HYPERSURFACES OF LOW DEGREE ARE RATIONALLY SIMPLY-CONNECTED

JASON MICHAEL STARR

Abstract. For a general hypersurface of degree d in projective n-space, if n ≥ d^{2} the spaces of 2-pointed rational curves on the hypersurface are rationally connected; thus the hypersurfaces are rationally simply connected. This paper proves stronger versions of theorems in [HS05].

1. Introduction

A smooth projective variety is rationally connected if every pair of points is contained in a rational curve, cf. [Kol96], [Deb01]. This is analogous to the notion of path-connectedness in topology. A path-connected topological space is simply-connected if the space of based paths is path-connected. By analogy, Mazur suggested definining a rationally connected variety to be rationally simply-connected if the space of based, 2-pointed rational curves of fixed homology class is rationally connected. Unfortunately, this notion is a bit too strong. The condition should be imposed only if the homology class is suitably positive and if the basepoints are suitably general. Also, since the space of 2-pointed rational curves is typically not compact, it is compactified using the Kontsevich moduli space, cf. [FP97].

Theorem 1.1. In characteristic 0, a general hypersurface of degree d in ℙ^{n} is rationally simply connected if n ≥ d^{2} and d ≥ 2. Precisely, for every e ≥ 2, the evaluation morphism ev : \overline{M}_{0,2}(X, e) → X ×X is surjective and a general fiber is irreducible, reduced and rationally connected. This also holds for (n, d) = (≥ 2, 1).

Tsen’s theorem proves that a Fano hypersurface defined over the function field of a curve has a rational point, cf. [Tse36]. Rational connectedness, introduced by Campana [Cam92] and Kollár-Miyaoka-Mori [KMM92b], is the natural notion for generalizing Tsen’s theorem, cf. [GHS03], [dJS03] and [GHMS05]. The analogue of Tsen’s theorem over a higher-dimensional base is the Tsen-Lang theorem, cf. [Lan52]. To generalize the Tsen-Lang theorem, the correct notion should be some version of higher rational connectedness

In truth, the best definition of rational simple-connectedness is not yet clear. At issue is a precise definition of sufficiently positive homology class. However, if Pic(X) = ℤ there is only one possible definition: a collection of homology classes β ∈ H_{2}(X, ℤ) is sufficiently positive if for one ample invertible sheaf L, and hence every ample invertible sheaf, there exists an integer e_{0} = e_{0}(L) such that β is in the collection if c_{1}(L) · β ≥ e_{0}. This is the notion that appears in Theorem 1.1.

Date: February 27, 2006.

Principle 1.2 (de Jong). A proper, smooth variety defined over the function field of a surface over an algebraically closed field of characteristic 0 has a rational point if the base-change to the algebraic closure of the function field is rationally simply-connected (in a slightly stronger sense than used here) and if a certain Brauer obstruction vanishes.

In proving the principle, there are technical difficulties related to singularities of spaces of rational curves, the geometric analogue of weak approximation, and the Brauer obstruction (as opposed to a Brauer obstruction). In a forthcoming paper with A. J. de Jong, de Jong’s strategy for proving Principle 1.2 is given, and Principle 1.2 is proved under some additional hypotheses.

Acknowledgments: A re-investigation of [HS05] leading to this paper was motivated by Theorem 11.1 of A. J. de Jong. I am grateful to de Jong for many useful conversations.

2. Sketch of the proof

The proof of Theorem 1.1 uses ideas from [HS05]. It is an induction argument. The induction step uses the fiber-by-fiber connected sum of 2 families of marked rational curves. Three new results go into the proof. First, very twisting morphisms are systematically studied leading to a simpler version of the induction argument, Proposition 3.10. The other two results concern the base case for the induction. Proposition 4.6 proves Theorem 1.1 in case e = 2. The last result is existence of a very twisting morphism ζ : ℙ^{1} → \overline{M}_{0,1}(X, 1). Some notation is necessary to state the result.

Notation 2.1. Let n ≥ 2. The scheme \overline{M}_{0,1}(ℙ^{n}, 1) is the partial flag variety Flag(1, 2; n + 1). There are morphisms ev : \overline{M}_{0,1}(ℙ^{n}, 1) → ℙ^{n} and pr : \overline{M}_{0,1}(ℙ^{n}, 1) → \overline{M}_{0,0}(ℙ^{n}, 1) = Grass(2, n + 1). Denote by h the divisor class ev^{∗}C_{1}(O_{ℙ^{n}}(1)). Denote by x the divisor class pr^{∗}C_{1}(O_{Grass}(1)), where O_{Grass}(1) is the invertible sheaf giving the Plücker embedding. The NEF cone of \overline{M}_{0,1}(ℙ^{n}, 1) is the set of divisor classes ℤ_{≥0}\{x, h\}.

The degree of a very twisting morphism ζ is large: If n = d^{2} then deg(ζ^{∗}h) ≥ d^{2}−d−1 and deg(ζ^{∗}x) ≥ 2(d^{2}−d−1). So it is unreasonable to try to directly construct a very twisting morphism. The strategy is to instead consider a nodal, reducible, genus 0 curve C, and a morphism ζ : C → \overline{M}_{0,1}(X, 1) whose restriction to every irreducible component of C is a twisting morphism of minimal degree.

The curve C deforms to a smooth genus 0 curve, i.e., to ℙ^{1}. If C has many irreducible components, a deformation of ζ to a morphism from ℙ^{1} may be a very twisting morphism. Proposition 5.7 gives a criterion for this. The essential case is n = d^{2}. Then the criterion is surjectivity of the derivative map between the Zariski tangent spaces of two parameter spaces. Sections 8, 9, 10, and particularly Proposition 10.1, prove the derivative map is surjective. This is the heart of the article and consists of an involved deformation theory analysis.

3. Twisting and very twisting morphisms

Hypothesis 3.1. Let K be an algebraically closed field. All schemes, algebraic spaces and Deligne-Mumford stacks are defined over K, and all morphisms and 1-morphisms commute with the morphisms to Spec K.

Every rank r, locally free O_{ℙ^{1}}-module is isomorphic to a direct sum of invertible sheaves by Grothendieck’s lemma [Har77, Exer. V.2.6], i.e., E ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{r}) for a sequence of integers a_{1} ≤ ⋯ ≤ a_{r}.

Definition 3.2. The negativity of E is the largest integer n (possibly 0) such that a_{i} < 0 for all i ≤ n. The nullity of E is the largest integer z (possibly 0) such that a_{i} = 0 for all n < i ≤ n + z. The positivity of E is the difference p := r − n − z. If n(E) = 0, then E is generated by global sections. If n(E) + z(E) = 0 and p(E) > 0, then E is ample.

Let π : Y → Z be a morphism of finite type Deligne-Mumford stacks. Denote by Z^{0} the maximal open substack of Z that is smooth over K. Denote by Y^{0} the maximal open substack of π^{−1}(Z^{0}) on which π is smooth. Denote by T_{π} the locally free sheaf on Y^{0} that is the dual of the sheaf of relative differentials of π, i.e., T_{π} is the vertical tangent bundle of π.

Definition 3.3. A non-constant morphism f : ℙ^{1} → Y is π-relatively free, resp. π-relatively very free, if

(i) f(ℙ^{1}) ⊂ Y^{0}, (ii) f^{∗}π^{∗}T_{Z^{0}} is generated by global sections, and (iii) f^{∗}T_{π} is generated by global sections (resp. ample).

For a π-relatively free morphism f, the nullity of f is z(f^{∗}T_{π}), and the positivity of f is p(f^{∗}T_{π}).

Lemma 3.4. The smooth locus of the 1-morphism Hom(ℙ^{1}, π) : Hom(ℙ^{1}, Y) → Hom(ℙ^{1}, Z) contains the locus of π-relatively free morphisms.

Proof. Associated to a π-relatively free morphism f there is a short exact sequence,

0 ⟶ f^{∗}T_{π} ⟶ f^{∗}T_{Y} ⟶^{dπ} f^{∗}π^{∗}T_{Z} ⟶ 0.

Because the outer terms are generated by global sections, in particular h^{1}(ℙ^{1}, f^{∗}T_{π}) = h^{1}(ℙ^{1}, f^{∗}T_{Y}) = h^{1}(ℙ^{1}, f^{∗}π^{∗}T_{Z}) = 0. Therefore both [f] ∈ Hom(ℙ^{1}, Y) and [π ∘ f] ∈ Hom(ℙ^{1}, Z) are smooth points, and H^{0}(ℙ^{1}, dπ) : H^{0}(ℙ^{1}, f^{∗}T_{Y}) → H^{0}(ℙ^{1}, f^{∗}π^{∗}T_{Z}) is surjective. Because this is the derivative of Hom(ℙ^{1}, π) at [f], the Jacobian criterion implies Hom(ℙ^{1}, π) is smooth at [f]. □

Remark 3.5. The smooth locus of ℙ^{1} ×Hom(ℙ^{1}, Y) → Y contains ℙ^{1} ×{f} for every π-relatively free morphism f. For every open subset U ⊂ Y whose closure intersects Image(f), every sufficiently small deformation f_{ϵ} of f is π-relatively free and Image(f_{ϵ}) intersects U.

The following proposition is the relative version of the criterion [Kol96, Theorem IV.5.8].

Proposition 3.6. Let π : Y → Z be a proper morphism of irreducible, finite type Deligne-Mumford stacks, and denote by |π| the induced morphism of coarse moduli spaces.

(i) If there exists a π-relatively free morphism f with positivity > 0, then every irreducible component of the geometric generic fiber of |π| is uniruled.

(ii) If also char(K) = 0, the dimension of the MRC quotient of the geometric generic fiber of |π| is at most the nullity of f, i.e., the fiber dimension of the MRC quotient morphism is at least the positivity of f.

Proof. (i): This is essentially [GHMS05, Lem. 4.4], which in turn relies on the Rigidity Lemma, cf. [Mum70, p. 43]. Denote Y_{f} = ℙ^{1} ×_{π∘f,Z,π} Y and denote the projection by π_{f} : Y_{f} → ℙ^{1}. The morphism f uniquely determines a section σ of π_{f}. The projection π_{f} is smooth along σ(ℙ^{1}). The sheaf σ^{∗}T_{π_{f}} = f^{∗}T_{π} is generated by global sections.

Denote by M ⊂ Hom(ℙ^{1}, Y_{f}) the locally closed substack of the Hom stack parametrizing morphisms τ : ℙ^{1} → Y_{f} with π_{f} ∘ τ = Id_{f}. By [Kol96, Prop. II.3.5], the smooth locus of ev : ℙ^{1} × M → Y_{f} contains ℙ^{1} × {[σ]}, and the fiber dimension of ev is h^{0}(ℙ^{1}, σ^{∗}T_{π_{f}} (−1)). The proof is unchanged in the case of Deligne-Mumford stacks.

Because the positivity of f is > 0, the fiber dimension of ev is > 0. By [GHMS05, Lem. 4.4], for every t ∈ ℙ^{1} there exists a rational curve containing σ(t) and contained in |π_{f}|^{−1}(t). The hypothesis K = ℂ from [GHMS05] is not used in the proof of [GHMS05, Lem. 4.4].

For every t ∈ ℙ^{1} there is a rational curve containing f(t) contained in a fiber of |π|. By Remark 3.5, the morphism ℙ^{1} × Hom(ℙ^{1}, Y) → Y is smooth at (t, [f]). So there is a dense open subset U of Y^{0} such that for every geometric point y ∈ U, there is a rational curve containing y and contained in a fiber of |π|.

(ii): By [Hir64], assume Y and Z are smooth and irreducible. The relative MRC quotient for Deligne-Mumford stacks is a datum (Q, ρ, U, ψ) of a smooth, irreducible algebraic space Q, a proper morphism ρ : Q → Z whose geometric generic fiber is not uniruled, an open subset U ⊂ Y whose complement has codimension ≥ 2, and a generically smooth 1-morphism ψ : U → Q such that π|_{U} is 2-equivalent to ρ ∘ ψ. Existence of the relative MRC quotient for Deligne-Mumford stacks is proved in [Sta04].

Because f : ℙ^{1} → Y is π-relatively free, it is free. By [Kol96, Prop. II.3.7], the morphism f can be deformed so that f(ℙ^{1}) is contained in U ∩ Y^{0} and f(ℙ^{1}) intersects the smooth locus of ψ. By the Jacobian criterion, at every point y ∈ Y^{0} the derivative map dπ_{y} : T_{Y} ⊗κ(y) → T_{Z} ⊗κ(π(y)) is surjective. The derivative map dρ_{ψ(y)} : T_{Q} ⊗ κ(ψ(y)) → T_{Z} ⊗ κ(π(y)) factors dπ_{y}. So dρ_{ψ(y)} is also surjective. By the Jacobian criterion, ρ is smooth at ψ(y), i.e., ρ is smooth along ψ(Y^{0}). In particular, ρ is smooth along ψ(f(ℙ^{1})).

There is a sheaf homomorphism T_{π}|_{U} → ψ^{∗}T_{ρ} whose restriction to the smooth locus of ψ is surjective. So there is a sheaf homomorphism α : f^{∗}T_{π} → f^{∗}ψ^{∗}T_{ρ} whose cokernel is a torsion sheaf. Because f^{∗}T_{π} is generated by global sections, also Image(α) and f^{∗}ψ^{∗}T_{ρ} are generated by global sections. Since the geometric generic fiber of ρ is not uniruled, by (i) the positivity of ψ ∘ f is 0. This implies that Image(α) = f^{∗}ψ^{∗}T_{ρ} and that f^{∗}ψ^{∗}T_{ρ} is a quotient of f^{∗}T_{π}/H^{0}(ℙ^{1}, f^{∗}T_{π}(−1)) ⊗ O_{ℙ^{1}}(1). Therefore the rank of f^{∗}ψ^{∗}T_{ρ} is at most the nullity of f. In other words, the dimension of the geometric generic fiber of ρ is at most the nullity of f. □

The main definition of both [HS05] and this paper is the following. It is slightly different than in [HS05] because it is used differently here.

Definition 3.7. Let X be a quasi-projective scheme, and denote by X^{0} the smooth locus of X. Let r ≥ 0 and e > 0 be integers. Let ζ : ℙ^{1} → \overline{M}_{0,r}(X^{0}, e) be a 1-morphism. The 1-morphism ζ is twisting, resp. very twisting, if

(i) the morphism ev : \overline{M}_{0,r}(X^{0}, e) → (X^{0})^{r} is unobstructed at every geometric point of the image of ζ,

(ii) the morphism ζ is ev-relatively free, resp. ev-relatively very free, (iii) for every i = 1, …, r, the degree of ζ^{∗}ψ_{i} is nonpositive, and (iv) the image under ζ of the geometric generic point of ℙ^{1} is a stable

map with irreducible domain. Let T be a Deligne-Mumford stack and let ζ : ℙ^{1} × T → \overline{M}_{0,r}(X^{0}, e) be a 1-morphism. The morphism ζ is twisting relative to T, resp. very twisting relative to T, if for every geometric point of T the restriction of ζ over this point is twisting, resp. very twisting.

Remark 3.8. (i) If r = 0, the morphism ev is just the structure morphism to Spec (K). So in this case, ζ is ev-relatively free, resp. ev-relatively very free, iff it is free, resp. very free.

(ii) Condition (i) implies that ev is smooth along the image of ζ. If char(K) = 0 and X is a general hypersurface of degree d < \frac{n+1}{2} in ℙ^{n}, the smooth locus of ev is precisely the set of points where ev is unobstructed. But for some schemes the unobstructed locus is strictly smaller than the smooth locus, e.g., for a general hypersurface of degree n in ℙ^{n} and e = 4.

(iii) The 1-morphism ζ is equivalent to a datum ((π : Σ → ℙ^{1}, σ_{1}, …, σ_{r}), g : Σ → X^{0}). The class ζ^{∗}ψ_{i} is simply the divisor class of σ_{i}^{∗}O_{Σ}(−σ_{i}(ℙ^{1})). Therefore condition (iii) states that the self-intersection (σ_{i}(ℙ^{1}) · σ_{i}(ℙ^{1}))_{Σ} is nonnegative for every i = 1, …, r.

(iv) Often the morphism (π, g) : Σ → ℙ^{1} × X^{0} is unramified and is étale locally a regular embedding, i.e., the sheaf homomorphism d(π, g)^{†} : (π, g)^{∗}Ω_{ℙ^{1}×X^{0}} → Ω_{Σ} is surjective and the kernel is locally free. In particular, this is true if e = 1. If d(π, g)^{†} is surjective and the kernel is locally free, denote by N_{(π,g)} the dual of the kernel. The morphism ev is unobstructed along ζ(ℙ^{1}) iff R^{1}π_{∗}(N_{(π,g)}(−(σ_{1}(ℙ^{1}) + ⋯ + σ_{r}(ℙ^{1})))) is 0 and then ζ^{∗}T_{ev} is the locally free sheaf π_{∗}(N_{(π,g)}(−(σ_{1}(ℙ^{1}) + ⋯ + σ_{r}(ℙ^{1})))).

(v) By Remark 3.5, if there exists a 1-morphism ζ : ℙ^{1} → \overline{M}_{0,r}(X, e) that satisfies (i)–(iii), and if ζ(ℙ^{1}) intersects the closure of the open set parametrizing stable maps with irreducible domain, then a small deformation of ζ is a 1-morphism satisfying (i)–(iv).

If r ≥ 2 consideration of the Picard group of Σ implies that deg(ζ^{∗}ψ_{i}) = 0 for all i = 1, …, r. However if r = 1 – the main case of interest – then −deg(ζ^{∗}ψ) can be arbitrarily positive.

Proposition 3.9. Let ζ_{0} : ℙ^{1} → \overline{M}_{0,1}(X^{0}, e) be a very twisting morphism. Denote a_{0} = −deg(ζ_{0}^{∗}ψ).

(i) If a_{0} is odd, there exists an integer a_{1} and for every integer a ≥ a_{1} there is a very twisting morphism ζ_{a} : ℙ^{1} → \overline{M}_{0,1}(X^{0}, e) such that a = −deg(ζ_{a}^{∗}ψ).

(ii) If a_{0} is even, there exists an integer a_{1} and for every even integer a ≥ a_{1} there is a very twisting morphism ζ_{a} : ℙ^{1} → \overline{M}_{0,1}(X^{0}, e) such that a = −deg(ζ_{a}^{∗}ψ).

Proof. (i): Since ζ_{0} is very twisting, ζ_{0}^{∗}T_{ev} ≅ O_{ℙ^{1}}(b_{1}) + ⋯ + O_{ℙ^{1}}(b_{t}) for integers 1 ≤ b_{1} ≤ ⋯ ≤ b_{t}. Define,

a_{1} = 2a_{0} ⌈\frac{a_{0} + b_{1}}{2b_{1}} ⌉ .

Let a ≥ a_{1} be an integer. There are 2 cases depending on whether a is even or odd.

If a is even, then a = q(2a_{0}) + r for an integer q and an integer r satisfying 0 ≤ r < 2a_{0}. Because a is even, r = 2r′ for an integer r′ satisfying 0 ≤ r′ < a_{0}. Define m = 2q. Because a ≥ a_{1}, m ≥ \frac{a_{0}+b_{1}}{b_{1}} , and thus mb_{1} − r′ > b_{1} > 0.

If a is odd, then a + a_{0} = q(2a_{0}) + r for an integer q and an integer r satisfying 0 ≤ r < 2a_{0}. Because a + a_{0} is even, r = 2r′ for an integer r′ satisfying 0 ≤ r′ < a_{0}. Define m = 2q − 1. Because a ≥ a_{1}, m ≥ \frac{a_{0}}{b_{1}} , and thus mb_{1} − r′ > 0.

In each case, a = ma_{0} + 2r′ where 0 ≤ r′ < a_{0} and where mb_{1} − r′ > 0. Let h : ℙ^{1} → ℙ^{1} be a finite morphism of degree m. The morphism ζ_{0} ∘ h : ℙ^{1} → \overline{M}_{0,1}(X, e) is very twisting. Moreover, (ζ_{0} ∘h)^{∗}T_{ev} ≅ O_{ℙ^{1}}(mb_{1})⊕⋯⊕ O_{ℙ^{1}}(mb_{t}) and −deg((ζ_{0} ∘ h)^{∗}ψ) = ma_{0}.

Denote by ((π : Σ → ℙ^{1}, σ : ℙ^{1} → Σ), g : Σ → X) the datum giving rise to ζ_{0} ∘ h. There exists a section σ′ : ℙ^{1} → Σ such that the divisor σ′(ℙ^{1}) ⊂ Σ is in the linear equivalence class of |σ(ℙ^{1}) + r′π^{−1}(0)|. Denote by ζ_{a} : ℙ^{1} → \overline{M}_{0,1}(X, e) the 1-morphism associated to the datum ((π : Σ → ℙ^{1}, σ′ : ℙ^{1} → Σ), g : Σ → X).

For every s ∈ ℙ^{1}, the obstruction group of ev at ζ_{0}(h(s)) is the hypercohomology group

H^{2}(π^{−1}(s), L_{(π,g)}^{∨} ⊗ O_{π^{−1}(s)}(−σ(s))),

where L_{(π,g)} is the cotangent complex of the morphism (π, g) : Σ → ℙ^{1} × X and where L_{(π,g)}^{∨} is the object RHom_{O_{Σ}}(L_{(π,g)}, O_{Σ}) in the derived category of quasi-coherent sheaves on Σ. By hypothesis, the obstruction group is 0. By construction, O_{π^{−1}(s)}(−σ′(s)) ≅ O_{π^{−1}(s)}(−σ′(s)). Therefore, the obstruction group of ev at ζ_{a}(s) is 0. So ζ_{a} satisfies (i) of Definition 3.7.

Moreover, ζ_{a}^{∗}T_{ev} is the cohomology sheaf,

H^{1}Rπ_{∗}(L_{(π,g)}^{∨}(−σ′(ℙ^{1}))) = H^{1}Rπ_{∗}(L_{(π,g)}^{∨}(−σ(ℙ^{1}))) ⊗ O_{ℙ^{1}}(−r′),

i.e., h^{∗}ζ_{0}^{∗}T_{ev} ⊗ O_{ℙ^{1}}(−r′). So ζ_{a}^{∗}T_{ev} ≅ O_{ℙ^{1}}(mb_{1} − r′) ⊕ ⋯ ⊕ O_{ℙ^{1}}(mb_{t} − r′). By construction, mb_{1} − r′ > 0. Therefore ζ_{a}^{∗}T_{ev} is ample. So ζ_{a} satisfies (ii) of Definition 3.7 for a very twisting morphism.

Finally, (σ′(ℙ^{1}) · σ′(ℙ^{1}))_{Σ} = (σ(ℙ^{1}) · σ(ℙ^{1}))_{Σ} + 2r′. Therefore,

−deg(ζ_{a}^{∗}ψ) = −deg(g^{∗}ζ_{0}^{∗}ψ) + 2r′ = ma_{0} + 2r′ = a.

So ζ_{a} satisfies (iii) of Definition 3.7 for a very twisting morphism. Because ζ_{0} satisfies (iv) of Definition 3.7, also ζ_{a} satisfies (iv). Therefore ζ_{a} is very twisting and −deg(ζ_{a}^{∗}ψ) = a.

(ii): This is exactly as in the even case of (i). The odd case of (i) does not work because a + a_{0} is not even. □

The main result about very twisting morphisms is the following theorem, which forms the induction step in the proof of Theorem 1.1.

Proposition 3.10. Let X be a quasi-projective variety. Let e_{1}, e_{2} > 0 be integers. Let r equal 1, resp. 2. Let ζ_{1} : ℙ^{1} → \overline{M}_{0,r}(X, e_{1}) be an ev-relatively very free morphism mapping a general point of ℙ^{1} to a stable map with irreducible domain and with ev ∘ ζ_{1} nonconstant (resp. ev_{1} ∘ ζ_{2} nonconstant). Let ζ_{2} : ℙ^{1} → \overline{M}_{0,1}(X, e_{2}) be a very twisting morphism.

Assume that \overline{M}_{0,0}(X, ϵ) is irreducible for all positive integers ϵ that are sufficiently divisible. Assume one of the following,

(i) −deg(ζ_{1}^{∗}ψ) > 0 (resp. −deg(ζ_{1}^{∗}ψ_{1}) > 0), or (ii) for every general degree e_{1} morphism g : ℙ^{1} → X there exists a

twisting morphism ζ_{g} : ℙ^{1} → \overline{M}_{0,1}(X, e_{2}) such that ev ∘ ζ_{g} = g. Then there exists an ev-relatively very free morphism ζ : ℙ^{1} → \overline{M}_{0,r}(X, e_{1} + e_{2}) mapping a general point of ℙ^{1} to a stable map with irreducible domain and with ev ∘ ζ_{1} nonconstant (resp. ev_{1} ∘ ζ_{2} nonconstant).

Moreover, if r = 1 and ζ_{1} is very twisting, then there exists such a ζ that is very twisting and has −deg(ζ^{∗}ψ) = 0.

Proof. To ease notation, if r = 1 denote ev also by ev_{1}. By Proposition 3.9, assume that −deg(ζ_{2}^{∗}ψ) > 0. If r = 1 and ζ_{1} is very twisting, also assume that −deg(ζ_{1}^{∗}ψ) > 0, i.e., (i) applies.

Let ϵ be a positive integer such that \overline{M}_{0,0}(X, ϵ) is irreducible. After precomposing ζ_{1} and ζ_{2} with finite morphism ℙ^{1} → ℙ^{1}, assume that deg(ev ∘ ζ_{1}) = deg(ev ∘ ζ_{2}) = ϵ. Denote by Hom(ℙ^{1}, X)_{ϵ} the open subset of Hom(ℙ^{1}, X) parametrizing morphisms of degree ϵ; this is irreducible by hypothesis. There are morphisms,

Hom(ℙ^{1}, ev_{1}) :Hom(ℙ^{1}, \overline{M}_{0,r}(X, e_{1})) ⟶ Hom(ℙ^{1}, X)
Hom(ℙ^{1}, ev) : Hom(ℙ^{1}, \overline{M}_{0,1}(X, e_{2})) ⟶ Hom(ℙ^{1}, X)

By Lemma 3.4, Hom(ℙ^{1}, ev_{1}) is smooth at [ζ_{1}] and Hom(ℙ^{1}, ev) is smooth at [ζ_{2}]. So the image of each is a dense open subset of Hom(ℙ^{1}, X)_{ϵ}. Therefore, after deforming ζ_{1} and ζ_{2}, assume that also ev_{1} ∘ ζ_{1} = ev ∘ ζ_{2}.

Denote by ((π_{2} : Σ_{2} → ℙ^{1}, σ_{2,1}), g_{2}) the datum associated to ζ_{2}. If r = 1, denote by ((π_{1} : Σ_{1} → ℙ^{1}, σ_{1,1}), g_{1}) the datum associated to ζ_{1}. If r = 2, denote by ((π_{1} : Σ_{1} → ℙ^{1}, σ_{1,1}, σ_{1,2}), g_{1}) the datum associated to ζ_{1}. Denote the two cases in the statement as Case (i) and Case (ii) respectively. In Case (ii) a similar argument to the last paragaph proves, after deforming ζ_{1} and ζ_{2} further, there exists a dense open subset U ⊂ ℙ^{1} such that for every closed point s ∈ U,

(i) π_{1}^{−1}(s) is irreducible, and

(ii) there exists a twisting morphism ζ_{s} : π_{1}^{−1}(s) → \overline{M}_{0,1}(X, e_{2}) such that ζ_{s}(σ_{1,1}(s)) = ζ_{2}(s).

In Case (i), denote C = C_{0} = σ_{1,1}(ℙ^{1}). In Case (ii), denote C_{0} = σ_{1,1}(ℙ^{1}). Let δ > 0 be an integer such that (C_{0} · C_{0})_{Σ_{1}} + δ > 0. Let s_{1}, …, s_{δ} ∈ U be distinct points. For i = 1, …, δ, denote C_{i} = π_{1}^{−1}(s_{i}). Denote C = C_{0} ∪ C_{1} ∪ ⋯ ∪ C_{δ}. In each case, the linear system |C| on Σ_{1} contains an irreducible curve that intersects σ_{1,1}(ℙ^{1}) (resp. σ_{1,1}(ℙ^{1}) ∪ σ_{1,2}(ℙ^{1})) transversely in finitely many points. Let D ⊂ ℙ_{t}^{1} × Σ_{1} be a divisor such that pr_{ℙ^{1}}^{−1}(0)∩D ⊂ Σ_{1} equals C and such that for general t ∈ ℙ_{t}^{1}, pr_{ℙ^{1}}^{−1}(t)∩D ⊂ Σ_{1} is an irreducible divisor that intersects σ_{1,1}(ℙ^{1}) (resp. σ_{1,1}(ℙ^{1}) ∪ σ_{1,2}(ℙ^{1})) transversely in finitely many points (the subscript “t” in ℙ_{t}^{1} is to distinguish ℙ_{t}^{1} from the target of π_{1}).

Denote by ζ_{C} : C → \overline{M}_{0,1}(X, e_{2}) the unique morphism such that ζ_{C} ∘ σ_{1,1} = ζ_{2} and such that ζ_{C}|_{C_{i}} = ζ_{s_{i}} for i = 1, …, δ in Case (ii). There is a 1morphism of relative Hom stacks,

êv : Hom_{ℙ_{t}^{1}}(D, ℙ_{t}^{1} × \overline{M}_{0,1}(X, e_{2})) → Hom_{ℙ_{t}^{1}}(D, ℙ_{t}^{1} × X).

There is a point (0, [ζ_{C}]) ∈ Hom_{ℙ_{t}^{1}}(D, ℙ_{t}^{1} × \overline{M}_{0,1}(X, e_{2})) lying over 0 ∈ ℙ_{t}^{1}. By the same argument as in the proof of Lemma 3.4, the morphism êv is smooth at (0, [ζ_{C}]).

The following morphism defines a section ĝ_{1} of Hom_{ℙ_{t}^{1}}(D, ℙ_{t}^{1} × X) → ℙ_{t}^{1},

(pr_{ℙ_{t}^{1}}, g_{1} ∘ pr_{Σ_{1}}) : D → ℙ_{t}^{1} × X.

Denote by pr_{ℙ_{t}^{1}} : H → ℙ_{t}^{1} the fiber product,

H = ℙ_{t}^{1} ×_{ĝ_{1},êv} Hom_{ℙ_{t}^{1}}(D, ℙ_{t}^{1} × \overline{M}_{0,1}(X, e_{2})).

Again, (0, [ζ_{C}]) defines a point of H lying over 0 ∈ ℙ_{t}^{1}. And pr_{ℙ_{t}^{1}} is smooth at (0, [ζ_{C}]) by base-change. Therefore the image of pr_{ℙ_{t}^{1}} is a dense open subset of ℙ_{t}^{1}. Let t ∈ ℙ_{t}^{1} be a general point, and define D ⊂ Σ_{1} to be pr_{ℙ_{t}^{1}}^{−1} ∩ D ⊂ Σ_{1}. Then there exists a morphism ζ_{D} : D → \overline{M}_{0,1}(X, e_{2}) that is a deformation of ζ_{C}. If t and ζ_{D} are general, then ζ_{D} is a very twisting morphism. The curve D ⊂ Σ_{1} is the image of a section σ_{1,0} : ℙ^{1} → Σ_{1}. Denote by ζ̃_{2} : ℙ^{1} → \overline{M}_{0,1}(X, e_{2}) the 1-morphism ζ̃ = ζ_{D} ∘ σ_{1,0}.

By hypothesis, D ∩ σ_{1,1}(ℙ^{1}) (resp. D ∩ (σ_{1,1}(ℙ^{1}) ∪ σ_{1,2}(ℙ^{1}))) is transverse. Denote by ν : Σ̃_{1} → Σ_{1} the blowing up of Σ_{1} at the finitely many intersection points. Denote by σ̃_{1,i} : ℙ^{1} → Σ̃_{1} the strict transform of σ_{1,i} for i = 0, 1 (resp. i = 0, 1, 2). In Case (i), (D · σ_{1,1}(ℙ^{1}))_{Σ_{1}} = (D · D)_{Σ_{1}} = −deg(ζ_{1}^{∗}ψ_{1}) > 0. If r = 2, also (D · σ_{1,2}(ℙ^{1}))_{Σ_{1}} = 0. In Case (ii), (D · σ_{1,1}(ℙ^{1}))_{Σ_{1}} = −deg(ζ_{1}^{∗}ψ_{1}) + δ > 0 and (D · D)_{Σ_{1}} = −deg(ζ_{1}^{∗}ψ_{1}) + 2δ > 0. If r = 2, also (D · σ_{1,2}(ℙ^{1}))_{Σ_{1}} = δ. In each case, for the strict transform D̃ ⊂ Σ̃_{1}, (D̃ · D̃)Σ̃ ≥ 0; it is precisely 0 except in Case (ii) if r = 1.

Denote by π̃_{1} = π_{1} ∘ ν and denote g̃_{1} = g_{1} ∘ ν. If r = 1, denote by ζ̃_{1} : ℙ^{1} → \overline{M}_{0,r+1}(X, e_{1}) the 1-morphism associated to the datum ((π̃_{1} : Σ̃_{1} → ℙ^{1}, σ̃_{1,0}, σ̃_{1,1}), g̃_{1}). If r = 2, denote by ζ̃_{1} : ℙ^{1} → \overline{M}_{0,r+1}(X, e_{1}) the 1morphism associated to the datum ((π̃_{1} : Σ̃_{1} → ℙ^{1}, σ̃_{1,0}, σ̃_{1,1}, σ̃_{1,2}), g̃_{1}). In each case, observe that ev_{0} ∘ ζ̃_{1} = ev ∘ ζ̃_{2}.

The following terminology is from [BM96]. Denote by τ the following genus 0 stable A-graph. There are two vertices v_{1} and v_{2} of degree e_{1} and e_{2} respectively. If r = 1, there are two flags, f_{0} and f_{1}, attached to v_{1}. If r = 2, there are three flags, f_{0}, f_{1} and f_{2}, attached to v_{1}. There is one flag attached to v_{2} and together with f_{0} it forms an edge connecting v_{1} to v_{2}. If r = 1, then f_{1} is a tail attached to v_{1}. If r = 2, both f_{1} and f_{2} are tails attached to v_{1}. Denote by \overline{M}(X, τ) the associated Behrend-Manin stack. In a natural way the pair (ζ̃_{1}, ζ̃_{2}) determines a 1-morphism ζ : ℙ^{1} → \overline{M}(X, τ): ζ̃_{1} corresponds to v_{1}, σ_{1,i} corresponds to f_{i} for each i, and ζ̃_{2} corresponds to v_{2}.

There is a canonical contraction of τ to a stable A-graph with a single vertex of degree e_{1} + e_{2} and r tails. Denote by ι : \overline{M}(X, τ) → \overline{M}_{0,r}(X, e_{1} + e_{2}) the associated 1-morphism. There is a canonical combinatorial morphism that is the inclusion of the maximal subgraph of τ that contains only the vertex v_{1}. Denote by α : \overline{M}(X, τ) → \overline{M}_{0,r+1}(X, e_{1}) the associated 1-morphism. Finally, denote by β : \overline{M}_{0,r+1}(X, e_{1}) → \overline{M}_{0,r}(X, e_{1}) the isogeny obtained by removing the tail f_{0}. Observe that ev ∘ β ∘ α is 2-equivalent to ev ∘ ι.

Of course α ∘ ζ = ζ̃_{1} and β ∘ α ∘ ζ = ζ_{1}. Denote by ι ∘ ζ by ζ as well. The hypotheses that ζ_{1} and ζ_{2} are ev-relatively free imply that ι is unramified and is étale locally a regular embedding of codimension 1 at every point of ζ(ℙ^{1}). Denote by ζ^{∗}N_{ι} the locally free sheaf that is the dual of the pullback of the conormal sheaf of ι. The hypotheses on ζ_{1} and ζ_{2} imply that α is unobstructed at every point of ζ(ℙ^{1}) and β is unobstructed at every point of α(ζ(ℙ^{1})). Denote by ζ^{∗}T_{α} the locally free sheaf that is the dual of the pullback of the sheaf of relative differentials of α. Denote by ζ^{∗}α^{∗}T_{β} the locally free sheaf that is the dual of the pullback of the sheaf of relative differentials of β. Finally, ev is unobstructed at every point of β(α(ζ(ℙ^{1}))) because ζ_{1} is ev-relatively free. Denote by ζ^{∗}α^{∗}β^{∗}T_{ev} the locally free sheaf that is the dual of the pullback of the sheaf of relative differentials of ev. It follows that also ev : \overline{M}_{0,r}(X, e_{1} + e_{1}) → X^{r} is unobstructed at every point of ζ(ℙ^{1}). Denote by ζ^{∗}T_{ev} the locally free sheaf that is the dual of the pullback of the sheaf of relative differentials of ev.

There is a filtration of ζ^{∗}T_{ev},

ζ^{∗}T_{ev} = F^{0} ⊃ F^{1} ⊃ F^{2} ⊃ F^{3} ⊃ F^{4} = (0),

with associated graded sheaves F^{0}/F^{1} = ζ^{∗}N_{ι}, F^{1}/F^{2} = ζ^{∗}α^{∗}β^{∗}T_{ev} = ζ_{1}^{∗}T_{ev}, F^{2}/F^{3} = ζ^{∗}α^{∗}T_{β}, and F^{3}/F^{4} = ζ^{∗}T_{α} = ζ̃_{2}^{∗}T_{ev}.

First, ζ^{∗}N_{ι} ≅ O_{ℙ^{1}}(−ζ̃_{1}^{∗}ψ_{0}) ⊗ O_{ℙ^{1}}(−ζ̃_{2}^{∗}ψ). As seen above, −deg(ζ̃_{1}^{∗}ψ_{0}) = (D̃ · D̃)Σ̃_{1} ≥ 0. And −deg(ζ̃_{2}^{∗}ψ) ≥ −deg(ζ_{2}^{∗}ψ) > 0. Therefore ζ^{∗}N_{ι} is an ample invertible sheaf. Next ζ_{1}^{∗}T_{ev} is ample because ζ_{1} is ev-relatively very free. Next, ζ^{∗}α^{∗}T_{β} ≅ σ_{1,0}^{∗}O_{Σ_{1}}(σ_{1,0}(ℙ^{1})) is an ample invertible sheaf of degree (D ·D)_{Σ} > 0. Therefore ζ^{∗}α^{∗}T_{β} is an ample invertible sheaf. Finally, ζ^{∗}T_{α} = ζ̃_{2}^{∗}T_{ev} is ample because ζ̃_{2} is very twisting. Because each of the associated graded sheaves is ample, also ζ^{∗}T_{ev} is ample.

Also ev ∘ ζ = ev ∘ ζ_{1} is free because ζ_{1} is ev-relatively very free. Therefore ζ is ev-relatively very free. Of course ζ maps every point of ℙ^{1} to a stable map with reducible domain. Because ζ_{1} and ζ_{2} are ev-relatively free, each of these stable maps deforms to a stable map with irreducible domain. So by Remark 3.5, a small deformation of ζ maps a general point of ℙ^{1} to a stable map with irreducible domain. Because ev_{1} ∘ ζ = ev_{1} ∘ ζ_{1}, it is nonconstant.

Finally, if r = 1 and ζ_{1} is very twisting, this is Case (i). By construction, the self-intersection of σ̃_{1,1}(ℙ^{1}) ⊂ Σ̃_{1} is 0. Therefore ζ is very twisting and −deg(ζ^{∗}ψ) = 0.

□

A similar argument proves the following result, which shows that if there exists a twisting morphism ℙ^{1} → \overline{M}_{0,2}(X, e), then for every r ≥ 2 and every e′ sufficiently divisible, there exists a twisting morphism ℙ^{1} → \overline{M}_{0,r}(X, e′).

Proposition 3.11. Assume that \overline{M}_{0,0}(X, ϵ) is irreducible for all ϵ sufficiently divisible. Let r ≥ 0, s_{0} ≥ 0, s_{1}, …, s_{r} > 0, and e_{0}, e_{1}, …, e_{r} ≥ 0 be integers. Let ζ_{0} : ℙ^{1} → \overline{M}_{0,s_{0}}(X, e_{0}) and ζ_{i} : ℙ^{1} → \overline{M}_{0,s_{i}}(X, e_{i}) be twisting morphisms for i = 1, …, r. Then there exists a twisting morphism ζ : ℙ^{1} → \overline{M}_{0,s}(X, e) where s = s_{0} +s_{1} +⋯+s_{r} −r and e = e_{0} +e_{1} +⋯+e_{r}.

Sometimes existence of a very twisting morphism ζ : ℙ^{1} → \overline{M}_{0,1}(X, e) implies existence of a twisting morphism ζ′ : ℙ^{1} → \overline{M}_{0,2}(X, e′). This is explained in the following proposition.

Notation 3.12. Let ((π : Σ → ℙ^{1}, σ : ℙ^{1} → Σ), g : Σ → X) be a datum giving rise to a 1-morphism ζ : ℙ^{1} → \overline{M}_{0,1}(X, e). Assume that the geometric generic fiber of π is irreducible, that (σ(ℙ^{1})·σ(ℙ^{1}))_{Σ} = 0, that g∘σ : ℙ^{1} → X is very free, and that g is unramified and étale locally a regular embedding. Denote by N the locally free sheaf on Σ that is the dual of the kernel of dg^{†} : g^{∗}Ω_{X} → Ω_{Σ}. The linear system |σ(ℙ^{1})| is a base-point-free pencil. Denote by ρ : Σ → ℙ^{1} the associated morphism. Denote by τ_{1}, τ_{2} : ℙ^{1} → Σ sections of ρ whose images are general fibers of π. Denote by ζ′ : ℙ^{1} → \overline{M}_{0,2}(X, e′) the 1-morphism associated to the datum ((ρ : Σ → ℙ^{1}, τ_{1}, τ_{2}), g).

Definition 3.13. If ζ satisfies the hypotheses of Notation 3.12, define a rotation of ζ to be the 1-morphism ζ′ associated to any choice of τ_{1} and τ_{2}.

Proposition 3.14. If ζ is very twisting and satisfies the hypotheses of Notation 3.12, then every rotation of ζ is twisting.

Proof. The object L_{(π,g)}^{∨} in the derived category of quasi-coherent sheaves on Σ is represented by the complex,

\begin{matrix}0 & & 1 \\ L_{(π,g)}^{∨} : & π^{∗}T_{ℙ^{1}} & ⟶ & N \end{matrix}

Denote by O_{Σ}(−D) the invertible sheaf O_{Σ}(−σ(ℙ^{1}) − τ_{1}(ℙ^{1}) − τ_{2}(ℙ^{1})). The object L_{(π,g)}^{∨} ⊗ O_{Σ}(−D) fits into a distinguished triangle,

N (−D)[−1] ⟶ L_{(π,g)}^{∨}(−D) ⟶ π^{∗}T_{ℙ^{1}}(−D)[0] ⟶ N (−D)[0]

Now π^{∗}T_{ℙ^{1}}(−D) has vanishing h^{0}, h^{1} and h^{2}, because R^{i}π_{∗}O_{Σ}(−D) is (0) for i = 1, 2. Therefore the hypercohomology of L_{(π,g)}^{∨}(−D) equals the hypercohomology of N (−D)[−1]. Because ζ is very twisting, h^{i}(Σ, N (−D)[−1]) = h^{i}(Σ, L_{(π,g)}^{∨}(−D)) = 0 for i ≥ 2.

The object L_{(}^{∨}ρ, g) in the derived category of quasi-coherent sheaves on Σ is represented by the complex,

\begin{matrix}0 & & 1 \\ L_{(ρ,g)}^{∨} : & ρ^{∗}T_{ℙ^{1}} & ⟶ & N \end{matrix}

And ρ^{∗}T_{ℙ^{1}}(−D) has vanishing h^{0}, h^{1} and h^{2}: ρ_{∗} is zero, and R^{1}ρ^{∗} is O_{ℙ^{1}}(−1). Therefore the hypercohomology of L_{(ρ,g)}^{∨}(−D) equals the hypercohomology of N (−D)[1]. In particular, h^{i}(Σ, L_{(ρ,g)}^{∨}(−D)) = h^{i}(Σ, N (−D)[−1]) = 0 for i ≥ 2.

Consider the object C = Rρ_{∗}L_{(π,g)}^{∨}(−τ_{1}(ℙ^{1}) − τ_{2}(ℙ^{1})). For all i ≥ 3, the cohomology sheaf H^{i}C is (0), i.e., C is quasi-isomorphic to a complex concentrated in degrees ≤ 2. Moreover, because this is a family of stable maps, for all i ≤ 0, the cohomology sheaf H^{i}C is (0), i.e., C is quasi-isomorphic to a complex concentrated in degrees [1, 2].

Because g ∘ σ is very free, ζ′ maps the geometric generic point of ℙ^{1} to an unobstructed point of ev. Therefore H^{2}C is a torsion sheaf, which is just H^{2}Rρ_{∗}L_{(π,g)}^{∨}(−D) ⊗ O_{ℙ^{1}}(1). Because it is torsion, it is isomorphic to H^{2}C ⊗ O_{ℙ^{1}}(−1) = H^{2}Rρ_{∗}L_{(π,g)}^{∨}(−D). There is a Leray spectral sequence computing the hypercohomology of L_{(π,g)}^{∨}(−D) in terms of the hypercohomology of Rρ_{∗}L_{(π,g)}^{∨}(−D). In the spectral sequence, for every differential whose domain is H^{0}(ℙ^{1}, H^{2}Rρ_{∗}L_{(π,g)}^{∨}(−D)), the target vector space is (0). Thus there is an injective homomorphism,

H^{0}(ℙ^{1}, H^{2}Rρ_{∗}L_{(π,g)}^{∨}(−D)) → H^{2}(Σ, L_{(π,g)}^{∨}(−D)).

Since h^{2}(Σ, L_{(π,g)}^{∨}(−D)) = 0, the torsion sheaf H^{2}Rρ_{∗}L_{(π,g)}^{∨}(−D) = (0). So also H^{2}C = (0), which proves that ev is unobstructed at every geomeric point of ζ′(ℙ^{1}). This is (i) of Definition 3.7.

Moreover, C is quasi-isomorphic to the complex concentrated in degree 1, namely (ζ′)^{∗}T_{ev}[−1]. To prove that (ζ′)^{∗}T_{ev} is generated by global sections, it suffices to prove that

h^{1}(ℙ^{1}, (ζ′)^{∗}T_{ev} ⊗ O_{ℙ^{1}}(−1)) = 0.

This equals h^{2}(ℙ^{1}, C ⊗ O_{ℙ^{1}}(−1)). Because C is concentrated in a single degree, the Leray spectral sequence computing hypercohomology of L_{(π,g)}^{∨}(−D) in terms of hypercohomology of C ⊗ O_{ℙ^{1}}(−1) degenerates. In particular,

h^{2}(ℙ^{1}, C ⊗ O_{ℙ^{1}}(−1)) = h^{2}(Σ, L_{(π,g)}^{∨}(−D)) = 0.

Therefore (ζ′)^{∗}T_{ev} is generated by global sections. This is (ii) of Definition 3.7

Finally, (τ_{i}(ℙ^{1}) · τ_{i}(ℙ^{1}))_{Σ} = 0 for i = 1, 2, because τ_{i}(ℙ^{1}) is a fiber of π. Therefore ζ^{∗}ψ_{i} has degree 0 for i = 1, 2. This is (iii) of Definition 3.7. □

4. A very free family of pointed conics

Hypothesis 4.1. In this section, it is assumed that char(K) = 0.

Let X ⊂ ℙ^{n} be a general hypersurface of degree d ≤ n − 2. The following terminology is from [BM96]. Denote by τ the stable A-graph that has two degree 1 vertices v_{1}, v_{2}, and edge joining v_{1} to v_{2}, and two tails f_{1}, f_{2} attached to v_{1} and v_{2} respectively. Denote by \overline{M}(X, τ) the associated Behrend-Manin stack, which is in fact a scheme. This stack parametrizes data ((L_{1}, x_{1}, y), (L_{2}, x_{2}, y)) consisting of lines L_{1}, L_{2} ⊂ X a point y ∈ L_{1} ∩ L_{2}, and points x_{i} ∈ L_{i} (possibly equal to y). There is an evaluation morphism ev : \overline{M}(X, τ) → X^{2} sending a datum to (x_{1}, x_{2}).

Proposition 4.2. The scheme \overline{M}(X, τ) is an integral, projective scheme of dimension 3n − 2d − 1. Every irreducible component of the singular locus \overline{M}(X, τ)_{sing} has dimension ≤ 2n − d − 1.

Proof. Denote by τ_{0} the graph obtained from τ by removing the tails f_{1} and f_{2}. Then \overline{M}(X, τ_{0}) = \overline{M}_{0,1}(X, 1) ×_{X} \overline{M}_{0,1}(X, 1). By [Kol96, Theorem V.4.3], \overline{M}_{0,1}(X, 1) is smooth, projective and irreducible of dimension 2n−d−2. By [HRS04, Thm. 2.1], ev : \overline{M}_{0,1}(X, 1) → X is flat of relative dimension n−d−1, and every geometric fiber is connected. By straightforward computation, the singular locus of ev has codimension ≥ n − d. Therefore pr_{1} : \overline{M}(X, τ_{0}) → \overline{M}_{0,1}(X, 1) is flat of relative dimension n − d − 1, every geometric fiber is connected, and the singular locus of pr_{1} has codimension ≥ n − d (being the preimage under the flat morphism pr_{2} of the singular locus of ev). Thus \overline{M}(X, τ_{0}) is an integral, projective scheme of dimension 3n−2d−3, and the singular locus has dimension ≤ 2n−d−3. The morphism \overline{M}(X, τ) → \overline{M}(X, τ_{0}) is the fiber product of two ℙ^{1}-bundles. □

Remark 4.3. If d = 1 or d = 2, then \overline{M}(X, τ) is smooth.

Proposition 4.4. If n ≥ d^{2} and d ≥ 2, the geometric generic fiber of ev : \overline{M}(X, τ) → X × X is smooth, connected, nonempty and rationally connected of dimension n + 1 − 2d.

Proof. Let (x_{1}, x_{2}) ∈ X×X be a pair contained in no common line contained in X. Consider the morphism ev_{y} : ev^{−1}(x_{1}, x_{2}) → X by ((L, x_{1}, y), (L_{2}, x_{2}, y)) ↦ y. Because x_{1}, x_{2} are contained in no common line contained in X, this morphism is a closed immersion. Denote by Y_{x_{1},x_{2}} ⊂ ℙV the image closed subscheme.

Let V be a vector space of dimension n + 1 such that ℙ^{n} = ℙV. Let L_{1}, L_{2} ⊂ V be 1-dimensional subspaces corresponding to x_{1} and x_{2}. For i = 1, 2 denote by E_{1} the rank 2 locally O_{ℙV} -module E_{i} = (L_{i}^{∨} ⊗O_{ℙV})⊕O_{ℙV} (1). Denote by ϕ_{i} : V^{∨} ⊗ O_{ℙV} → E_{i} the unique sheaf homomorphism such that pr_{1} ∘ ϕ_{i} : V^{∨} ⊗ O_{ℙV} → L_{i}^{∨} ⊗ O_{ℙV} and pr_{2} ∘ ϕ_{i} : V^{∨} ⊗ O_{ℙV} → O_{ℙV} (1) are the canonical surjections. There is an induced sheaf homomorphism,

Sym^{d}ϕ_{i} : Sym^{d}(V^{∨}) ⊗ O_{ℙV} → Sym^{d}(E_{i}) ≅ ⨁_{k=0}^{d}(L_{i}^{∨})^{⊗(d−k)} ⊗ O_{ℙV} (k).

For each k = 0, …, d, denote by ϕ_{i,k} composition of Sym^{d}ϕ_{i} with projection onto the k^{th} direct summand.

Let F ∈ Sym^{d}(V^{∨}) be a defining equation of X. It is straightforward that Y_{x_{1},x_{2}} is the scheme of simultaneous zeroes of ϕ_{i,k}(F) for i = 1, 1 ≤ k ≤ d and i = 2, 1 ≤ k ≤ d − 1. Because n + 1 − 2d ≥ 1, Y_{x_{1},x_{2}} is nonempty and connected, and every irreducible component has dimension ≥ n + 1 − 2d. In particular, this implies that ev : \overline{M}(X, τ) → X^{2} is surjective.

By Proposition 4.2 and Remark 4.3, the dimension of the singular locus of \overline{M}(X, τ) is less than 2n − 2 = dim(X^{2}). Hence the morphism ev is generically smooth of relative dimension n + 1 − 2d. So for (x_{1}, x_{2}) ∈ X^{2} a general pair, Y_{x_{1},x_{2}} is a complete intersection. By the adjunction formula, the dualizing sheaf is the restriction to Y_{x_{1},x_{2}} of the invertible sheaf,

⋀^{n+1}(V^{∨})⊗O_{ℙV} (−n−1)⊗⨂_{k=1}^{d} ((L_{1}^{∨})^{⊗(d−k)} ⊗ O_{ℙV} (k))⊗⨂_{k=1}^{d−1} ((L_{2}^{∨})^{⊗(d−k)} ⊗ O_{ℙV} (k)) .

In other words, ω_{Y} ≅ O_{Y} (−n − 1 + d^{2}). Because n ≥ d^{2}, ω_{Y}^{∨} is ample, i.e., Y_{x_{1},x_{2}} is a Fano manifold. By [KMM92a], [Cam92], Y_{x_{1},x_{2}} is rationally connected. □

Remark 4.5. If n ≥ 3 and d = 1, the proposition is still true. In this case, for any pair of distinct points x_{1}, x_{2} ∈ X, the morphism ev_{y} : ev^{−1}(x_{1}, x_{2}) → X is the blowing up of X at x_{1} and x_{2}.

Proposition 4.6. Let X ⊂ ℙ^{n} be a general hypersurface of degree d. If either n ≥ d^{2} and d ≥ 2 or n ≥ 3 and d = 1, there exists a ev-relatively very free 1-morphism ζ : ℙ^{1} → \overline{M}_{0,2}(X, 2) mapping a general point of ℙ^{1} to a stable map with irreducible domain and with ev_{1} ∘ ζ nonconstant.

Proof. The scheme X^{2} is smooth, projective and rationally connected. Therefore there exists a family of very free rational curves dominating X^{2}, say f : D × ℙ^{1} → X^{2}. For each geometric point t ∈ D, denote by pr_{t} : \overline{M}(X, τ)_{t} → ℙ^{1} the fiber product,

\overline{M}(X, τ)_{t} = ℙ^{1} ×_{f_{t},X^{2},ev} \overline{M}(X, τ).

By Proposition 4.2 and Remark 4.3, the image under ev of the singular locus of \overline{M}(X, τ) has codimension ≥ 2. Therefore a general very free rational curve does not intersect this locus. By generic smoothness, for a general point t ∈ D, \overline{M}(X, τ)_{t} is smooth (although pr_{t} is only generically smooth). By Proposition 4.4 and Remark 4.5, if t is general then the geometric generic fiber of pr_{t} is rationally connected, i.e., pr_{t} is a rationally connected fibration over a curve. By [GHS03], there exists a section σ_{0} of pr_{t}, and σ_{0}(ℙ^{1}) is contained in the smooth locus of pr_{t}. Denote by σ̂_{0} : ℙ^{1} → \overline{M}(X, τ) the composition of σ_{0} with the projection.

Consider the morphism ι : \overline{M}(X, τ) → \overline{M}_{0,2}(X, 2). Every smooth point of ev is an unobstructed point. Every unobstructed point of ev is in the open subset where ι is unramified and étale locally a regular embedding of codimension 1. In particular, the image of σ̂_{0} is in this open set. Denote by N_{ι} the locally free sheaf on this open set that is the normal bundle of ι.

If (x_{1}, x_{2}) ∈ X^{2} is general, ev^{−1}(x_{1}, x_{2}) = Y_{x_{1},x_{2}} is contained in the locus where ι is unramified and étale locally a regular embedding of codimension 1. The restriction of the normal bundle of ι to Y_{x_{1},x_{2}} is O_{ℙV} (2). So for a very free curve in Y_{x_{1},x_{2}}, not only is the relative tangent bundle T_{ev} ample, also N_{ι} is ample. Form a comb whose handle is σ_{0}, and whose teeth are very free curves in fibers of Y_{x_{1},x_{2}}. By [Kol96, II.7.10, II.7.11], after attaching sufficiently many teeth, the comb deforms to a section σ of pr_{t} such that both σ^{∗}T_{ev} is ample and σ^{∗}N_{ι} is ample. Denote by σ̂ : ℙ^{1} → \overline{M}(X, τ) the induced morphism.

To distinguish it from ev : \overline{M}(X, τ) → X^{2}, denote by ev_{a} : \overline{M}_{0,2}(X, 2) → X^{2} the evaluation morphism. On the open set where ι is unramified and étale locally a regular embedding of codimension 1, there is an exact sequence of locally free sheaves,

0 ⟶ T_{ev} ⟶ ι^{∗}T_{ev_{a}} ⟶ N_{ι} ⟶ 0

Therefore, σ^{∗}ι^{∗}T_{ev_{a}} is ample. So ι ∘ σ : ℙ^{1} → \overline{M}_{0,2}(X, 2) is a ev_{a}-relatively very free morphism such that ev_{a} ∘ ισ̂ is nonconstant. Of course a general point of ℙ^{1} is mapped to a stable map with reducible domain. But this stable map deforms to a stable map with irreducible domain. So by Remark 3.5, a small deformation ζ of ι ∘ σ̂ is a ev_{a}-relatively very free morphism mapping a general point of ℙ^{1} to a stable map with irreducible domain and such that ev_{a} ∘ ζ is nonconstant. □

5. Minimal twisting morphisms

Definition 5.1. Let X be a quasi-projective variety with smooth locus X^{0}. A twisting morphism ζ_{0} : ℙ^{1} → \overline{M}_{0,1}(X^{0}, 1) is a minimal twisting morphism if deg(ζ_{0}^{∗}h) = 1 and deg(ζ_{0}^{∗}x) = 2, cf. Notation 2.1. For a base scheme D, a twisting morphism relative to D, ζ_{0} : D × ℙ^{1} → \overline{M}_{0,1}(X^{0}, 1) is a minimal twisting relative to D if the restriction to every geometric point of D is a minimal twisting morphism.

Denote by N ⊂ Hom(ℙ^{1}, \overline{M}_{0,1}(X^{0}, 1)) the locally closed subscheme parametrizing minimal twisting morphisms.

Remark 5.2. (i) In fact N is an open subset.

(ii) Let ((π : Σ → ℙ^{1}, σ), g) be the datum associated to a twisting morphism ζ_{0}. Then ζ_{0} is a minimal twisting morphism iff g : Σ → X is a closed immersion whose image is a smooth quadric surface and g ∘ σ : ℙ^{1} → X is a closed immersion whose image is a line.

(iii) Unless X is ruled by linear spaces over a non-uniruled variety, a minimal twisting morphism is minimal in the following sense: deg(ζ_{0}^{∗}x) ≥ 2 and deg(ζ_{0}^{∗}h) ≥ 1 for every twisting morphism ζ_{0}, cf. [HS05, Rmk. 5.12]. A variety is called quadric type if a minimal twisting family is minimal in this sense.

(iv) A smooth hypersurface is quadric type iff the degree is ≥ 2.

Notation 5.3. Let X be a quasi-projective variety and denote by X^{0} the smooth locus of X. Denote by \overline{M}_{0,1}(X, 1)_{ev} the maximal open subscheme of \overline{M}_{0,1}(X^{0}, 1) on which the obstruction group of ev : \overline{M}_{0,1}(X^{0}, 1) → X^{0} is zero. In other words, a pointed line [L, x] ∈ \overline{M}_{0,1}(X^{0}, 1) is in \overline{M}_{0,1}(X, 1)_{ev} iff T_{X} ⊗ O_{L} is generated by global sections. Denote by T_{ev,X} the locally free sheaf on \overline{M}_{0,1}(X, 1) that is the dual of the sheaf of relative differentials of ev.

By definition, if ζ_{0} : ℙ^{1} → \overline{M}_{0,1}(X, 1) is a twisting family then ζ_{0}(ℙ^{1}) ⊂ \overline{M}_{0,1}(X, 1)_{ev} so that ζ_{0}^{∗}T_{ev,X} is defined. The twisting family ζ_{0} is very twisting iff ζ_{0}^{∗}T_{ev,X} is ample. Typically this is not the case. Let X be of quadric type. Then for every connected component of N there exist nonnegative integers a, b such that ζ_{0}^{∗}T_{ev,X} ≅ O_{ℙ^{1}}^{a} ⊕ O_{ℙ^{1}}(1)^{b}. The family ζ_{0} is very twisting iff b > 0 and a = 0.

Notation 5.4. Let X be a quasi-projective morphism, and denote by X^{0} the smooth locus of X. Let s ∈ ℙ^{1} be a point, and let [L, x] ∈ \overline{M}_{0,1}(X^{0}, 1)_{ev} be a point. Denote by M_{s,[L,x]} the locally closed subscheme of Hom(ℙ^{1}, \overline{M}_{0,1}(X^{0}, 1)) that parametrizes minimal twisting families ζ_{0} : ℙ^{1} → \overline{M}_{0,1}(X^{0}, 1) such that ζ_{0}(s) = [L, x]. Denote by M the locally closed subscheme of ℙ^{1} × \overline{M}_{0,1}(X^{0}, 1)_{ev} × N that parametrizes triples (s, [L, x], ζ_{0}) such that ζ_{0} is a minimal twisting family with ζ_{0}(s) = [L, x], i.e., M is the graph of the evaluation morphism ev : ℙ^{1} × N → \overline{M}_{0,1}(X^{0}, 1)_{ev}.

For every ζ_{0} ∈ M_{s,[L,x]}, ζ_{0}^{∗}T_{ev,X} ⊗ κ(s) equals T. For every connected component M_{s,[L,x],i} of M_{s,[L,x]}, there is a pair of nonnegative integers a, b such that ζ_{0}^{∗}T_{ev,X} ≅ O_{ℙ^{1}}^{a} ⊕ O_{ℙ^{1}}(1)^{b} for every ζ_{0} in M_{s,[L,x],i}. The subbundle of ζ_{0}^{∗}T_{ev,X} spanned by O_{ℙ^{1}}(1)^{b} restricts to a b-dimensional subspace of T. This defines a morphism from M_{s,[L,x],i} to the Grassmannian of T.

Notation 5.5. For each connected component M_{s,[L,x],i} ⊂ M_{s,[L,x]}, denote by q_{s,[L,x],i} : M_{s,[L,x],i} → Grass(b, T) the morphism defined above. The morphism q_{s,[L,x],i} is called spanning if there is no proper subspace T′ ⊂ T such that the image of q_{s,[L,x],i} is contained in Grass(b, T′). Denote by G(T) the disjoint union over all b of Grass(b, T). Denote by q_{s,[L,x]} : M_{s,[L,x]} → G(T) the disjoint union over all connected components M_{s,[L,x],i} of the morphism q_{s,[L,x],i}. The morphism q_{s,[L,x]} is spanning if at least one q_{s,[L,x],i} is spanning.

Lemma 5.6. Let X be a quasi-projective variety of quadric type, and denote by X^{0} the smooth locus of X.

(i) The schemes M and N are smooth, and the projection morphism pr_{1,2} : M → ℙ^{1} × \overline{M}_{0,1}(X^{0}, 1)_{ev} is smooth.

(ii) For each connected component M_{s,[L,x],i} ⊂ M_{s,[L,x]}, the morphism q_{s,[L,x],i} is spanning iff the pullback morphism,

q_{s,[L,x],i}^{∗} : H^{0}(Grass(b, T), O_{Grass}(1)) = T^{∨} → H^{0}(M_{s,[L,x],i}, q_{s,[L,x],i}^{∗}O_{Grass}(1)),

is injective. (iii) There exists an open subset U ⊂ ℙ^{1} ×\overline{M}_{0,1}(X^{0}, 1)_{ev} such that for ev-

ery (s, [L, x]) ∈ ℙ^{1} ×\overline{M}_{0,1}(X^{0}, 1)_{ev}, q_{s,[L,x]} is spanning iff (s, [L, x]) ∈ U.

Proof. (i): By [Kol96, Thm. II.1.7], the projection morphism pr_{1,2} : M → ℙ^{1} × \overline{M}_{0,1}(X^{0}, 1)_{ev} is smooth if,

h^{1}(ℙ^{1}, ζ_{0}^{∗}T_{\overline{M}_{0,1}(X^{0},1)_{ev}} ⊗ O_{ℙ^{1}}(−s)) = 0,

for every geometric point (s, [L_{s}, x_{s}], ζ_{0}) ∈ M. There is a short exact sequence,

0 → ζ_{0}^{∗}T_{ev,X}⊗O_{ℙ^{1}}(−s) → ζ_{0}^{∗}T_{\overline{M}_{0,1}(X^{0},1)_{ev}}⊗O_{ℙ^{1}}(−s) → ζ_{0}^{∗}ev^{∗}T_{X^{0}}⊗O_{ℙ^{1}}(−s) → 0.

Because ev ∘ ζ_{0} : ℙ^{1} → X^{0} is a twisting line, h^{1} of the third term is 0. Because ζ_{0}^{∗}T_{ev,X} is generated by global sections, h^{1} of the first term is 0. Therefore, by the long exact sequence in cohomology, h^{1} of the middle term is 0.

The scheme X^{0} is smooth by definition. The morphism ev : \overline{M}_{0,1}(X^{0}, 1)_{ev} → X^{0} is smooth by definition. Hence \overline{M}_{0,1}(X^{0}, 1)_{ev} is smooth. Hence the product ℙ^{1} × \overline{M}_{0,1}(X^{0}, 1) is smooth. Because pr_{1,2} is smooth, the scheme M is smooth. Because M is the graph of a morphism, projection pr_{1,3} : M → ℙ^{1} × N is an isomorphism. Because ℙ^{1} × N is smooth, the scheme N is smooth.

(ii): This follows from the definition by taking duals of T and T′.

(iii): It is not difficult to see that the set U is constructible. Therefore it suffices to prove it is stable under generization. Let (R, 𝔪) be a DVR containing K, and let s : Spec (R) → ℙ^{1} and [L, x] : Spec (R) → \overline{M}_{0,1}(X, 1)_{ev} be given. Denote by pr_{R} : M_{R} → Spec (R) the fiber product,

M_{R} = Spec (R) ×_{(s,[L,x]),ℙ^{1}×\overline{M}_{0,1}(X,1),pr_{1,2}} M.

Because pr_{1,2} is smooth, also pr_{R} is smooth. After replacing R by a finite unramified cover by a DVR, assume that for every connected component of M_{R}, the geometric generic fiber is also connected. Let M_{R,i} be a connected component such that q_{s,[L,x],i} is spanning.

There is a pullback homomorphism of R-modules,

q_{R}^{∗} : T^{∨} ⊗_{K} R → H^{0}(M_{R,i}, q_{R}^{∗}O_{G(T)}(1)).

Because q_{s,[L,x],i} is spanning, the kernel of the closed fiber of q_{R}^{∗} is injective. Because pr_{R} is smooth, in particular it is flat. Therefore the target R-module is flat, hence torsion-free. So the kernel of q_{R}^{∗} is a saturated submodule of T^{∨}. Because the kernel of the closed fiber is zero, also the kernel of the generic fiber is zero. □

Proposition 5.7. Let X be a quasi-projective variety of quadric type, and denote by X^{0} the smooth locus of X. Denote by U the open set from Lemma 5.6.

(i) If U is nonempty, then there exists a very twisting family ζ′ : ℙ^{1} → \overline{M}_{0,1}(X, 1) such that ev ∘ ζ′ : ℙ^{1} → X^{0} is free.

(ii) If U is nonempty, then for a general deformation Y of X there exists a very twisting family ζ′ : ℙ^{1} → \overline{M}_{0,1}(Y^{0}, 1) such that ev ∘ ζ′ : ℙ^{1} → Y^{0} is free.

Proof. (i): The idea is to construct a map ζ from a comb C to \overline{M}_{0,1}(X, 1) such that ζ deforms to a morphism ζ′ : ℙ^{1} → \overline{M}_{0,1}(X, 1) that is very twisting.

Define C_{0} to be a copy of ℙ^{1}. By hypothesis, there exists a minimal twisting morphism ζ_{0} : C_{0} → \overline{M}_{0,1}(X, 1) such that the graph of ζ_{0} intersects U. There are nonnegative integers a, b such that ζ_{0}^{∗}T_{ev,X} ≅ O_{C_{0}}^{a} ⊕ O_{C_{0}}(1)^{b}. If a = 0, then ζ_{0} is already very twisting. Therefore assume that a > 0.

Let s_{1}, … s_{a} ∈ C_{0} be distinct points and denote by [L_{1}, x_{1}], …, [L_{a}, x_{a}] their images under ζ_{0}. Of course ζ_{0}^{−1}(U) is infinite, so there exist points such that each pair (s_{i}, [L_{i}, x_{i}]) is in U. For each i, denote T_{i} = T_{ev,X} ⊗ κ([L_{i}, x_{i}]). And denote by T′_{i} ⊂ T_{i} the subspace spanned by the image of O_{C_{0}}(1)^{b} under restriction to the fiber at s_{i}. For i = 1, …, a, the quotient vector space T_{i}/T′_{i} is canonically isomorphic to H^{0}(C_{0}, ζ_{0}^{∗}T_{ev,X}/O_{C_{0}}(1)^{b}). Denote this common vector space by T/T′.

Because each of (s_{i}, [L_{i}, x_{i}]) is spanning, for each i = 1, …, a there exists a curve C_{i} that is a copy of ℙ^{1} and a minimal twisting morphism ζ_{i} : C_{i} → \overline{M}_{0,1}(X, 1) that is in M_{s_{i},[L_{i},x_{i}]}, and an invertible subsheaf L_{i} ≅ O_{C_{i}}(1) ⊂ ζ_{i}^{∗}T_{ev,X} such that the images of L_{i} ⊗ κ(s_{i}) in T/T′ span T/T′.

Define (ι_{i} : C_{i} ↪ C)_{0≤i≤a} to be the initial family of morphisms such that for i = 1, …, a, ι_{i}(s_{i}) = ι_{0}(s_{i}). Then C is a connected, proper, nodal curve whose irreducible components are C_{0}, …, C_{a}. Moreover, the dual graph of C is a tree, the vertex of C_{0} has valence a, and for i = 1, …, a the vertex of C_{i} has valence 1 and is connected only to the vertex of C_{0}. The reducible curve C is a comb, the curve C_{0} is the handle, and the curves C_{1}, …, C_{a} are the teeth, cf. [Kol96, Defn. II.7.7].

By the universal property of C, there is a unique morphism ζ : C → \overline{M}_{0,1}(X^{0}, 1)_{ev} such that the restriction to each C_{i} is ζ_{i}. By [HS05, Lem. 4.5], ζ : C → \overline{M}_{0,1}(X^{0}, 1)_{ev} is twisting.

There exists a DVR R containing K and a proper, flat morphism ρ : C → Spec (R) whose closed fiber is C, whose generic fiber is ℙ^{1}, and such that C is regular. Consider the relative Hom scheme,

H_{R} = Hom_{Spec (R)}(C, Spec (R) × \overline{M}_{0,1}(X^{0}, 1)_{ev}).

The morphism ζ determines a point in the closed fiber of H_{R}. Because the restriction of ζ^{∗}T_{\overline{M}_{0,1}(X,1)} to every irreducible component of C is generated by global sections, a leaf induction argument proves that h^{1}(C, ζ^{∗}T_{\overline{M}_{0,1}(X,1)}) = 0. By [Kol96, Thm. II.1.7], the projection morphism H_{R} → Spec (R) is smooth at [ζ]. Therefore, after replacing R by a finite unramified cover by a DVR, there exists a section, i.e., there exists a morphism ζ̃ : C → \overline{M}_{0,1}(X^{0}, 1)_{ev}. Denote by ζ′ : ℙ^{1} → \overline{M}_{0,1}(X^{0}, 1)_{ev} the base-change of ζ̃ over the geometric generic point of Spec (R) (also base change K to the algebraic closure of the fraction field of R).

The claim is that ζ′ is very twisting. By [HS05, Lem. 4.6], ζ′ is twisting. Therefore it only remains to prove that (ζ′)^{∗}T_{ev,X} is ample. Consider the locally free sheaf ζ̃^{∗}T_{ev,X}. Define T ⊂ ζ̃^{∗}T_{ev,X}(C_{1}+⋯+C_{a}) to be the kernel of the surjective sheaf homomorphism,

ζ̃^{∗}T_{ev,X}(C_{1}+⋯+C_{a}) → ⨁_{i=1}^{a} ζ̃^{∗}T_{ev,X}(C_{1}+⋯+C_{a})|_{C_{i}}/L_{i}⊗O_{C}(C_{1}+⋯+C_{a})|_{C_{i}}.

In other words T is the elementary transform up of ζ̃^{∗}T_{ev,X} along C_{1}, …, C_{a} determined by L_{1}, …, L_{a}.

The sheaf T is locally free, and contains ζ̃^{∗}T_{ev,X} as a subsheaf. The restriction of T to C_{0} is the sheaf of meromorphic sections of ζ_{0}^{∗}T_{ev,X} that have simple poles in the direction of L_{i} ⊗ κ(s_{i}) for each i = 1, …, a. By construction, this sheaf is isomorphic to O_{C_{0}}(1)^{a+b}. Also the restriction of T to C_{i} fits into a short exact sequence,

0 ⟶ L_{i} ⟶ ζ_{i}^{∗}T_{ev,X} ⟶ T |_{C_{i}} ⟶ L_{i}(−s_{i}) ⟶ 0.

In particular, T |_{C_{i}} is generated by global sections. Therefore, by [HS05, Lem 2.11], the sheaf T |_{C} is deformation ample. By [HS05, Lem. 2.9], the sheaf T on C is relatively deformation ample over Spec (R). By [HS05, Lem. 2.8], the restriction of T to the geometric generic fiber of C is ample. But, of course, the restriction of T to the geometric generic fiber is (ζ′)^{∗}T_{ev,X}. Therefore ζ′ is very twisting. Also, because ζ^{∗}ev^{∗}T_{X} is generated by global sections, also (ζ′)^{∗}ev^{∗}T_{X} is generated by global sections, i.e., ev ∘ ζ′ : ℙ^{1} → X^{0} is free.

(ii): Because ev ∘ ζ′ : ℙ^{1} → X^{0} is free, it is a point at which the relative Kontsevich moduli space is smooth over the base of the deformation. Therefore ev ∘ ζ′ deforms to Y. By [HS05, Prop. 4.8], the very twisting morphism ζ′ also deforms to Y. □

Because of Proposition 5.7, to prove Theorem 1.1 it suffices to prove that there exists a hypersurface X, a point s, and a point [L, x] ∈ \overline{M}_{0,1}(X^{0}, 1) such that q_{s,[L,x]} is spanning. The boundary case is when n = d^{2}. This case is the most difficult, and implies the result for all n ≥ d^{2}. In this case the subspace of T has dimension 1, i.e., q_{s,[L,x]} is a morphism M_{s,[L,x]}E → ℙT. To prove that q_{s,[L,x]} is spanning, it suffices to prove there exists an element ζ_{0} ∈ M_{s,[L,x]} at which q_{s,[L,x]} is smooth – this will even prove that the image of q_{s,[L,x]} contains a dense open subset of ℙT. To see q_{s,[L,x]} is smooth at ζ_{0}, it suffices to prove that the derivative of q_{s,[L,x]} at ζ_{0} is surjective. This is a deformation theory computation that is the heart of this note.

Before proceeding to this computation, the following proposition shows that the inequality n ≥ d^{2} in Theorem 1.1 is necessary.

Proposition 5.8. Let K be an algebraically closed field of characteristic 0. Let (d, n) be a pair of positive integers such that d ≤ \frac{n}{2}. Let X ⊂ ℙ^{n} be a hypersurface of degree d, and denote by X^{0} the smooth locus of X. If there exists a very twisting morphism ζ : ℙ^{1} → \overline{M}_{0,1}(X^{0}, 1), then n ≥ d^{2}.

Proof. By [HS05, Defn. 4.3 (ii)], the stable map ev ∘ ζ : ℙ^{1} → X^{0} is unobstructed. This implies that the relative Kontsevich moduli space of the universal family of degree d hypersurfaces over ℙSym^{d}(K^{n+1}) is smooth over ℙSym^{d}(K^{n+1}) at ([X], [ev∘ζ]). So for a general deformation of X, the stable map deforms as well. By [HS05, Prop. 4.8], also the very twisting family deforms. So for a general hypersurface, there is also a very twisting morphism. Therefore assume that X is general (this is made precise in the next paragraph).

By [HRS04], if X is general then every Kontsevich moduli space is irreducible. Also, for every degree e there is a free rational curve of degree e: because d ≤ n there is a free line L on X, and a finite degree e morphism ℙ^{1} → L is free. The locus of free rational curves is open. Therefore, a general point of the Kontsevich moduli space parametrizes a free rational curve. By the same argument in the last paragraph, ζ can be deformed so that ev ∘ ζ : ℙ^{1} → X is a free rational curve.

Consider the Picard group of \overline{M}_{0,1}(ℙ^{n}, 1). There is the projection morphism pr : \overline{M}_{0,1}(ℙ^{n}, 1) → \overline{M}_{0,0}(ℙ^{n}, 1) = Grass(2, n + 1). And there is the projection morphism ev : \overline{M}_{0,1}(ℙ^{n}, 1) → ℙ^{n}. Denote by x the first Chern class of pr^{∗}O_{Grass}(1), and denote by h the first Chern class of ev^{∗}O_{ℙ^{n}}(1). For \overline{M}_{0,1}(ℙ^{n}, 1), Chow^{1} = ℤ\{x, h\}. There is a tautological class ψ = x − 2h. For any family of pointed lines parametrized by a base B, say ζ_{ℙ^{n}} = (π : Σ → B, σ : B → Σ, g : Σ → ℙ^{n}), C_{1}(σ^{∗}O_{Σ}(σ(B))) = −ζ^{∗}ψ.

It is straightforward to compute that,

C_{1}(T_{ev,ℙ^{n}}) = nx − (n − 1)h = (n + 1)x + nψ.

Denote the universal family of pointed lines by,

ζ̃ = (π̃ : Σ̃ → \overline{M}_{0,1}(ℙ^{n}, 1), σ̃ : \overline{M}_{0,1}(ℙ^{n}, 1) → Σ̃, g̃: Σ̃ → ℙ^{n}).

It is straightforward to compute that,

C_{1}(π̃_{∗}(g^{∗}O_{ℙ^{n}}(d) ⊗ OΣ̃(−Image(σ̃)))) = \frac{d(d + 1)}{2}x − dh = d^{2}h + \frac{d(d + 1)}{2}ψ.

Therefore, if ζ_{X} : B → \overline{M}_{0,1}(X, 1)_{ev} is a morphism and ζ_{ℙ^{n}} is the associated morphism to \overline{M}_{0,1}(ℙ^{n}, 1), then,

\begin{matrix}C_{1}(ζ_{X}^{∗}T_{ev,X}) = C_{1}(ζ_{ℙ^{n}}^{∗}T_{ev,ℙ^{n}}) − C_{1}(π_{∗}(g^{∗}O_{ℙ^{n}}(−d) ⊗ O_{Σ}(−σ(B)))) = \\ (n − \frac{d(d+1)}{2})ζ^{∗}x − (n − d − 1)ζ^{∗}h = (n + 1 − d^{2})ζ^{∗}h + (n − \frac{d(d+1)}{2})ζ^{∗}ψ. \end{matrix}

In particular, this holds for the very twisting family ζ : ℙ^{1} → \overline{M}_{0,1}(X, 1).

There exists a short exact sequence,

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

Because ζ is very twisting, the first term is positive. Because ev∘ζ is free, the third term is semipositive. Therefore ζ^{∗}T_{\overline{M}_{0,1}(X,1)} is semipositive. Because ζ^{∗}pr^{∗}T_{\overline{M}_{0,0}(X,1)} is a quotient of ζ^{∗}T_{\overline{M}_{0,1}(X,1)}, it is also semipositive. The derivative d(pr ∘ ζ) : O_{ℙ^{1}}(2) → ζ^{∗}pr^{∗}T_{\overline{M}_{0,0}(X,1)} is zero iff pr ∘ ζ is constant. Therefore either C_{1}(ζ^{∗}T_{\overline{M}_{0,1}(X,1)}) has positive degree or pr ∘ ζ is constant.

The scheme \overline{M}_{0,1}(X, 1) is simply the Fano scheme of lines on X. The canonical bundle of the Fano scheme is straightforward to compute, giving,

pr^{∗}C_{1}(T_{\overline{M}_{0,0}(X,1)}) = (n + 1 − \frac{d(d + 1)}{2})x.

So either n + 1 > \frac{d(d+1)}{2}, or else pr ∘ ζ is constant. In the second case deg(ζ^{∗}x) = 0, so that,

deg(C_{1}(ζ^{∗}T_{ev,X})) = −(n − d − 1)deg(ζ^{∗}h).

Since d ≤ n − 1 this is nonpositive, contradicting that ζ^{∗}T_{ev,X} is ample. Therefore n + 1 > \frac{d(d+1)}{2}, i.e. n − \frac{d(d+1)}{2} ≥ 0.

Because ζ is very twisting, σ^{∗}O_{Σ}(σ(ℙ^{1})) has nonnegative degree, i.e., deg(ζ^{∗}ψ) ≤

0. So (n − \frac{d(d+1)}{2})deg(ζ^{∗}ψ) ≤ 0. Therefore,

deg(C_{1}(ζ^{∗}T_{ev,X})) ≤ (n + 1 − d^{2})deg(ζ^{∗}h).

Because ζ^{∗}T_{ev,X} is ample, (n + 1 − d^{2})deg(ζ^{∗}h) > 0. And deg(ζ^{∗}h) ≥ 0. Therefore n + 1 − d^{2} > 0, i.e., n ≥ d^{2}. □

6. Notation

In this section, the notation for the computation is introduced. A specific homogeneous polynomial G of degree d is given, and the hypersurface X is V (G) ⊂ ℙ^{n}. A minimal twisting family is specified,

ζ_{0} = (π : Σ → B, σ : B → Σ, f_{0} : Σ → X).

In a later section it is proved that ζ_{0} is a smooth point of Hom(ℙ^{1}, \overline{M}_{0,1}(X, 1)). An Artin local scheme D is specified that is a closed subscheme of the firstorder neighborhood of [ζ_{0}] in Hom(ℙ^{1}, \overline{M}_{0,1}(X, 1)). The family ζ_{0} is “thickened” to a family,

ζ = ((1, π) : D × Σ → D × B, (1, σ) : D × B → D × Σ, f : D × Σ → X).

For each closed point s ∈ B, the pointed line ζ_{0}(s) is denoted by [L_{s}, x_{s}], and the scheme M_{s,[L_{s},x_{s}]} from Notation 5.4 is denoted by M_{s}. Similarly, the morphism q_{s,[L_{s},x_{s}]} from Notation 5.5 is denote by q_{s}. The subscheme D ∩ M_{s} ⊂ D is denoted by D_{s}. It turns out that to prove dq_{s}|_{ζ_{0}} is surjective, it suffices to restrict to the Zariski tangent space T_{0}D_{s} ⊂ T_{[ζ_{0}]}M_{s}. This restriction is denoted d′q_{s}.

Let K be an algebraically closed field with char(K) ≠ 2. Let d ≥ 3 be an integer and denote n = d^{2}. Denote by I_{d} the set with d^{2} − 4 elements,

{(i, j) ∈ ℤ^{2}|0 ≤ i, j ≤ d − 1, (i, j) ≠ (0, 0), (0, 1), (1, 0), (1, 1)}.

Denote by V the (n + 1)-dimensional K-vector space with ordered basis,

(a_{0}, a_{1}, a_{2}, a_{3}) ∪ (b_{(i,j)}|(i, j) ∈ I_{d}) ∪ (c).

Denote the dual ordered basis of V^{∨} by,

(X_{0}, X_{1}, X_{2}, X_{3}) ∪ (Y_{(i,j)}|(i, j) ∈ I_{d}) ∪ (Z).

Denote by V_{a} the subspace of V generated by a_{0}, …, a_{3}.

Given a pair of nonnegative integers, i and j, denote by k = k(i, j) the minimum. Denote by G ∈ Sym^{d}(V^{)} the homogeneous polynomial,

G = (X_{0}X_{3}−X_{1}X_{2})X_{3}^{d−2}+ ∑_{(i,j)∈I_{d}} X_{0}^{k}X_{1}^{i−k}X_{2}^{j−k}X_{3}^{d−1−i−j+k}Y_{(i,j)}+ ∑_{(i,j)∈I_{d}} Y_{(i,j)}Zγ_{(i,j)}+Z^{2}γ_{z},

where the elements γ_{(i,j)} and γ_{z} are homogeneous polynomials of degree d − 2 in the variables X_{0}, X_{1}, X_{2}, X_{3}. Denote by X ⊂ ℙV the hypersurface defined by G.

Denote by Σ the surface,

ℙ_{s}^{1} × ℙ_{t}^{1} = {([S_{0} : S_{1}], [T_{0} : T_{1}])|[S_{0} : S_{1}], [T_{0} : T_{1}] ∈ ℙ^{1}}.

For each pair of integers (i, j), denote by O_{Σ}(i, j) the invertible sheaf pr_{ℙ_{s}^{1}}^{∗}O_{ℙ_{s}^{1}}(i)⊗ pr_{ℙ_{t}^{1}}^{∗}O_{ℙ_{t}^{1}}(j).

Denote by B the curve ℙ_{s}^{1}. Denote by π : Σ → B projection onto the first factor. Denote by σ : B → Σ the section of π,

σ([S_{0} : S_{1}]) = ([S_{0} : S_{1}], [1 : 0]).

Denote by E the K-vector space with ordered basis,

(u_{(i,j)}^{0}, u_{(i,j)}^{1}, u_{(i,j)}^{2}, u_{(i,j)}^{3}|(i, j) ∈ I_{d}) ∪ (v^{0}, v^{1}, v^{2}, v^{3}).

Denote by D′ the local, Artin K-scheme,

D′ = Spec Sym^{•}(E)/Sym^{2}(E) · Sym^{•}(E).

There is a sheaf homomorphism, V^{∨} ⊗_{K} O_{D′} → V_{a}^{∨} ⊗_{K} O_{D′} by,

\{ \begin{matrix}X_{l} & ↦ & X_{l}, & l = 0, 1, 2, 3 \\ Y_{(i,j)} & ↦ & ∑_{l=0}^{3} u_{(i,j)}^{l}X_{l}, & (i, j) ∈ I_{d} \\ Z & ↦ & ∑_{l} v^{l}X_{l} \end{matrix}

This induces a morphism of schemes, ϕ′ : D′×V_{a} → D′×V. There is a unique morphism of schemes ψ′ : D′ × Σ → D′ × ℙV_{a} such that (ψ′)^{∗}O_{ℙV_{a}}(1) = O_{Σ}(1, 1) and such that,

X_{0} ↦ S_{0}T_{0}, X_{1} ↦ S_{0}T_{1}, X_{2} ↦ S_{1}T_{0}, X_{3} ↦ S_{1}T_{1}.

Define f′ : D′ × Σ → D′ × ℙV to be the composition f′ = ϕ′ ∘ ψ′.

If (i, j) is a pair that is not in I_{d}, define u_{(i,j)}^{l} to be 0. Denote by D ⊂ D′ the closed subscheme whose ideal sheaf is,

I(D) = ⟨u_{(i−1,j−1)}^{0} + u_{(i−1,j)}^{1} + u_{(i,j−1)}^{2} + u_{(i,j)}^{3}|0 ≤ i, j ≤ d⟩.

Define ϕ to be the restriction of ϕ′ to D × V, define ψ to be the restriction of ψ′ to D × Σ, and define f to be the composition f = ϕ ∘ ψ. Define ϕ_{0}, ψ_{0} and f_{0} to be the restrictions to the closed point 0 ∈ D.

Lemma 6.1. The preimage f^{∗}G in H^{0}(D × Σ, f^{∗}O_{ℙV} (d)) equals 0. Therefore the morphism f factors through D × X ⊂ D × ℙV.

Proof. For every (i, j) ∈ I_{d}, f^{∗}Y_{(i,j)} ∈ 𝔪 · H^{0}(D × Σ, f^{∗}O_{ℙV} (d)). Similarly for f^{∗}Z. Because 𝔪^{2} = 0, this implies that every term that involves Y_{(i,j)}Z or Z^{2} pulls back to 0. So these terms are ignored. Without these terms the preimage of f^{∗}G is,

\begin{matrix}(S_{0}T_{0} · S_{1}T_{1} − S_{0}T_{1} · S_{1}T_{0})(S_{1}T_{1})^{d−2}+ \\ ∑_{(i,j)∈I_{d}} S_{0}^{i}S_{1}^{d−1−i}T_{0}^{j}T_{1}^{d−1−j}(u_{(i,j)}^{0}S_{0}T_{0} + u_{(i,j)}^{1}S_{0}T_{1} + u_{(i,j)}^{2}S_{1}T_{0} + u_{(i,j)}^{3}S_{1}T_{1}). \end{matrix}

The first term is 0 because S_{0}T_{0} · S_{1}T_{1} = S_{0}T_{1} · S_{1}T_{0}. Gathering like monomials S_{0}^{i}S_{1}^{d−i}T_{0}^{j}T_{1}^{d−j}, and using the notation that u_{(i,j)}^{l} = 0 if (i, j) ∉ I_{d}, the remaining terms sum to,

∑_{0≤i,j≤d}(u_{(i−1,j−1)}^{0} + u_{(i−1,j)}^{1} + u_{(i,j−1)}^{2} + u_{(i,j)}^{3})S_{0}^{i}S_{1}^{d−i}T_{0}^{j}T_{1}^{d−j}.

By definition, each element u_{(i−1,j−1)}^{0} + u_{(i−1,j)}^{1} + u_{(i,j−1)}^{2} + u_{(i,j)}^{3} equals 0 in O_{D}. Therefore f^{∗}G equals 0. □

Denote,

ζ_{ℙV} = ((1, π) : D × Σ → D × B, (1, σ) : D × B → D × Σ, f : Σ → ℙV), ζ_{X} = ((1, π) : D × Σ → D × B, (1, σ) : D × B → D × Σ, f : Σ → X),

ζ_{ℙV,0} = (π : Σ → B, σ : B → Σ, f_{0} : Σ → ℙV),

ζ_{X,0} = (π : Σ → B, σ : B → Σ, f_{0} : Σ → X)

For each s = [s_{0} : s_{1}] ∈ ℙ_{s}^{1}, denote by D_{s} ⊂ D the closed subscheme whose ideal is,

I(D_{s}) = ⟨s_{0}u_{(i,j)}^{0}+s_{1}u_{(i,j)}^{2}, s_{0}u_{(i,j)}^{1}+s_{1}u_{(i,j)}^{3}|(i, j) ∈ I_{d}⟩+⟨s_{0}v^{0}+s_{1}v^{2}, s_{0}v^{1}+s_{1}v^{3}⟩.

Denote by Λ_{s} ⊂ V_{a} ⊂ V the subspace,

span{s_{0}a_{0} + s_{1}a_{2}, s_{0}a_{1} + s_{1}a_{3}}.

Denote L_{s} = ℙΛ_{s} ⊂ ℙV, and denote x_{s} ∈ L_{s} the point [s_{0}a_{0} + s_{1}a_{2}] ∈ L_{s}.

Lemma 6.2. The closed subscheme D_{s} ⊂ D is the maximal subscheme over which f : D_{s} × π^{−1}(s) → D_{s} × ℙV factors through D_{s} × L_{s} ⊂ D_{s} × ℙV.

Proof. The image of D × π^{−1}(s) under ϕ is V_{s}, where V_{s} ⊂ D × V is the vector subbundle with generators,

s_{0}a_{0} + s_{1}a_{2} + ∑_{(i,j)∈I_{d}}(s_{0}u_{(i,j)}^{0} + s_{1}u_{(i,j)}^{2})b_{(i,j)} + (s_{0}v^{0} + s_{1}v^{2})c,
s_{0}a_{1} + s_{1}a_{3} + ∑_{(i,j)∈I_{d}}(s_{0}u_{(i,j)}^{1} + s_{1}u_{(i,j)}^{3})b_{(i,j)} + (s_{0}v^{1} + s_{1}v^{3})c

Therefore the maximal closed subscheme D_{s} such that D_{s} ×_{D} V_{s} is contained in D_{s} × Λ_{s} is the closed subscheme where the coefficients of b_{(i,j)} and c in the generators equal 0, i.e., the closed subscheme D_{s} defined above. □

7. Description of some coherent sheaves

Denote by ev : \overline{M}_{0,1}(ℙV, 1) → ℙV the evaluation morphism. Denote by T_{ev,ℙV} the locally free sheaf on \overline{M}_{0,1}(ℙV, 1) that is the dual of the sheaf of relative differentials of ev. Denote by ev : \overline{M}_{0,1}(X, 1) → X the evaluation morphism and denote by \overline{M}_{0,1}(X, 1)_{ev} ⊂ \overline{M}_{0,1}(X, 1) the open subscheme where ev is smooth. Denote by T_{ev,X} the locally free sheaf on \overline{M}_{0,1}(X, 1)_{ev} that is the dual of the sheaf of relative differentials of ev.

The families ζ_{ℙV} and ζ_{X} in the last section define morphisms ζ_{ℙV} : D ×B → \overline{M}_{0,1}(ℙV, 1) and ζ_{X} : D × B → \overline{M}_{0,1}(X, 1). In this section it is proved that the image of ζ_{X} is contained in \overline{M}_{0,1}(X, 1)_{ev}, and explicit descriptions are given for ζ_{ℙV}^{∗} T_{ev,ℙV} and ζ_{X}^{∗}T_{ev,X}. This description will be the basis for the computation of dq_{s} in the next section.

Denote by ψ̃ : D × Σ → D × B × ℙV_{a} the unique morphism such that pr_{1,3} ∘ ψ̃ = ψ and pr_{2} ∘ ψ̃ = π ∘ pr_{Σ}. Similarly, denote by f̃ : D × Σ → D×B×ℙV the unique morphism such that pr_{1,3}∘f̃ = f and pr_{2}∘f̃ = π∘pr_{Σ}. Denote by ψ̃: D × B × ℙV_{a} → D × B × ℙV the unique morphism such that pr_{1,3} ∘ ψ̃ = ψ ∘ pr_{1,3} and pr_{2} ∘ ψ̃ = pr_{2}.

Denote by Nϕ̃ the normal bundle of the closed immersion ϕ̃. Denote by

Nf̃ the normal bundle of the closed immersion f̃. There is a short exact sequence of locally free sheaves on D × Σ,

0 → Nϕ̃ ⟶^{dψ̃} Nf̃ → V ⊗_{K} f^{∗}O_{ℙV} (1)/ψ(V_{a}) · f^{∗}O_{ℙV} (1) → 0.

Using the Euler exact sequence for the tangent bundle of ℙV and the tangent bundle of ℙ_{t}^{1}, the locally free sheaf Nϕ̃ is the cokernel of the sheaf homomorphism,

pr_{Σ}^{∗}O_{Σ}(0, 1){t_{0}^{∨}, t_{1}^{∨}} → pr_{Σ}^{∗}O_{Σ}(1, 1){a_{0}, a_{1}, a_{2}, a_{3}},

defined by,

t_{0}^{∨} ↦ s_{0}a_{0} + s_{1}a_{2}, t_{1}^{∨} ↦ s_{0}a_{1} + s_{1}a_{3}.

The cokernel is the sheaf homomorphism,

pr_{Σ}^{∗}O_{Σ}(1, 1){a_{0}, a_{1}, a_{2}, a_{3}} → pr_{Σ}^{∗}O_{Σ}(2, 1){e_{0}, e_{1}},

defined by,

a_{0} ↦ s_{1}e_{0}, a_{1} ↦ s_{1}e_{1}, a_{2} ↦ −s_{0}e_{0}, a_{3} ↦ −s_{0}e_{1}.

This cokernel is identifed with Nϕ̃.

There is a canonical surjective sheaf homomorphism f^{∗}T_{ℙV} → Nf̃. Moreover, by the Euler exact sequence for ℙV, there is a canonical sheaf homomorphism V ⊗_{K} O_{ℙV} (1) → T_{ℙV}. Together, these give a canonical sheaf homomorphism V ⊗_{K} pr_{Σ}O_{Σ}(1, 1) → Nf̃. For each element v of V, also

\begin{matrix}Element & ∂G_{0} & ∂G_{𝔪} \\ c & 0 & ∑_{l}(2v^{l}γ_{z} + ∑_{(i,j)} u_{(i,j)}^{l}γ_{(i,j)})f^{∗}X_{l} \\ b_{(i,j)} & S_{0}^{i}S_{1}^{d−1−i} · T_{0}^{j}T_{1}^{d−1−j} & − \\ a_{0} & S_{1}^{d−1} · T_{1}^{d−1} & − \\ a_{1} & −S_{1}^{d−1} · T_{0}T_{1}^{d−2} & − \\ a_{2} & −S_{0}S_{1}^{d−2} · T_{1}^{d−1} & − \\ a_{3} & S_{0}S_{1}^{d−2} · T_{0}T_{1}^{d−2} & − \\ e_{0} & S_{1}^{d−2} · T_{1}^{d−2} & − \\ e_{1} & −S_{1}^{d−2} · T_{0}T_{1}^{d−2} & − \end{matrix}

Figure 1. The homomorphism ∂G

denote by v the induced sheaf homomorphism pr_{Σ}^{∗}O_{Σ}(1, 1) → Nf̃, and the induced global section of Nf̃ ⊗ pr_{Σ}^{∗}O_{Σ}(−1, −1).

In particular there are global sections of Nf̃ ⊗ pr_{Σ}^{∗}O_{Σ}(−1, −1): c and b_{(i,j)},

for (i, j) ∈ I_{d}. The images of these global sections in V ⊗_{K} O_{D×Σ}/f̃(V_{a}) give a basis as a free O_{D×Σ}-module. With the isomorphism from the last paragraph this defines an isomorphism,

Nf̃ ≅ pr_{Σ}^{∗}O_{Σ}(2, 1){e_{0}, e_{2}} ⊕ pr_{Σ}^{∗}O_{Σ}(1, 1){b_{(i,j)}|(i, j) ∈ I_{d}} ⊕ pr_{Σ}^{∗}O_{Σ}(1, 1){c}.

The locally free sheaf ζ_{ℙV}^{∗} T_{ev,ℙV} is canonically isomorphic to (1×π)_{∗}Nf̃(−D× σ(B)), cf. [HS05, Sec. 3]. Therefore,

ζ_{ℙV}^{∗} T_{ev,ℙV} ≅ pr_{B}^{∗}O_{B}(2){e_{0}, e_{2}}⊕pr_{B}^{∗}O_{B}(1){b_{(i,j)}|(i, j) ∈ I_{d}}⊕pr_{B}^{∗}O_{B}(1){c}.

The morphism T_{ℙ_{V}} |_{X} → N_{X/ℙV} induces a sheaf homomorphism,

N_{f}(−D × σ(B)) → f^{∗}(O_{ℙV} (d))(−D × σ(B)).

Applying (1 × π)_{∗}, this leads to a sheaf homomorphism,

∂G : ζ_{ℙV}^{∗} T_{ev,ℙV} → pr_{B}^{∗}O_{B}(d){t_{0}^{j}t_{1}^{d−1−j}|0 ≤ j ≤ d − 1}.

By the deformation theory of stable maps, this 2-term complex is quasiisomorphic to the good [0, 1] truncation of ℝHom of the pullback by ζ_{X} of the relative cotangent complex of ev : \overline{M}_{0,1}(X, 1) → X. In particular, the image of ζ_{X} is contained in \overline{M}_{0,1}(X, 1)_{ev} iff ∂G is a surjective sheaf homomorphism. And in this case, ζ_{X}^{∗}T_{ev,X} is isomorphic to the kernel of ∂G, cf. [HS05, Rmk. 4.4].

There is a canonical splitting O_{D} ≅ K ⊕ 𝔪O_{D}. The domain and target of ∂G are pullbacks under pr_{B} of locally free sheaves on B. Therefore ∂G has a canonical splitting ∂G = ∂G_{0} + ∂G_{𝔪}. By Nakayama’s lemma, to prove that ∂G is surjective, it suffices to prove that ∂G_{0} is surjective after reducing modulo 𝔪. The homomorphism ∂G is computed in Figure 1.

Some explanation of Figure 1 is in order. The map being described in the first 6 rows is the map on global sections induced by the sheaf homomorphism V ⊗_{K} O_{D×B} → pr_{B}^{∗}O_{B}(d − 1){T_{0}^{j}T_{1}^{d−1−j}|0 ≤ j ≤ d − 1} obtained by composing ∂G with the canonical surjection V ⊗_{K} pr_{B}^{∗}O_{B}(1) → ζ_{ℙV}^{∗} T_{ev,ℙV} described above, and then twisting by O_{B}(−1). Each of these rows is obtained by taking the partial derivative of G with respect to the corresponding dual coordinate, and the pulling back by f^{∗}.

The last two rows are the images of e_{0} and e_{1} under the map on global section obtained from ∂G by twisting by O_{B}(−2). The images are uniquely determined by the images of the a_{l}s together with the relation between e_{0}, e_{1} and the a_{l}s.

The images of b_{(i,j)} and e_{0}, e_{1} under ∂G_{𝔪} are not given because they will not be used. The image of c will be used.

Proposition 7.1. (i) The sheaf homomorphism ∂G is surjective. Moreover, the sheaf homomorphism obtained by twisting by pr_{B}^{∗}O_{B}(−1) is surjective on global sections.

(ii) The image of ζ_{X} is contained in \overline{M}_{0,1}(X, 1)_{ev}, and ζ_{X}^{∗}T_{ev,X} is gen-

erated by global sections. (iii) The morphism ζ_{X} : D × B → \overline{M}_{0,1}(X, 1) is a minimal twisting

family parametrized by D. (iv) For every s ∈ B, the restriction of ζ to D_{s} defines a morphism from

D_{s} to M_{s}.

Proof. (i): Because pr_{B}^{∗}O_{B}(d){T_{0}^{j}T_{1}^{d−1−j}|0 ≤ j ≤ d − 1} is generated by global sections even after twisting by pr_{B}^{∗}O_{B}(−1), it suffices to prove the sheaf homomorphism obtained from ∂G after twisting by pr_{B}^{∗}O_{B}(−1) is surjective on global sections. Every monomial S_{0}^{i}S_{1}^{d−1−i}T_{0}^{j}T_{1}^{d−1−j} in H^{0}(B, O_{B}(d − 1)){T_{0}^{j}T_{1}^{d−1−j}|0 ≤ j ≤ d − 1} occurs as the image under ∂G_{0} of either b_{(i,j)}, if (i, j) ∈ I_{d}, or of S_{0}e_{0}, S_{1}e_{0}, S_{0}e_{1} or S_{1}e_{1}.

(ii): As discussed above, because ∂G is surjective the image of ζ_{X} is contained in \overline{M}_{0,1}(X, 1)_{ev} and ζ_{X}^{∗}T_{ev,X} is the kernel of ∂G. Because ∂G is surjective on global sections after twisting by pr_{B}^{∗}O_{B}(−1), H^{1}(D ×B, ζ_{X}^{∗}T_{ev,X}) is a subspace of H^{1}(D × B, ζ_{ℙV}^{∗} T_{ev,ℙV}). But this is zero. Therefore ζ_{X}^{∗}T_{ev,X} is generated by global sections.

(iii): By (ii), the image of ζ_{X} is contained in \overline{M}_{0,1}(X, 1)_{ev}, and the pullback of T_{ev,X} is generated by global sections. By construction, f : D×Σ → D×X is a closed immersion whose image is a family of smooth quadric surfaces. Also f ∘ (1, σ) : D × B → D × X is a closed immersion whose image is a family of lines.

The only thing left to check is that the image of f ∘ (1, σ) : D × B → D × X is a family of free lines. The computation so far is symmetric with respect to the involution that permutes ℙ_{s}^{1} and ℙ_{t}^{1}. By (ii), every fiber of pr_{ℙ_{s}^{1}} :

ℙ_{s}^{1} × ℙ_{t}^{1} → ℙ_{s}^{1} is mapped under f to a free line. So also every fiber of pr_{ℙ_{t}^{1}} is mapped to a free line. Since the image of σ is the fiber over [1 : 0] ∈ ℙ_{t}^{1}, f ∘ (1, σ) : D × ℙ_{s}^{1} → D × X maps to a family of free lines.

(iv): This follows from (iii) and Lemma 6.2. □

Notation 7.2. A kernel of ∂G is,

ι : pr_{B}^{∗}O_{B}{f_{(i,j)}|(i, j) ∈ I_{d}, i ≤ d−2}⊕pr_{B}^{∗}O_{B}{g_{0}, g_{1}}⊕pr_{B}^{∗}O_{B}(1){h} → ζ_{ℙV}^{∗} T_{ev,ℙV} ,

defined by,

\{ \begin{matrix}f_{(i,j)} & ↦ & S_{0}b_{(i,j)} − S_{1}b_{(i+1,j)} & + & f_{(i,j),𝔪} \\ g_{0} & ↦ & S_{0}^{2}e_{0} − S_{1}b_{(2,0)} & + & g_{0,𝔪} \\ g_{1} & ↦ & S_{0}^{2}e_{1} + S_{1}b_{(2,1)} & + & g_{1,𝔪} \\ h & ↦ & c & + & h_{𝔪} \end{matrix} ,

for some choice of

f_{(i,j),𝔪}, g_{l,𝔪} ∈ 𝔪·H^{0}(D×B, ζ_{ℙV}^{∗} T_{ev,ℙV}), and h_{𝔪} ∈ 𝔪·H^{0}(D×B, ζ_{ℙV}^{∗} T_{ev,ℙV} ⊗pr_{B}^{∗}O_{B}(−1)).

There are many choices of f_{(i,j),𝔪} and g_{l,𝔪} such that ι is an isomorphism to the kernel of ∂G. The element h_{𝔪} is more canonical, and this is crucial to the computation. The space of global sections,

H^{0}(D × B, ζ_{X}^{∗}T_{ev,X} ⊗ pr_{B}^{∗}O_{B}(−1)),

is a free O_{D}-module of rank 1. So h is unique up to multiplication by an element in 1 + 𝔪. Therefore h_{𝔪} is a well-defined global section of,

𝔪⊗_{K}(ζ_{ℙV,0}^{∗}T_{ev,ℙV} ⊗ O_{B}(−1)/O_{B}{c}) = 𝔪⊗_{K}Hom_{O_{B}}(O_{B}(1){c}, ζ_{ℙV,0}^{∗}T_{ev,ℙV} /O_{B}(1){c}).

A lift of h_{𝔪} to 𝔪 ⊗_{K} ζ_{ℙV,0}^{∗}T_{ev,ℙV} ⊗ O_{B}(−1) is any global section,

∑_{(i,j)∈I_{d}} h_{𝔪,(i,j)}b_{(i,j)} + h_{𝔪,0}e_{0} + h_{𝔪,1}e_{1},

whose image under ∂G_{0} equals −∂G_{𝔪}(c).

8. The derivative map

In the last section, a canonical global section h_{𝔪} was identified in,

𝔪O_{D} ⊗_{K} Hom_{O_{B}}(O_{B}(1){c}, ζ_{ℙV,0}^{∗}T_{ev,ℙV} /O_{B}(1){c}).

In this section, h_{𝔪} is restricted to 𝔪O_{D_{s}} and it is shown that this gives rise to the restriction to T_{0}D_{s} of the derivative map dq_{s}|_{ζ_{0}}; this restriction is denoted d′q_{s}.

Fix s ∈ B. Restrict from D to the closed subscheme D_{s}. By Lemma 6.2, ζ_{X}(D_{s} × {s}) equals ζ_{0}(s) equals [(L_{s}, x_{s})]. Therefore, the map ι|_{D_{s}} gives a map of O_{D_{s}}-modules,

pr_{B}^{∗}O_{B}(1){h} ⊗_{O_{B}} κ(s) → T_{L_{s},x_{s}} = H^{0}(L_{s}, N_{L_{s}/X}(−x_{s})).

This in turn induces a K-linear map,

d′q_{s} : T_{0}D_{s} = Hom_{K}(𝔪O_{D_{s}}, K) → Hom_{K}(O_{B}(1){h}⊗κ(s), T_{L_{s},x_{s}}/O_{B}(1){h}⊗κ(s)).

The map d′q_{s} has another description. The element h_{𝔪} from the last section gives a global section h_{𝔪,s} of,

𝔪O_{D_{s}} ⊗_{K} Hom_{O_{B}}(O_{B}(1){c}, ζ_{ℙV,0}^{∗}T_{ev,ℙV} /O_{B}(1){c}).

This global section restricts to give an element in,

𝔪O_{D_{s}} ⊗_{K} Hom_{O_{B}}(O_{B}(1){c}, ζ_{ℙV,0}^{∗}T_{ev,ℙV} /O_{B}(1){c}) ⊗_{O_{B}} κ(s).

The image of this element in,

𝔪O_{D_{s}} ⊗_{K} O_{B}(d − 1){T_{0}^{j}T_{1}^{d−1−j}|0 ≤ j ≤ d − 1} ⊗_{O_{B}} κ(s),

is zero. Therefore this is an element in,

𝔪O_{D_{s}} ⊗_{K} Hom_{O_{B}}(O_{B}(1){h}, ζ_{X,0}^{∗}T_{ev,X}/O_{B}(1){h}) ⊗_{O_{B}} κ(s).

By adjointness of ⊗ and Hom, this induces the homomorphism d′q_{s} above.

The map d′q_{s} can be made more explicit. Let θ be an element in T_{0}D_{s} = Hom_{K}(𝔪O_{D_{s}}, K). For each (i, j) ∈ I_{d}, the element ⟨h_{𝔪,(i,j)}, θ⟩ is just an element in K. For l = 0, 1, the element ⟨h_{𝔪,l}, θ⟩ is an element of H^{0}(B, O_{B}(1)). The image in H^{0}(B, O_{B}(1) ⊗ O_{B}(−s)) is just an element in K. So d′q_{s}(θ) is really just a vector of elements in K. All of these elements of K are uniquely determined by,

−⟨∂G_{𝔪}(c), θ⟩ ∈ H^{0}(B, O_{B}(d − 1){T_{0}^{j}T_{1}^{d−1−1}|0 ≤ j ≤ d − 1}).

The algorithm for computing d′q_{s} is the following. Step 1 Choose an ordered basis for 𝔪O_{D_{s}} and a dual basis for T_{0}D_{s}. Step 2 For each basis element w of 𝔪O_{D_{s}} with dual basis element θ, com-

pute ⟨∂G_{𝔪}(c), θ⟩, i.e., compute the coefficient of w in the expression ∂G_{𝔪}(c) in the vector space, 𝔪O_{D_{s}} ⊗H^{0}(B, O_{B}(d−1){T_{0}^{j}T_{1}^{d−1−j}|0 ≤ j ≤ d − 1}).

Step 3 Find the unique element in H^{0}(B, ζ_{ℙV,0}^{∗}T_{ev,ℙV} ⊗ O_{B}(−1)) that is mapped by ∂G_{0} to −⟨∂G_{𝔪}(c), θ⟩, i.e., compute all the coefficients of b_{(i,j)}, and of S_{0}e_{0}, S_{1}e_{0}, S_{0}e_{1} and S_{1}e_{1}.

Step 4 Compute the image of this element in ζ_{ℙV,0}^{∗}T_{ev,ℙV} ⊗ O_{B}(−1) ⊗_{O_{B}} κ(s). In essence, consider the element modulo H^{0}(B, O_{B}(1){e_{0}, e_{1}}⊗

O_{B}(−s)).

Step 5 Compute the element of ζ_{X,0}^{∗}T_{ev,X} ⊗O_{B}(−1)/O_{B}{h}⊗_{O_{B}} κ(s) that is mapped by ι ⊗ κ(s) to this element, i.e., compute all the coefficients of f_{(i,j)}, g_{0} and g_{1}.

Proposition 8.1. There exist polynomials γ_{z} and γ_{(i,j)}, for (i, j) ∈ I_{d}, and there exists s ∈ ℙ_{s}^{1} such that q_{s} is smooth at ζ_{X}|_{0}.

Index  Polynomial  f^{∗}Polynomial
γ_{(i,j)},
(i, j) ∈ I_{d}, i, j ≤ d − 2 C_{(i,j)}X_{0}^{k}X_{1}^{i−k}X_{2}^{j−k}X_{3}^{d−2−i−j+k}  C_{(i,j)}S_{0}^{i}S_{1}^{d−2−i} · T_{0}^{j}T_{1}^{d−2−j}
γ_{(d−1,j)},
2 ≤ j ≤ d − 1  C_{j}X_{1}^{j−1}X_{3}^{d−1−j}  C_{j}S_{0}^{j−1}S_{1}^{d−1−j}T_{1}^{d−2}
γ_{(d−1,1)}  C_{1,a}X_{1}X_{3}^{d−3} + C_{1,b}X_{2}X_{3}^{d−3}  C_{1,a}S_{0}S_{1}^{d−3} · T_{1}^{d−2} + C_{1,b}S_{1}^{d−2} · T_{0}T_{1}^{d−3}
γ_{z}  C_{z,a}X_{0}X_{3}^{d−3} + C_{z,b}X_{1}X_{3}^{d−3} C_{z,a}S_{0}S_{1}^{d−3} · T_{0}T_{1}^{d−3} + C_{z,b}S_{0}S_{1}^{d−3} · T_{1}^{d−2}
γ_{(i,d−1)},
0 ≤ i ≤ d − 2  0  0
γ_{(d−2,d−1)}  0  0
γ_{(d−1,0)}  0  0

Figure 2. The polynomials γ

By Lemma 5.6, the domain and target of q_{s} are smooth. So, by the Jacobian criterion, q_{s} is smooth at ζ_{0} iff dq_{s}|_{ζ_{0}} is surjective. To prove this is surjective, it suffices to prove the restriction d′q_{s} is surjective.

9. Proof of the proposition

The point s is [1 : 0]. The polynomials γ_{(i,j)} and γ_{z} are given in Figure 2. The terms C_{(i,j)}, C_{j}, C_{1,a}, C_{1,b}, C_{z,a} and C_{z,b} are elements of K that are generic (this is made more precise later). For any index (i, j) not appearing above, C_{(i,j)} is defined to be 0. The remainder of this section carries out the algorithm of the previous section.

Step 1: Define J_{d} = {(i, j) ∈ I_{d}|(i, j) ≠ (i, 0), (0, 2), (1, 2)}. There is an isomorphism of K-vector spaces,

𝔪O_{D_{s}} → K{v^{2}, v^{3}} ∪ {w_{(i,j)}|(i, j) ∈ J_{d}},

given in Figure 3.

Step 2: With respect to the isomorphism above, h_{𝔪} equals,

h_{𝔪} = ∑_{(i,j)∈J_{d}} w_{(i,j)} · ⟨h_{𝔪}, w_{(i,j)}^{∨}⟩ + v^{1} · ⟨h_{𝔪}, (v^{1})^{∨}⟩ + v^{2} · ⟨h_{𝔪}, (v^{2})^{∨}⟩

given in Figure 4. Each computation in Figure 4 is straightforward, and is left to the reader.

Step 3: For each w in the basis of 𝔪O_{D_{s}}, an element in H^{0}(B, ζ_{ℙV,0}^{∗}T_{ev,ℙV} ⊗ O_{B}(−1)) that maps under ∂G_{0} to −⟨h_{𝔪}, w^{∨}⟩ is given in Figure 5

Step 4: The images of these elements in ζℙV, 0^{∗}T_{ev,ℙV} ⊗ O_{B}(−1) ⊗_{O_{B}} κ(s) are obtained by setting S_{1} equal to 0. This only effects the term for w_{(d−1,1)}.

Variable Image  Range
u_{(i,j)}^{0}  0  −
u_{(i,j)}^{1}  0  −
u_{(i,j)}^{2} −w_{(i,j+1)}  j ≤ d − 2
u_{(i,d−1)}^{2}  0  −
u_{(i,j)}^{3}  w_{(i,j)}  (i, j) ∈ J_{d}
u_{(i,j)}^{3}  0  (i, j) = (i, 0), (0, 2), (1, 2)
v^{0}  0  −
v^{1}  0  −
v^{2}  v^{2}  −
v^{3}  v^{3}  −
Figure 3. Basis for 𝔪O_{D_{s}}
Element w  ⟨h_{𝔪}, w^{∨}⟩

w_{(i,j)},

(i, j) ∈ J_{d}, i ≤ d − 2  (C_{(i,j)} − C_{(i,j−1)})S_{0}^{i}S_{1}^{d−1−i}T_{0}^{j}T_{1}^{d−1−j}

w_{(d−1,j)},

3 ≤ j ≤ d − 1  C_{j}S_{0}^{j−1}S_{1}^{d−j}T_{1}^{d−1} − C_{j−1}S_{0}^{j−2}S_{1}^{d+1−j}T_{0}T_{1}^{d−2}
w_{(d−1,2)}  −C_{1,a}S_{0}S_{1}^{d−2}T_{0}T_{1}^{d−2} − C_{1,b}S_{1}^{d−1}T_{0}^{2}T_{1}^{d−3} + C_{2}S_{0}S_{1}^{d−2}T_{1}^{d−1}
w_{(d−1,1)}  C_{1,a}S_{0}S_{1}^{d−2}T_{1}^{d−1} + C_{1,b}S_{1}^{d−1}T_{0}T_{1}^{d−2}
v^{2}  C_{z,a}S_{0}S_{1}^{d−2}T_{0}^{2}T_{1}^{d−3} + C_{z,b}S_{0}S_{1}^{d−2}T_{0}T_{1}^{d−2}
v^{3}  C_{z,a}S_{0}S_{1}^{d−2}T_{0}T_{1}^{d−2} + C_{z,b}S_{0}S_{1}^{d−2}T_{1}^{d−1}

Figure 4. Computation of ⟨h_{𝔪}, w^{∨}⟩

Step 5: The vector space map, ζ_{X,0}^{∗}T_{ev,X}⊗O_{B}(−1)/O_{B}{h}⊗_{O_{B}}κ(s) → ζ_{ℙV,0}^{∗}T_{ev,ℙV} ⊗O_{B}(−1)/O_{B}{c}⊗_{O_{B}}κ(s), is given by,

\{ \begin{matrix}f_{(i,j)} & ↦ & S_{0}b_{(i,j)} \\ g_{0} & ↦ & S_{0}^{2}e_{0} \\ g_{1} & ↦ & S_{0}^{2}e_{1} \end{matrix}

For each basis element w ∈ 𝔪O_{D_{s}}, d′q_{s}(w^{∨}) is the preimage of the element in the previous table. This is given in Figure 6.

Element w  −∂G_{0}^{−1}⟨h_{𝔪}, w^{∨}⟩
w_{(i,j)},
(i, j) ∈ J_{d}, i ≤ d − 2  (−C_{(i,j)} + C_{(i,j−1)})b_{(i,j)}
w_{(d−1,j)},
4 ≤ j ≤ d − 1  −C_{j}b_{(j−1,0)} + C_{j−1}b_{(j−2,1)}
w_{(d−1,3)}  −C_{3}b_{(2,0)} − C_{2}S_{0}e_{1}
w_{(d−1,2)}  −C_{1,a}S_{0}e_{1} + C_{1,b}b_{(0,2)} − C_{2}S_{0}e_{0}
w_{(d−1,1)}  −C_{1,a}S_{0}e_{0} + C_{1,b}S_{1}e_{1}
v^{2}  −C_{z,a}b_{(1,2)} + C_{z,b}S_{0}e_{1}
v^{3}  C_{z,a}S_{0}e_{1} − C_{z,b}S_{0}e_{0}

Figure 5. Computation of −∂G_{0}^{−1}⟨h_{𝔪}, w^{∨}⟩

Element w  d′q_{s}(w^{∨})
w_{(i,j)},
(i, j) ∈ J_{d}, i ≤ d − 2  (−C_{(i,j)} + C_{(i,j−1)})\frac{1}{S_{0}} f_{(i,j)}
w_{(d−1,j)},
4 ≤ j ≤ d − 1  −C_{j} \frac{1}{S_{0}} f_{(j−1,0)} + C_{j−1} \frac{1}{S_{0}} f_{(j−2,1)}
w_{(d−1,3)}  −C_{3} \frac{1}{S_{0}} f_{(2,0)} − C_{2} \frac{1}{S_{0}} g_{1}
w_{(d−1,2)}  −C_{1,a} \frac{1}{S_{0}} g_{1} + C_{1,b} \frac{1}{S_{0}} f_{(0,2)} − C_{2} \frac{1}{S_{0}} g_{0}
w_{(d−1,1)}  −C_{1,a} \frac{1}{S_{0}} g_{0}
v^{2}  −C_{z,a} \frac{1}{S_{0}} f_{(1,2)} + C_{z,b} \frac{1}{S_{0}} g_{1}
v^{3}  C_{z,a} \frac{1}{S_{0}} g_{1} − C_{z,b} \frac{1}{S_{0}} g_{0}

Figure 6. Computation of d′q_{s}

From the first row, the image of d′q_{s} contains \frac{1}{S_{0}} f_{(i,j)} for all (i, j) ∈ J_{d} and i ≤ d − 2. In particular, (j − 2, 1) is such an index for 4 ≤ j ≤ d − 1. So, from the second row, the term −C_{j−1} \frac{1}{S_{0}} f_{(j−2,1)} is already in the image of d′q_{s}. Therefore \frac{1}{S_{0}} f_{(j−1,0)} is in the image of d′q_{s} for 4 ≤ j ≤ d − 1.

Among the terms f_{(i,j)}, the only ones not covered by the first 2 rows are (i, j) = (0, 2), (1, 2) and (2, 0).

From the fifth row, \frac{1}{S_{0}} g_{0} is in the image of d′q_{s}. Together with the last row, then \frac{1}{S_{0}} g_{1} is in the image of d′q_{s}. With the fourth row, \frac{1}{S_{0}} f_{(0,2)} is in the image. With the sixth row, \frac{1}{S_{0}} f_{(1,2)} is in the image. And with the third row, \frac{1}{S_{0}} f_{(2,0)} is in the image. Therefore d′q_{s} is surjective.

10. A spanning minimal twisting family

If d = 1 or 2, any minimal twisting family is already very twisting.

Proposition 10.1. Let X ⊂ ℙ^{n} be a general hypersurface of degree d ≥ 3. If n ≥ d^{2}, there exists a point s ∈ ℙ^{1}, a point [L, x] ∈ \overline{M}_{0,1}(X, 1) and a minimal twisting morphism ζ_{X,0} : (ℙ^{1}, s) → (\overline{M}_{0,1}(X, 1), [L, x]) such that q_{s,[L,x]} is spanning at ζ_{X,0}.

Let d ≥ 3 and let n′ ≥ d^{2}. Denote n = d^{2}. Choose a linear subspace ℙ^{n} ⊂ ℙ^{n′}. Let X ⊂ ℙ^{n} be the hypersurface from Section 9. Let X′ ⊂ ℙ^{n} be a cone over X. Let s = [1 : 0] and let ζ_{X} : D_{s} × B → \overline{M}_{0,1}(X, 1) be the minimal twisting family from Section 6. This is also a minimal twisting family for X′; denote by ζ_{X′} : D_{s} × B → \overline{M}_{0,1}(X′, 1) the corresponding morphism. There is a short exact sequence,

0 ⟶ ζ_{X}^{∗}T_{ev,X} ⟶ ζ_{X′}^{∗}T_{ev,X′} ⟶ pr_{B}^{∗}O_{B}(1)^{n′−n} ⟶ 0.

Therefore, the quotient of T_{ev,X′} ⊗ κ([L_{s}, x_{s}]) by pr_{B}^{∗}O_{B}(1)^{n′+1−d^{2}} is canonically isomorphic to the quotient of T_{ev,X} ⊗ κ([L_{s}, x_{s}]) by pr_{B}^{∗}O_{B}(1)^{n+1−d^{2}}. By Proposition 8.1, d′q_{s} surjects to this quotient. Therefore the morphism q_{s} for X′ is smooth at ζ_{0}; i.e., q_{s} is spanning. By Proposition 5.7, for a general hypersurface of degree d in ℙ^{n′}, there exists a very twisting family of lines.

11. Can we do better?

What is the weakest inequality such that the spaces \overline{M}_{0,0}(X, e) are rationally connected? What is the weakest inequality such that there exists a very twisting 1-morphism ζ : ℙ^{1} → \overline{M}_{0,1}(X, 1)? The weakest inequality such that there exists a very twisting morphism is n ≥ d^{2}, cf. Proposition 5.8. But rational connectedness holds under slightly weaker hypotheses. The basic idea is contained in the following result.

Theorem 11.1 (de Jong). Let K be an algebraically closed field with char(K) = 0. Let (d, n) be positive integers such that either d = 1 and n ≥ 2 or d ≥ 2 and n+1 ≥ d^{2}. Let X ⊂ ℙ^{n} be a general hypersurface of degree d. For every e ≥ 1, every irreducible component of \overline{M}_{0,0}(X, e) is uniruled (or is a point if (d, n, e) = (1, 2, 1)).

Proof. If d = 1 or d = 2, this is easy and follows from a stronger result of Kim and Pandharipande, cf. [KP01]. In those cases there is an action of SL_{n}, resp. SO_{n+1}, on \overline{M}_{0,0}(X, e). With the one exception of (d, n, e) = (1, 2, 1), the stabilizer of a general point is a subgroup of positive codimension. Because SL_{n} and SO_{n+1} are rational, the orbit of a general point is a positive dimensional, unirational variety; in particular it is uniruled. Because the orbit of a general point is uniruled, \overline{M}_{0,0}(X, e) is uniruled.

Let d ≥ 3. By [HS05, Prop. 6.5], there exists a twisting family ζ : ℙ^{1} → \overline{M}_{0,1}(X, 1) such that ev ∘ ζ : ℙ^{1} → X is a line. Let g : ℙ^{1} → ℙ^{1} be a finite morphism of degree e. Then ζ ∘ g : ℙ^{1} → \overline{M}_{0,1}(X, 1) is also twisting; i.e., there exists a degree e stable map ev ∘ ζ ∘ g : ℙ^{1} → X that is twistable, cf. [HS05, Def. 4.7]. By [HS05, Prop. 4.8], there is a nonempty open subset of \overline{M}_{0,0}(X, e) of twistable stable maps. By [HRS04], \overline{M}_{0,0}(X, e) is irreducible, therefore this open set is dense. If h : ℙ^{1} → X is in this open set, then there exists a twisting morphism ξ : ℙ^{1} → \overline{M}_{0,1}(X, 1) such that ev ∘ ξ : ℙ^{1} → X equals h. Let (π : Σ → ℙ^{1}, σ : ℙ^{1} → Σ, g : Σ → X) be the family inducing ζ. Then σ(B) ⊂ Σ deforms in its linear equivalence class. This deformation defines a non-constant, rational transformation ℙ^{1} −−→ \overline{M}_{0,0}(X, 1) whose image contains [h]. Therefore \overline{M}_{0,0}(X, 1) is uniruled. □

A. J. de Jong’s proof is different, and proves a stronger result: for every smooth hypersurface, every irreducible component of \overline{M}_{0,0}(X, e) is uniruled. A. J. de Jong’s theorem motivated a re-investigation of [HS05, Lem. 7.4] and the proof of Theorem 1.1.

In a forthcoming paper, by elaborating on the proof of Theorem 11.1, de Jong and I prove that if n ≥ d^{2} + 1, then the spaces of curves are rationally connected. However, that proof does not give the existence of very twisting families of lines.

This may seem inconsequential, but it is actually important for another purpose: de Jong and I have a method of generalizing the Tsen-Lang theorem. In that method, existence of a very twisting family of lines plays an important role. As the Tsen-Lang theorem holds for n ≥ d^{2} but does not hold for n = d^{2} + 1, non-existence of a very twisting family of lines appears to be signficant.

In the other direction, there is reason to expect that \overline{M}_{0,0}(X, e) is not uniruled if n + 2 ≤ d^{2}. Proposition 11.2. [Sta03] Let K be an algebraically closed field with char(K) = 0. Let (d, n) be positive integers such that d < min(n − 3, \frac{n+1}{2}) and n + 2 ≤ d^{2} (if n ≥ 6, the conditions are d ≤ \frac{n}{2} and n + 2 ≤ d^{2}). Let X ⊂ ℙ^{n} be a general hypersurface of degree d. For every e ≫ 0 the canonical divisor class on \overline{M}_{0,0}(X, e) is big. If e ≤ n−d, then the coarse moduli space \overline{M}_{0,0}(X, e) is of general type (the singularities are canonical).

References

[BM96] K. Behrend and Yu. Manin. Stacks of stable maps and Gromov-Witten invariants. Duke Math. J., 85(1):1–60, 1996.

[Cam92] F. Campana. Connexité rationnelle des variétés de Fano. Ann. Sci. École Norm. Sup. (4), 25(5):539–545, 1992.

[Deb01] Olivier Debarre. Higher-dimensional algebraic geometry. Universitext. Springer-Verlag, New York, 2001.

[dJS03] A. J. de Jong and J. Starr. Every rationally connected variety over the function field of a curve has a rational point. Amer. J. Math., 125(3):567–580, 2003.

[FP97] W. Fulton and R. Pandharipande. Notes on stable maps and quantum cohomology. In Algebraic geometry—Santa Cruz 1995, volume 62 of Proc. Sympos. Pure Math., pages 45–96. Amer. Math. Soc., Providence, RI, 1997.

[GHMS05] Tom Graber, Joe Harris, Barry Mazur, and Jason Starr. Rational connectivity and sections of families over curves. Ann. Sci. École Norm. Sup. (4), 38:671– 692, 2005.

[GHS03] Tom Graber, Joe Harris, and Jason Starr. Families of rationally connected varieties. J. Amer. Math. Soc., 16(1):57–67 (electronic), 2003.

[Har77] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.

[Hir64] Heisuke Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. of Math. (2) 79 (1964), 109–203; ibid. (2), 79:205–326, 1964.

[HRS04] Joe Harris, Mike Roth, and Jason Starr. Rational curves on hypersurfaces of low degree. J. Reine Angew. Math., 571:73–106, 2004.

[HS05] Joe Harris and Jason Starr. Rational curves on hypersurfaces of low degree. II. Compos. Math., 141(1):35–92, 2005.

[KMM92a] János Kollár, Yoichi Miyaoka, and Shigefumi Mori. Rational connectedness and boundedness of Fano manifolds. J. Differential Geom., 36(3):765–779, 1992.

[KMM92b] János Kollár, Yoichi Miyaoka, and Shigefumi Mori. Rationally connected varieties. J. Algebraic Geom., 1(3):429–448, 1992.

[Kol96] János Kollár. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 1996.

[KP01] B. Kim and R. Pandharipande. The connectedness of the moduli space of maps to homogeneous spaces. In Symplectic geometry and mirror symmetry (Seoul, 2000), pages 187–201. World Sci. Publishing, River Edge, NJ, 2001.

[Lan52] Serge Lang. On quasi algebraic closure. Ann. of Math. (2), 55:373–390, 1952.

[Mum70] David Mumford. Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, No. 5. Published for the Tata Institute of Fundamental Research, Bombay, 1970.

[Sta03] Jason Starr. Kodaira dimensions of spaces of rational curves on low degree hypersurfaces. in preparation, 2003.

[Sta04] Jason Starr. The maximal free rational quotient. preprint, 2004.

[Tse36] C. Tsen. Quasi-algebraische-abgeschlossene Funktionenkörper. J. Chinese Math., 1:81–92, 1936.

Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139

E-mail address: jstarr@math.mit.edu