7.4, W_{0}(a, b, c) · S_{(0,c)} = S_{(a,b+c)}. Hence IN (Image(μ_{W,c})) = S_{(a,b+c)}, and therefore Image(μ_{W,c}) = S_{(a,b+c)}. □

Remark 7.7. The most important case is a = d − 1, b = (d − 1)(k − 1) and c = k − 3 for positive integers d ≥ 1 and k ≥ 3 (d will be the degree of the hypersurface X ⊂ ℙ^{n}, and k will be the degree of the curve C_{0} ⊂ X). In particular, if d ≥ 2 and k ≥ 2d, then b − 1 = (d − 1)(k − 2) + d − 2, a + b − 1 = (d − 1)(k − 2) + 2d − 3 and 0 ≤ d − 2, 2d − 3 ≤ k − 3. Hence r(i) = d − 1 for i = 0, …, a and N(a, b, c) = d(d + 1) = d^{2} + d. Moreover, min_{k≥3} N(d − 1, (d − 1)(k − 1), (k − 3)) = d^{2} + d. This is the origin of the term “d^{2} + d” in Theorem 1.1.

7.2. Cohomology Results.

Notation 7.8. Let N_{d}, ℙ^{N_{d}}, and X ⊂ ℙ^{N_{d}} × ℙ^{n} be as in Notation 6.2. Let k be any integer with 3 ≤ k ≤ \frac{n}{2} (only the case k = 2d will be used later). Let R^{k}(ℙ^{n}) ⊂ Hilb_{ℙ^{n}}^{kt+1} denote the open subscheme parametrizing curves C_{0} ⊂ ℙ^{n} that are projectively equivalent to a degree k rational normal curve C_{0} ⊂ ℙ^{k} ⊂ ℙ^{n}. Let R^{k}(X) ⊂ ℙ^{N_{d}} × R^{k}(ℙ^{n}) denote the parameter space for pairs ([X], [C_{0}]) such that C_{0} ⊂ X. Let Q(t) = \frac{1}{2}(t + 1)((2k − 1)t + 2) denote the Hilbert polynomial of a rational normal scroll of degree 2k − 1 in ℙ^{2k}. Let U ⊂ Hilb_{ℙ^{n}}^{Q(t)} denote the open subscheme parametrizing closed subschemes Σ ⊂ ℙ^{n} that are projectively equivalent to a rational normal scroll of degree 2k − 1 in ℙ^{2k} ⊂ ℙ^{n} and that are abstractly isomorphic to 𝔽_{1}. Let V ⊂ U × R^{k}(ℙ^{n}) denote the parameter space of pairs ([Σ], [C_{0}]) such that C_{0} ⊂ Σ and such that, via the isomorphism of Σ ≅ 𝔽_{1}, the invertible sheaf of C_{0} is O(1, 0). Let W ⊂ ℙ^{N_{d}} × U × R^{k}(ℙ^{n}) denote the parameter space for triples ([X], [Σ], [C_{0}]) where ([Σ], [C_{0}]) is in V and where Σ ⊂ X.

50 J. HARRIS AND J. STARR

Observe that R^{k} (ℙ^{n}) is a homogeneous space of PGL_{n+1}, and therefore is smooth and connected. Observe that the projection R^{k}(X) → R^{k}(ℙ^{n}) is a projective bundle of relative dimension N_{d} − (kd + 1). Observe that the projection map V → U factors as an open subset (with nonempty fibers) of a projective bundle over U of relative dimension 2 (more precisely, every fiber is isomorphic to the 𝔸^{2} of irreducible curves in the linear system |O(1, 0)|). Observe that the projection map W → V is a projective bundle of relative dimension N_{d} − Q(d).

For each triple ([X], [Σ], [C_{0}]) ∈ W, define ∂_{X,Σ} : ℂ^{n+1} → H^{0} (Σ, O_{Σ}(d − 1)) as in Section 6. More precisely, denote E = O_{W}^{⊕(n+1)}. Denote by G the unique quotient of H^{0}(ℙ^{n}, O_{ℙ^{n}}(d−1))⊗_{ℂ} O_{U} that is locally free and whose fiber at each point Σ is the quotient,

H^{0}(ℙ^{n}, O_{ℙ^{n}}(d − 1)) → H^{0} (Σ, O_{Σ}(d − 1)) .

Denote by F the locally free O_{W}-modules

pr_{ℙ^{N_{d}}}^{∗} (O_{ℙ^{N_{d}}} (1)) ⊗ pr_{U}^{∗}G.

Then there is a map of O_{W}-modules ∂ : E → F whose fiber at each point ([X], [Σ], [C_{0}]) is the map ∂_{X,Σ}. Denote by W^{o} ⊂ W the open subscheme parametrizing points ([X], [Σ], [C_{0}]) such that,

Image(∂_{X,Σ}) ⊂ H^{0}(Σ, O_{ℙ^{n}}(d − 3)|_{Σ}),

is a (k − 3)-generating linear system.

Let Σ ⊂ ℙ^{2k} ⊂ ℙ^{n} be a rational normal surface scroll of degree 2k − 1, and let f : 𝔽_{1} → Σ be an isomorphism. For each pair of nonnegative integers (a, b), denote by N(a, b) the locally free O_{𝔽_{1}}-module,

N(a, b) = f^{∗} (N_{Σ/ℙ^{n}} ⊗ O_{ℙ^{n}}(−1)|_{Σ}) ⊗ O(a, b),

and denote by N′(a, b) ⊂ N(a, b) the subsheaf,

N′(a, b) = f^{∗} (N_{Σ/ℙ^{2k}} ⊗ O_{ℙ^{n}}(−1)|_{Σ}) ⊗ O(a, b).

Lemma 7.9. (i) N′(0, 0) is generated by global sections and h^{i}(𝔽_{1}, N′(0, 0)) = 0 for i > 0, (ii) N(0, 0) is generated by global sections and h^{i}(𝔽_{1}, N(0, 0)) = 0 for i > 0, (iii) for every pair of nonnegative integers (a, b) and for every coherent sheaf F on 𝔽_{1} that is

generated by global sections and such that h^{i}(𝔽_{1}, F) = 0 for i > 0, F(a, b) := F ⊗ O(a, b) is generated by global sections and h^{i}(𝔽_{1}, F(a, b)) = 0 for i > 0.

In particular, for every pair of nonnegative integers (a, b), N(a, b) (resp. N′(a, b)) is generated by global sections and h^{i}(𝔽_{1}, N(a, b)) = 0 for i > 0 (resp. h^{i}(𝔽_{1}, N′(a, b)) = 0 for i > 0).

Proof. (i): The morphism pr_{ℙ^{1}} : 𝔽_{1} → ℙ^{1} is isomorphic over ℙ^{1} to projection from the projective bundle,

𝔽_{1} ≅ ℙ (O_{ℙ^{1}}(−(k − 1)) ⊕ O_{ℙ^{1}}(−k)) .

Under this isomorphism the invertible sheaf O(1, k−1) on Σ corresponds to the invertible sheaf O(1) on ℙ(O_{ℙ^{1}}(−(k − 1)) ⊕ O_{ℙ^{1}}(−k)) where O(1) is the universal invertible quotient of pr_{ℙ^{1}}^{∗}(O_{ℙ^{1}}(k − 1) ⊕ O_{ℙ^{1}}(k)). Up to projective equivalence, the morphism f : 𝔽_{1} → ℙ^{2k} is the closed immersion given by the complete linear system of O(1); in particular, f^{∗}O_{ℙ^{2k}}(1) ≅ O(1). Using this isomorphism, there is a short exact sequence of O_{𝔽_{1}}-modules,

0 → pr_{ℙ^{1}}^{∗}T_{ℙ^{1}} → pr_{ℙ^{1}}^{∗}(O_{ℙ^{1}}(1)^{⊕(2k−1)}) ⊗ f^{∗}O_{ℙ^{2k}}(1) → f^{∗}N_{Σ/ℙ^{2k}} → 0.

Twisting by f^{∗}O_{ℙ^{2k}}(−1), N′(0, 0) is a quotient of pr_{ℙ^{1}}^{∗}(O_{ℙ^{1}}(1)^{⊕(2k−1)}). Hence N′(0, 0) is generated by global sections. Also,

(pr_{ℙ^{1}})_{∗}(pr_{ℙ^{1}}^{∗}T_{ℙ^{1}} ⊗ f^{∗}O_{ℙ^{2k}}(−1)) = {0}, R^{1}(pr_{ℙ^{1}})_{∗}(pr_{ℙ^{1}}^{∗}T_{ℙ^{1}} ⊗ f^{∗}O_{ℙ^{2k}}(−1)) = {0}.

RATIONAL CURVES ON HYPERSURFACES II 51

Twisting the short exact sequence by f^{∗}O_{ℙ^{2k}}(−1) and forming the associated long exact sequence of higher direct images, R^{1}(pr_{ℙ^{1}})_{∗}(f^{∗}N_{Σ/ℙ^{2k}}(−1)) = {0}, and (pr_{ℙ^{1}})_{∗}(f^{∗}N_{Σ/ℙ^{2k}}(−1)) ≅ O_{ℙ^{1}}(1)^{⊕(2k−1)}. Computing the cohomology of N′(0, 0) via the Leray spectral sequence associated to pr_{ℙ^{1}} : 𝔽_{1} → ℙ^{1}, h^{i}(𝔽_{1}, N′(0, 0)) = 0 for i > 0.

(ii): There is a short exact sequence,

0 ⟶ N′(0, 0) ⟶ N(0, 0) ⟶ O(0, 0)^{⊕(n−2k)} ⟶ 0.

By (i), h^{i}(𝔽_{1}, O_{𝔽_{1}}) = 0 for i > 0. Therefore N(0, 0) is generated by global sections and h^{i}(𝔽_{1}, N(0, 0)) = 0 for i > 0.

(iii): Let F be a coherent sheaf on 𝔽_{1} such that F is generated by global sections and such that h^{i}(𝔽_{1}, F) = 0 for i > 0. It will be proved by double induction on (a, b) that for every pair of nonnegative integers (a, b), F(a, b) is generated by global sections and h^{i}(𝔽_{1}, F(a, b)) = 0 for i > 0.

The base case is b = 0 and is established by induction on a. For a = 0, the result follows by hypothesis. Let a > 0 and, by way of induction, suppose the result is proved for a − 1. Let D ⊂ 𝔽_{1} be a general member of the linear system |O(1, 0)|. Then D is a smooth curve isomorphic to ℙ^{1}. Since D is general, there is a short exact sequence,

0 ⟶ F(a − 1, 0) ⟶ F(a, 0) ⟶ F(a, 0)|_{D} ⟶ 0.

The sheaf F|_{D} is generated by global sections, and O_{𝔽_{1}}(a(e + f))|_{D} ≅ O_{ℙ^{1}}(a). Hence also F(a, 0)|_{D} is generated by global sections. By the induction assumption, h^{1}(𝔽_{1}, F(a − 1, 0)) = 0. By the long exact sequence of cohomology associated to the short exact sequence, every global section of F(a, 0)|_{D} is the image of a global section of F(a, 0). Hence F(a, 0) is generated by global sections. A coherent sheaf on ℙ^{1} that is generated by global sections has no higher cohomology. Combined with the induction assumption and the long exact sequence in cohomology associated to the short exact sequence above, h^{i}(𝔽_{1}, F(a, 0)) = 0 for i > 0. Therefore, for every a > 0, F(a, 0) is generated by global sections and h^{i}(𝔽_{1}, F(a, 0)) = 0 for i > 0.

Suppose that b > 0 and suppose the result is proved for b − 1. Let L ⊂ 𝔽_{1} be a general fiber of pr_{1}. Then L is smooth and isomorphic to ℙ^{1}. Since L is general, there is a short exact sequence,

0 ⟶ F(a, b − 1) ⟶ F(a, b) ⟶ F(a, b)|_{L} ⟶ 0.

Via the isomorphism L ≅ ℙ^{1}, O_{𝔽_{1}}(a(e + f) + bf)|_{L} ≅ O_{ℙ^{1}}(a). By almost identical arguments to those above, F(a, b) is generated by global sections and h^{i}(𝔽_{1}, F(a, b)) = 0 for i > 0. □

Let ([X], [Σ], [C_{0}]) be a point in W^{o} and let f : 𝔽_{1} → Σ be an isomorphism. For each pair of nonnegative integers (a, b), denote N_{X}(a, b) = f^{∗}(N_{Σ/X} ⊗ O_{ℙ^{n}}(−1)|_{Σ}) ⊗ O(a, b).

Lemma 7.10. (i) The hypersurface X is smooth along Σ.

(ii) For each pair of nonnegative integers (a, b), h^{i}(𝔽_{1}, N_{X}(a, b + k − 3)) = 0 for i > 0.

(iii) For every line of ruling L ⊂ Σ and every nonnegative integer a, h^{1}(L, N_{L/X}(a − 1)) = 0. (iv) For every nonnegative integer a, h^{1}(C_{0}, N_{C_{0}/X}(a − 2)) = 0.

(v) The projection morphism pr_{ℙ^{N_{d}}} : W → ℙ^{N_{d}} is smooth at ([X], [Σ], [C_{0}]).

(vi) For every line of ruling L ⊂ Σ, the projection morphism pr_{ℙ^{N_{d}}} : F (X) → ℙ^{N_{d}} is smooth at

([X], [L]). (vii) The projection morphism pr_{ℙ^{N_{d}}} : R^{k}(X) → ℙ^{N_{d}} is smooth at ([X], [C_{0}]). (viii) The projection morphism π : W → R^{k}(X) is smooth at ([X], [Σ], [C_{0}]).

52 J. HARRIS AND J. STARR

Proof. (i): Since the partial derivatives of a defining equation of X span a c-generating linear series, in particular they generate the sheaf O_{Σ}(d−1). Hence, there is no point of Σ at which all the partial derivatives vanish. By the Jacobian criterion, X is smooth along Σ.

(ii): There is a short exact sequence,

0 ⟶ N_{Σ/X} ⟶ N_{Σ/ℙ^{n}} ⟶ N_{X/ℙ^{n}}|_{Σ} ⟶ 0.

Denote α = a + (d − 1) and β = b + (d − 1)(k − 1) (these are different than α_{d}, α_{r}, β_{d} and β_{r}). There is a short exact sequence,

0 ⟶ N_{X}(a, b) ⟶ N(a, b) ⟶ O(α, β) ⟶ 0.

Since a, b ≥ 0, by Lemma 7.9, h^{i}(𝔽_{1}, N(a, b)) = 0 for i ≥ 0. By direct computation, h^{i}(𝔽_{1}, O(α, β)) = 0 for i ≥ 0. Hence h^{2}(𝔽_{1}, N_{X}(a, b)) = 0, and h^{1}(𝔽_{1}, N_{X}(a, b)) = 0 iff the following map is surjective,

H^{0}(𝔽_{1}, N(a, b)) → H^{0}(𝔽_{1}, O(α, β)).

There is a commutative diagram,

\begin{matrix}H^{0}(𝔽_{1}, N(a, b)) ⊗ H^{0}(𝔽_{1}, O(a′, b′)) & ⟶ & H^{0}(𝔽_{1}, N(a + a′, b + b′)) \\ ↓ & & ↓ \\ H^{0}(𝔽_{1}, O(α, β)) ⊗ H^{0}(𝔽_{1}, O(a′, b′)) & ⟶ & H^{0}(𝔽_{1}, O(α + a′, β + b′)) \end{matrix}.

By direct computation the bottom horizontal arrow is surjective if a′, b′ ≥ 0. Hence, if the left vertical arrow is surjective, then also the right vertical arrow is surjective; i.e., if h^{1}(𝔽_{1}, N_{X}(a, b)) = 0 then also h^{1}(𝔽_{1}, N_{X}(a + a′, b + b′)) = 0. Thus (ii), is reduced to the case a = 0, b = k − 3. In this case the commutative diagram above factors the following commutative diagram,

\begin{matrix}H^{0}(𝔽_{1}, T_{ℙ^{n}}) ⊗ H^{0}(𝔽_{1}, O(0, k − 3)) & ⟶ & H^{0}(𝔽_{1}, N(0, k − 3)) \\ ↓ & & ↓ \\ H^{0}(𝔽_{1}, O(d − 1, (d − 1)(k − 1))) ⊗ H^{0}(𝔽_{1}, O(0, k − 3)) & ⟶ & H^{0}(𝔽_{1}, O(α, β)) \end{matrix}.

By definition, the composition,

H^{0}(𝔽_{1}T_{ℙ^{n}}) ⊗ H^{0}(𝔽_{1}, O(0, k − 3)) → H^{0}(𝔽_{1}, O(α, β)),

is surjective iff the triple ([X], [Σ], [C_{0}]) is in W^{o}. Since ([X], [Σ], [C_{0}]) is in W^{o}, the right vertical arrow is surjective, i.e., h^{1}(𝔽_{1}, N_{X}(0, k − 3)) = 0.

(iii): There is a short exact sequence,

0 ⟶ N_{L/Σ}(a − 1) ⟶ N_{L/X}(a − 1) ⟶ N_{Σ/X}|_{L}(a − 1) ⟶ 0.

Since N_{L/Σ} ≅ O_{L}, for all a ≥ 0, h^{1}(L, N_{L/Σ}(a−1)) = 0. Therefore it suffices to prove h^{1}(L, N_{Σ/X}|_{L}(a− 1)) = 0. Since O(a − 1, b)|_{L} ≅ O_{L}(a − 1), there is a short exact sequence,

0 ⟶ N_{X}(a, k − 3) ⟶ N_{X}(a, k − 2) ⟶ N_{Σ/X}|_{L}(a − 1) ⟶ 0.

By (ii), for a ≥ 0 the higher cohomology of the first two terms vanishes. By the long exact sequence in cohomology associated to this short exact sequence, h^{1}(L, N_{Σ/X}|_{L}(a − 1)) = 0 for a ≥ 0.

(iv): The proof is almost identical to the proof of (iii).

(v): By [16, Prop. 2.14.2], the obstruction space for the relative Hilbert scheme Hilb_{X /ℙ^{N_{d}}}^{Q(t)} at the point ([X], [Σ]) is contained in H^{1}(Σ, N_{Σ/X}). If ([X], [Σ], [C_{0}]) is in W^{o}, then by (ii), h^{1}(Σ, N_{Σ/X}) =

RATIONAL CURVES ON HYPERSURFACES II 53

h^{1}(𝔽_{1}, N_{X}(1, k − 1)) = 0. By [16, Thm. 2.10], Hilb_{X /ℙ^{N_{d}}}^{Q(t)} → ℙ^{N_{d}} is smooth at ([X], [Σ]). The

projection W^{o} → Hilb^{Q(t)}(X /ℙ^{N_{d}}) is an open subset of a projective bundle, and so is smooth. Therefore the composite morphism W^{o} → ℙ^{N_{d}} is smooth at ([X], [Σ], [C_{0}]).

(vi): The proof is very similar to the proof of (v) and uses the vanishing, h^{1}(L, N_{L/X}) = 0, which was proved in (iii).

(vii): The proof is very similar to the proof of (v) and uses the vanishing, h^{1}(C_{0}, N_{C_{0}/X}) = 0, which was proved in (iv).

(viii): Since W^{o} → ℙ^{N_{d}} is smooth at ([X], [Σ], [C_{0}]) and since R^{k} → ℙ^{N_{d}} is smooth at ([X], [C_{0}]), to prove that π : W^{0} → R^{k}(X) is smooth at ([X], [Σ], [C_{0}]), it suffices to check that the derivative map dπ : T_{W^{0}/ℙ^{N_{d}}} → π^{∗}T_{R^{k}(X)/ℙ^{N_{d}}} is surjective at ([X], [Σ], [C_{0}]). This reduces to the statement that H^{0}(Σ, N_{Σ/X}) → H^{0}(C_{0}, N_{Σ/X}|_{C_{0}}) is surjective. The cokernel is contained in H^{1}(𝔽_{1}, N_{X}(0, k − 1)). By (iii), h^{1}(C_{0}, N_{Σ/X}|_{C_{0}}) = 0, therefore the derivative dπ is surjective at ([X], [Σ], [C_{0}]). □

Let ([X], [Σ], [C_{0}]) be a point of W. Denote by σ : C_{0} → Σ the inclusion and denote by pr_{C_{0}} : Σ → C_{0} the unique projection morphism such that σ is a section of pr_{C_{0}} (via the isomorphism Σ ≅ 𝔽_{1}, pr_{C_{0}} corresponds to pr_{ℙ^{1}}). Denote by g : Σ → X the inclusion. There is a family of stable maps ζ : C_{0} → \overline{M}_{0,1}(X, 1),

ζ = ((pr_{C_{0}} : Σ → C_{0}, σ), g : Σ → X).

Lemma 7.11. If ([X], [Σ], [C_{0}]) is in W^{o}, then ζ : C_{0} → \overline{M}_{0,1}(X, 1) is very twisting and very positive.

Proof. Very twisting: First the axioms of Definition 4.3 are verified. Since g ∘ σ : C_{0} → X is an embedding, Axiom (i) of Definition 4.3 is satisfied. By Lemma 7.10 (vii), the dimension of the obstruction group of \overline{M}_{0,0}(X, k) at [g ∘ σ : C_{0} → X] is 0, i.e., Axiom (ii) is satisfied.

The proof that Axiom (iii) holds is identical to the argument for Axiom (iii) in the proof of Lemma 6.4, with Lemma 6.3 (iv) replaced by Lemma 7.10 (iii).

As in the proof of Lemma 6.4, ζ^{∗}T_{ev} ≅ (pr_{C_{0}})_{∗}N_{X}(0, k−1). Hence ζ^{∗}T_{ev} is ample iff h^{1}(C_{0}, ζ^{∗}T_{ev}(−2)) = 0. By a Leray spectral sequence argument similar to the one in the proof of Lemma 6.4, h^{1}(C_{0}, ζ^{∗}T_{ev}(−2)) = h^{1}(𝔽_{1}, N_{X}(0, k − 3)), which, Lemma 7.10 (ii), equals 0. Therefore ζ^{∗}T_{ev} is an ample bundle, i.e., Axiom (iv) is satisfied.

Finally, observe that σ^{∗}O_{Σ}(σ) ≅ O_{C_{0}}(1) is ample, i.e., Axiom (v) is satisfied. Thus ζ is a very twisting family.

Very positive: Next the axioms of Definition 4.11 are verified. Axioms (i), (ii) and (iii) follow from Axioms (i), (ii) and (iii) of Definition 4.3, as proved above. There is a short exact sequence,

0 ⟶ ζ^{∗}T_{ev} ⟶ ζ^{∗}T_{\overline{M}_{0,1}(X,1)} ⟶ (g ∘ σ)^{∗} T_{X} ⟶ 0.

It is proved above that ζ^{∗}T_{ev} is ample. Moreover, by Lemma 7.10 (iv), N_{C_{0}/X} is ample. Of course T_{C_{0}} is ample. Therefore T_{X}|_{C_{0}} is ample by Lemma 2.10 (ii). Since the first and last term in the short exact sequence are ample, by Lemma 2.10 (ii), ζ^{∗}T_{\overline{M}_{0,1}(X,1)} is ample. Since ζ^{∗}pr^{∗}T_{\overline{M}_{0,0}(X,1)} is a quotient of ζ^{∗}T_{\overline{M}_{0,1}(X,1)}, by Lemma 2.10 (i), ζ^{∗}pr^{∗}T_{\overline{M}_{0,0}(X,1)} is ample; i.e., Axiom (iv) is satisfied.

Finally, σ^{∗}O_{Σ}(σ) ≅ O_{C_{0}}(1), which is ample, i.e., Axiom (v) is satisfied. Thus ζ is a very positive family. □

Proposition 7.12. (i) If d ≥ 3 and n ≥ d^{2}+d+1, then for k = 2d, W^{o} → R^{k}(X) is dominant,

and R^{k}(X) → ℙ^{N_{d}} is dominant. (ii) If d = 1 or 2 and if n ≥ 7, then for k = 3, W^{o} → R^{3}(X) is dominant, and R^{3}(X) → ℙ^{N_{d}}

is dominant.

Proof. By item (8) of lemma 7.10, it suffices to prove that W^{o} is nonempty. We have to find a pair ([X], [Σ]) such that for a = d − 1, b = (d − 1)(k − 1) and for c = k − 3, we have that the image of the derivative map

d_{X,Σ} : H^{0} ((ℙ^{n}, T_{ℙ^{n}}(−1)) → H^{0} (𝔽_{1}, O(a, b))  (13)

is a c-generating linear system.

Recall that S = ℂ[T_{0}, T_{1}, U, V] is the ℤ^{2}-graded Cox homogeneous coordinate ring of 𝔽_{1}. Denote by A_{d} the set of d^{2} + d monomials that occur in the linear system W_{0}(a, b, c), i.e.,

\begin{aligned}A_{d} = & \{U^{i}V^{d−1−i}T_{0}^{((d−1)(k−1)+i)−j(k−2)}T_{1}^{j(k−2)}|i = 0, …, d − 1, j = 1, …, r(i)\} \\ & ∪ \{U^{i}V^{d−1−i}T_{1}^{(d−1)(k−1)+i}| i = 0, …, d − 1\} \end{aligned}

where r(i) = d − 1 + ⌊\frac{d−2+i}{k−2}⌋.

(i), d ≥ 4: Suppose that d ≥ 4 and n ≥ d^{2} + d + 1. Denote by B_{d} the set of 4d − 1 monomials,

B_{d} =  { U^{d−1}  T_{0}^{(d−1)k−(k−2)j}  T_{1}^{(k−2)j} |j = 0, …, d − 1 }
∪ { U^{d−2} V  T_{0}^{(d−1)k−1−(k−2)j} T_{1}^{(k−2)j} |j = 0, …, d − 1 }
∪ { U^{d−3} V^{2}  T_{0}^{(d−1)k−2−(k−2)j} T_{1}^{(k−2)j} |j = 1, …, d − 1 }
∪ { U^{d−4} V^{3}  T_{0}^{(d−1)k−3−(k−2)j} T_{1}^{(k−2)j} |j = 1, …, d − 1 }
∪ {  V^{d−1} T_{0}^{(d−1)(k−1)}  }.

Denote by C_{d} the set of d^{2} − 3d + 1 monomials C_{d} = A_{d} − B_{d}. Denote by

{Y_{0}, …, Y_{k}} ∪ {Z_{0}, …, Z_{k−1}} ∪ {X_{M}|M ∈ C_{d}} ∪ {V_{l}|l = 1, …, n − (d^{2} + d + 1)} ,

a basis of H^{0}(ℙ^{n}, O_{ℙ^{n}}(1)), i.e., a basis of homogeneous coordinates on ℙ^{n}.

Denote by f : 𝔽_{1} → ℙ^{2k} ⊂ ℙ^{n} the morphism mapping ([T_{0} : T_{1}], [T_{0}U : T_{1}U : V]) ∈ 𝔽_{1} to the point in ℙ^{n} with coordinates X_{M} = 0, M ∈ C_{d}, with V_{l} = 0, l = 1, …, n − (d^{2} + d + 1), and with

Y_{0} = UT_{0}^{k}, …, Y_{i} = UT_{0}^{k−i}T_{1}^{i}, …, Y_{k} = UT_{1}^{k},
Z_{0} = V T_{0}^{k−1}, …, Z_{j} = V T_{0}^{k−1−j}T_{1}^{j}, …, Z_{k−1} = V T_{1}^{k−1}.

This is an embedding whose image Σ = f(𝔽_{1}) is a rational normal scroll of degree 2k − 1.

The pullback map H^{0} ((ℙ^{2k}, O_{ℙ^{2k}}(1)) → H^{0} (𝔽_{1}, O(1, k − 1)) is surjective by construction. And the natural map,

Sym^{d−1}H^{0}(𝔽_{1}, O(1, k − 1)) → H^{0}(𝔽_{1}, O(d − 1, (d − 1)(k − 1))),

is surjective. Therefore the pullback map

H^{0} (ℙ^{2k}, O_{ℙ^{2k}}(d − 1)) → H^{0} (𝔽_{1}, O (d − 1, (d − 1)(k − 1)))

is surjective. For each monomial M ∈ C_{d}, choose a polynomial G_{M}(Y_{0}, …, Y_{2k}) such that f^{∗}G_{M} = M.

coordinate  IN(∂_{X,Σ}(coordinate))  lower order terms in ∂_{X,Σ}(coordinate)
Z_{j+1},  U^{d−1}T_{0}^{(k−1)d−(k−2)j}T_{1}^{(k−2)j}  −U^{d−1}T_{0}^{(k−1)d−(k−2)(j+1)−1}T_{1}^{(k−2)(j+1)+1}

j = 0, …, d − 1

Z_{d} U^{d−1}T_{0}^{(k−1)d−(k−2)(d−1)}T_{1}^{(k−2)(d−1)} −U^{d−3}V^{2}T_{0}^{(k−1)(d−2)+(d−3)}T_{1}^{k−1}
−UV^{d−2}T_{0}^{(k−1)(d−2)+d}T_{1}^{d}
Y_{j+1}, −U^{d−2}V T_{0}^{(k−1)d−1−(k−2)j}T_{1}^{(k−2)j} +U^{d−2}V T_{0}^{(k−1)d−(k−2)(j+1)−2}T_{1}^{(k−2)(j+1)+1}

j = 0, …, d − 1

Y_{d}  −U^{d−2}V T_{0}^{(k−1)d−1−(k−2)(d−1)}T_{1}^{(k−2)(d−1)} +U^{d−4}V^{2}T_{0}^{(k−1)(d−2)+(d−4)}T_{1}^{k−1}
−V^{d−1}T_{0}^{(k−1)(d−1)−d}T_{1}^{d}
Z_{d+j}, U^{d−3}V^{2}× −U^{d−2}V^{2}×
j = 1, …, d − 2 T_{0}^{(k−1)(d−1)+(d−3)−(k−2)j}T_{1}^{(k−2)j}  T_{0}^{(k−1)(d−1)+(d−4)+(k−2)(j+1)}T_{1}^{(k−2)(j+1)}
Z_{k−1}  U^{d−3}V^{2}×
T_{0}^{(k−1)(d−1)+(d−3)−(k−2)(d−1)}T_{1}^{(k−2)(d−1)}
Y_{d+j}, −U^{d−4}V^{3}× U^{d−4}V^{3}×
j = 1, …, d − 2 T_{0}^{(k−1)(d−1)+(d−4)−(k−2)j}T_{1}^{(k−2)j}  T_{0}^{(k−1)(d−1)+(d−5)+(k−2)(j+1)}T_{1}^{(k−2)(j+1)}
Y_{k−1}  U^{d−4}V^{3}×
T_{0}^{(k−1)(d−1)+(d−4)−(k−2)(d−1)}T_{1}^{(k−2)(d−1)}
Y_{k}  V^{d−1}T_{0}^{(k−1)(d−1)}
X_{M},  M

M ∈ C_{d}

Z_{0}  −U^{d−1}T_{0}^{(k−1)(d−1)+(d−2)}T_{1}  UV^{d−2}T_{0}^{(k−1)(d−2)}T_{1}^{k}
Y_{0}  U^{d−2}V T_{0}^{(k−1)(d−1)+(d−3)}T_{1}
Figure 1. The map ∂_{X,Σ}

Consider the hypersurface X ⊂ ℙ^{n} with defining equation

F = ∑_{j=0}^{d−2} (Y_{j}Z_{j+1} − Y_{j+1}Z_{j})  Y_{0}^{d−2−j}Y_{k−3}^{i}  +
∑_{j=2}^{d−2} (Y_{d−1+j}Z_{d+j} − Y_{d+j}Z_{d−1+j}) Y_{k}^{j−2}Y_{3}^{d−2−l}Z_{3}Z_{4} +
(Y_{d−1}Z_{d} − Y_{d}Z_{d−1})  Y_{k−3}^{d−4}Y_{k−4}^{2}  +
(Y_{d}Z_{d+1} − Y_{d+1}Z_{d})  Y_{1}^{d−4}Z_{7}Z_{8}  +
(Y_{k−2}Z_{k−1} − Y_{k−1}Z_{k−2})  Y_{k}^{d−4}Z_{k−1}Z_{5}  +
(Y_{k}Z_{0} − Z_{d}Y_{d})  Z_{0}^{d−2}  +
∑M∈C_{d} G_{M}(Y_{0}, …, Y_{2k})X_{M}.

Observe that F is contained in the homogeneous ideal of Σ, i.e., Σ ⊂ X. The derivative map ∂_{X,Σ} acts on the coordinates Y_{j}, Z_{j}, X_{M} as in Figure 1. Each of the monomials in A_{d} occurs as the initial term of ∂_{X,Σ} acting on one coordinate. For every coordinate except Y_{0} and Z_{0}, the initial term of ∂_{X,Σ} is one of the monomials in A_{d}. By Lemma 7.6, the image of ∂_{X,Σ} is a (k − 3)-generating linear system.

56 J. HARRIS AND J. STARR

(i), d = 3: Suppose that d = 3 and n ≥ 3^{2} + 3 + 1 = 13. Denote by B_{3} the set of 4d − 1 = 11 monomials,

B_{d} = { U^{2} T_{0}^{12−(k−2)j} T_{1}^{(k−2)j} |j = 0, 1, 2 }
∪ { U V T_{0}^{11−(k−2)j} T_{1}^{(k−2)j} |j = 0, 1, 2 }
∪ {  V^{2} T_{0}^{10−(k−2)j} T_{1}^{(k−2)j} |j = 0, 1, 2 }
∪ { U V  T_{1}^{11}  }
∪ {  V^{2}  T_{1}^{10}  }.

Denote by C_{3} the singleton set consisting of the monomial M = U^{2}T_{1}^{12}. Denote by

{Y_{0}, …, Y_{6}} ∪ {Z_{0}, …, Z_{5}} ∪ {X_{M}|M ∈ C_{3}} ∪ {V_{l}|l = 1 …, n − 13} ,

a basis of H^{0}(ℙ^{n}, O_{ℙ^{n}}(1)), i.e., a basis of homogeneous coordinates on ℙ^{n}.

Denote by f : 𝔽_{1} → ℙ^{12} ⊂ ℙ^{n} the morphism mapping ([T_{0} : T_{1}], [T_{0}U : T_{1}U : V]) ∈ 𝔽_{1} to the point in ℙ^{n} with coordinates X_{M} = 0, M ∈ C_{d}, with V_{l} = 0, l = 1, …, n − 13, and with

Y_{0} = UT_{0}^{6}, …, Y_{i} = UT_{0}^{6−i}T_{1}^{i}, …, Y_{k} = UT_{1}^{6},
Z_{0} = V T_{0}^{5}, …, Z_{j} = V T_{0}^{5−j}T_{1}^{j}, …, Z_{5} = V T_{1}^{5}.

This is an embedding whose image Σ = f(𝔽_{1}) is a rational normal scroll of degree 11.

Consider the hypersurface X ⊂ ℙ^{n} with defining equation

F = ∑_{j=0}^{1} (Y_{j}Z_{j+1} − Y_{j+1}Z_{j}) Y_{0}^{1−j}Y_{3}^{i} +
(Y_{2}Z_{3} − Y_{3}Z_{2})  Y_{6}  +
(Y_{5}Z_{5} − Y_{6}Z_{4})  Z_{5}  +
(Y_{3}Z_{3} − Y_{6}Z_{0})  Z_{4}  +
(Y_{2}Z_{2} − Y_{4}Z_{0})  Z_{0}  +
Y_{6}^{2}X_{M}.

Observe that F is contained in the homogeneous ideal of Σ, i.e., Σ ⊂ X. It is straightforward to compute the action of ∂_{X,Σ} on the coordinates Y_{j}, Z_{j}, X_{M}. Every monomial in A_{3} occurs as the initial term of ∂_{X,Σ} acting on one coordinate. For every coordinate except Y_{0} and Z_{0}, the initial term of ∂_{X,Σ} is one of the monomials in A_{3}. By Lemma 7.6, the image of ∂_{X,Σ} is a (k−3)-generating linear system.

(ii), d = 2: Suppose that d = 2 and n ≥ 2^{2} + 2 + 1 = 7. Let k = 3. Denote by B_{2} the set of 6 monomials,

\begin{aligned}B_{2} = & & \{ & U & & T_{0}^{3−j} & T_{1}^{j} & |j = 0, 1, 2 & \} \\ & ∪ & \{ & & V & T_{0}^{2−j} & T_{1}^{j} & |j = 0, 1, 2 & \}. \end{aligned}

Denote by C_{2} the singleton set consisting of the monomial M = UT_{1}^{3}. Denote by

{Y_{0}, Y_{1}, Y_{2}, Y_{3}, Z_{0}, Z_{1}, Z_{2}, X_{M}} ∪ {V_{l}|l = 1, …, n − 7} ,

a basis of H^{0}(ℙ^{n}, O_{ℙ^{n}}(1)), i.e., a basis of homogeneous coordinates on ℙ^{n}.

Denote by f : 𝔽_{1} → ℙ^{6} ⊂ ℙ^{n} the morphism mapping ([T_{0} : T_{1}], [T_{0}U : T_{1}U : V]) ∈ 𝔽_{1} to the point in ℙ^{n} with coordinates X_{M} = 0, M ∈ C_{d}, with V_{l} = 0, l = 1, …, n − 7, and with

Y_{0} = UT_{0}^{3}, Y_{1} = UT_{0}^{2}T_{1}, Y_{2} = UT_{0}T_{1}^{2}, Y_{3} = UT_{1}^{3},
Z_{0} = V T_{0}^{2}, Z_{1} = V T_{0}T_{1}, Z_{2} = V T_{1}^{2}.

This is an embedding whose image Σ = f(𝔽_{1}) is a rational normal scroll of degree 6.

Consider the hypersurface X ⊂ ℙ^{n} with defining equation

F = (Y_{3}Z_{1} − Y_{2}Z_{2}) + (Y_{0}Z_{1} − Y_{1}Z_{0}) + Y_{3}X_{M}.

Observe that F is contained in the homogeneous ideal of Σ, i.e., Σ ⊂ X. It is straightforward to compute the action of ∂_{X,Σ} on the coordinates Y_{j}, Z_{j}, X_{M}. Every monomial in A_{2} occurs as the initial term of ∂_{X,Σ} acting on one coordinate. For every coordinate except Y_{0}, the initial term of ∂_{X,Σ} is one of the monomials in A_{2}. By Lemma 7.6, the image of ∂_{X,Σ} is a (k − 3)-generating linear system.

(ii), d = 1. Denote by

{Y_{0}, Y_{1}, Y_{2}, Y_{3}, Z_{0}, Z_{1}, Z_{2}, X_{M}} ∪ {V_{l}|l = 1, …, n − 7} ,

a basis of H^{0}(ℙ^{n}, O_{ℙ^{n}}(1)), i.e., a basis of homogeneous coordinates on ℙ^{n}.

Denote by f : 𝔽_{1} → ℙ^{6} ⊂ ℙ^{n} the morphism mapping ([T_{0} : T_{1}], [T_{0}U : T_{1}U : V]) ∈ 𝔽_{1} to the point in ℙ^{n} with coordinates X_{M} = 0, M ∈ C_{d}, with V_{l} = 0, l = 1, …, n − 7, and with

Y_{0} = UT_{0}^{3}, Y_{1} = UT_{0}^{2}T_{1}, Y_{2} = UT_{0}T_{1}^{2}, Y_{3} = UT_{1}^{3},
Z_{0} = V T_{0}^{2}, Z_{1} = V T_{0}T_{1}, Z_{2} = V T_{1}^{2}.

This is an embedding whose image Σ = f(𝔽_{1}) is a rational normal scroll of degree 6.

Consider the hypersurface X ⊂ ℙ^{n} with defining equation F = X_{M}. Observe that F is contained in the homogeneous ideal of Σ, i.e., Σ ⊂ X. The image of ∂_{X,Σ}(X_{M}) is a generator for S_{0,0}; i.e., the image of ∂_{X,Σ} is a (k − 3)-generating linear system. □

Together with Remark 7.1, Lemma 7.11 and Proposition 7.12 imply the following corollary.

Corollary 7.13. If X ⊂ ℙ^{n} is a general hypersurface of degree d and if n ≥ d^{2} + d + 1, then there exists a 1-morphism ζ : C_{0} → \overline{M}_{0,1}(X, 1) that is both very twisting and very positive.

8. Proof of the main theorem

As explained at the end of Section 1, if d < \frac{n+1}{2}, then for a general hypersurface X ⊂ ℙ^{n} of degree d, Hypothesis 1.5, Hypothesis 1.6, and Hypothesis 1.7 are satisfied. By Corollary 6.6, if d ≥ 2 and n + 1 ≥ d^{2}, or if d = 1 and n ≥ 3, then for a general hypersurface X ⊂ ℙ^{n} of degree d, Hypothesis 4.10 is satisfied. Finally, if n ≥ d^{2} + d + 1 then by Corollary 7.13 there exists a very twisting, very positive family ζ : C_{0} → \overline{M}_{0,1}(X, 1). Therefore (ζ, ζ) is an inducting pair.

By Theorem 5.13, for every e ≥ 1 there exists an inducting pair (ζ_{1}, \overline{ζ}_{e}). In particular, there exists a very positive 1-morphism \overline{ζ}_{e} : C → \overline{M}_{0,1}(X, e). As shown in the proof of Theorem 5.13, it may be assumed that C is smooth and that the image of C is contained in the smooth locus of the fine moduli locus. By Lemma 4.14 (i), \overline{ζ}_{e} is a very free morphism. By [11, Prop. 7.4], \overline{M}_{0,0}(X, e) is an irreducible variety. Therefore, by [16, Thm. IV.3.7], \overline{M}_{0,0}(X, e) is rationally connected.

References

[1] D. Abramovich and A. Vistoli. Compactifying the space of stable maps. Journal of the American Mathematical Society, 15:27–75, 2002. arXiv: math.AG/9908167.

[2] V. Alexeev. Moduli spaces m_{g,n}(w) for surfaces. In Higher dimensional complex varieties (Trento, 1994), pages 1–22. de Gruyter, 1996.

[3] K. Behrend. Gromov-Witten invariants in algebraic geometry. Inventiones Mathematica, 127:601–617, 1997. arXiv: math.AG/9601011.

[4] K. Behrend and B. Fantechi. The intrinsic normal cone. Inventiones Mathematica, 128:45–88, 1998. arXiv: math.AG/961010.

[5] K. Behrend and Y. Manin. Stacks of stable maps and Gromov-Witten invariants. Duke Math Journal, 85:1–60, 1996. arXiv: math.AG/9506023.

[6] S. Bosch, W. Lütkebohmert, and M. Raynaud. Néron Models, volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1990.

[7] O. Debarre. Higher-dimensional algebraic geometry. Universitext. Springer-Verlag, 2001.

[8] W. Fulton. Intersection Theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. Springer-Verlag, 2 edition, 1998.

[9] W. Fulton and R. Pandaharipande. Notes on stable maps and quantum cohomology. In Algebraic geometry – Santa Cruz 1995, pages 45–96. American Mathematical Society, 1995. arXiv: math.AG/9608011.

[10] A. Grothendieck and J. Dieudonné. Éléments de géométrie algébrique. IV, volume 20, 24, 28, 32 of Publications Mathématiques. Institute des Hautes Études Scientifiques., 1964-1967.

[11] J. Harris, M. Roth, and J. Starr. Rational curves on hypersurfaces of low degree. submitted Crelle’s Journal, preprint arXiv: math.AG/0203088.

[12] R. Hartshorne. Residues and Duality, Lecture notes of a seminar on the work of A. Grothendieck, volume 20 of Lecture Notes in Math. Springer-Verlag, 1966.

[13] R. Hartshorne. Algebraic Geometry, volume 52 of Graduate Texts in Mathematics. Springer-Verlag, 1977.

[14] J.-P. Jouanolou. Théorèmes de Bertini et Applications, volume 42 of Progess in Mathematics. Birkhäuser, 1983.

[15] B. Kim and R. Pandharipande. The connectedness of the moduli space of maps to homogeneous spaces. In Symplectic geometry and mirror symmetry, chapter 5. World Scientific, 2001. arXiv: math.AG/0003168.

[16] J. Kollár. Rational Curves on Algebraic Varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. Springer-Verlag, 1996.

[17] J. Kollár and S. Mori. Birational Geometry of Algebraic Varieties, volume 134 of Cambrige Tracts in Mathematics. Cambridge University Press, 1998.

[18] S. Lang. On quasi-algebraic closure. Ann. Math., 55:373–390, 1952.

[19] R. Pandharipande. Intersections of Q-divisors on Kontsevich’s moduli space. Trans. Amer. Math. Soc., 351:1481– 1505, 1999.

Department of Mathematics, Harvard University, Cambridge MA 02138

E-mail address: harris@math.harvard.edu

Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139

E-mail address: jstarr@math.mit.edu