RATIONAL CURVES ON HYPERSURFACES OF LOW DEGREE

JOE HARRIS, MIKE ROTH, AND JASON STARR

Abstract. We prove that for n > 2 and d < \frac{n+1}{2} , a general complex hypersurface X ⊂ ℙ^{n} of degree d has the property that for each integer e the scheme R_{e}(X) parametrizing degree e, smooth rational curves on X is an integral, local complete intersection scheme of “expected” dimension (n + 1 − d)e + (n − 4).

The techniques used in the proof include:

(1) Classical results about lines on hypersurfaces including a new result about flatness of the projection map from the space of pointed lines.

(2) The Kontsevich moduli space of stable maps, \overline{M}_{0,r}(X, e). In particular we use the deformation theory of stable maps, properness of the stack \overline{M}_{0,r}(X, e), and the decomposition of \overline{M}_{0,r}(X, e) described in [2].

(3) A version of Mori’s bend-and-break lemma.

1. Summary

1.1. Brief Summary. All schemes we consider will be ℂ-schemes and all morphisms will be morphisms of ℂ-schemes. All (absolute) products will be over ℂ.

For a projective scheme X over ℂ along with an ample line bundle L we define R_{e}(X) to be the open subscheme of the Hilbert scheme Hilb^{et+1}(X/k) which parametrizes smooth rational curves of degree e lying in X. Theorem 1.1. Let n > 2 be an integer and let d be a positive integer such that d < \frac{n+1}{2}. For a general hypersurface X ⊂ ℙ^{n} of degree d and for every integer e ≥ 1, the scheme R_{e}(X) is an integral, local complete intersection scheme of dimension (n + 1 − d)e + (n − 4).

The idea of the proof is as follows. There is an embedding of R_{e}(X) into the smooth scheme R_{e}(ℙ^{n}). Denote by π : U_{e}(ℙ^{n}) → R_{e}(ℙ^{n}) the universal family of rational curves in ℙ^{n} and by ρ : U_{e}(ℙ^{n}) → ℙ^{n} the evaluation morphism. Then R_{e}(X) is the scheme of zeroes of a section of the locally free sheaf π_{∗}ρ^{∗}O_{ℙ^{n}}(d). Thus to prove that R_{e}(X) is a local complete intersection scheme, it suffices to prove that the codimension of R_{e}(X) in R_{e}(ℙ^{n}) equals the rank of π_{∗}ρ^{∗}O_{ℙ^{n}}(d).

The remainder of the proof is a “deformation and specialization” argument: we embed the non-proper scheme R_{e}(X) as an open subscheme of a proper scheme which is still modular, i.e. we choose a “modular compactification”. Then we show that every generic point in R_{e}(X) specializes to a point in the “boundary” of the compactification. We use deformation theory to study the irreducible components of the boundary of the compactification. In particular we show that a general point of each irreducible component of the boundary is a unibranch point of the compactification whose local ring is reduced and has the expected dimension. This reduces the proof to a combinatorial argument. 1.2. Detailed Summary.

Date: November 30, 2005.

The third author was partially supported by an NSF Graduate Research Fellowship and a Sloan Dissertation Fellowship.

In the next few paragraphs we will give a detailed summary of the proof. Our compactification consists of the embedding of R_{e}(X) as an open subscheme in the Kontsevich moduli space \overline{M}_{0,0}(X, e) parametrizing stable maps to X. We recall the partition of \overline{M}_{0,0}(X, e) into locally closed subsets defined in [2]; we call this partition the Behrend-Manin decomposition (our partition differs slightly from that in [2]). In particular, the image of R_{e}(X) is a dense open subset of a component of this partition. We identify certain basic components as those components of the partition parametrizing stable maps such that each irreducible component of the domain curve is mapped to a line in X.

We prove a new result about lines on X. We define the incidence correspondence of pointed lines in X:

F_{0,1}(X) = {(p, l)|p a point, l a line, p ∈ l ⊂ X}.  (1)

We prove that for a general hypersurface X ⊂ ℙ^{n} of degree d ≤ n−1, the projection morphism F_{0,1}(X) → X is flat of relative dimension n−d−1. From this theorem it easily follows that each basic component B is an integral scheme whose general point is a unibranch point of \overline{M}_{0,0}(X, e) at which \overline{M}_{0,0}(X, e) is reduced of dimension (n+1−d)e+(n−4). Thus for each basic component B there is a unique irreducible component M(B) of \overline{M}_{0,0}(X, e) which contains B, and M(B) is reduced and has dimension (n + 1 − d)e + (n − 4).

Using a version of the bend-and-break lemma of Mori, we prove that every irreducible component of \overline{M}_{0,0}(X, e) is of the form M(B) for some basic component B. Using this fact and results about flatness, we bootstrap to prove that each evaluation map \overline{M}_{0,r}(X, e) → X is flat of the expected dimension and is generically unobstructed. This implies that each \overline{M}_{0,r}(X, e) (including r = 0) has the expected dimension and is generically smooth. Thus \overline{M}_{0,r}(X, e) is a reduced, local complete intersection stack of the expected dimension and it only remains to prove that \overline{M}_{0,r}(X, e) is irreducible, i.e. it remains to prove that all of the irreducible components M(B) are actually equal.

To prove that all of the irreducible components M(B) are equal, we observe that there is a combinatorially defined equivalence relation defined on the set of basic components B such that equivalent basic components, B ≅ B′ satisfy M(B) = M(B′). Thus we are reduced to a combinatorial argument which proves that all basic components are equivalent.

Along the way we generalize the strategy of proof above so that it could apply to smooth projective schemes X other than hypersurfaces X ⊂ ℙ^{n} of degree d < \frac{n+1}{2} (this is made completely explicit for complete intersections in ℙ^{n}). One is reduced to proving

(1) The evaluation morphism \overline{M}_{0,1}(X, e) → X is flat, generically unobstructed and the general fiber is geometrically irreducible.

(2) For each positive integer e at most the threshold degree of X, the evaluation morphism \overline{M}_{0,1}(X, e) → X is flat of the expected dimension.

(3) For each positive integer e at most the threshold degree of X, the stack \overline{M}_{0,0}(X, e) is irreducible.

The most difficult condition to verify seems to be (2), but it is our hope that this can be verified for a larger class of Fano schemes X than the hypersurfaces above.

In [8], Kim and Pandharipande proved irreducibility and rationality of the stacks \overline{M}_{0,r}(X, β) when X is a homogeneous variety for a linear algebraic group (and β is a numerical equivalence class of curves on X). In particular, when X is a linear or quadric hypersurface in ℙ^{n}, with n ≥ 4, it follows from [8, Corollary 1] that \overline{M}_{0,r}(X, e) is irreducible and from [8, Theorem 3] that \overline{M}_{0,r}(X, e) is rational. This paper can be seen as a generalization of the irreducibilty result [8, Corollary 1] to hypersurfaces X ⊂ ℙ^{n} of degree roughly d ≤ \frac{n}{2}. In a forthcoming paper, [6], we give a partial generalization of the rationality result of [8, Theorem 3] to hypersurfaces X ⊂ ℙ^{n} of degree roughly d ≤ \sqrt{n}. 1.3. Notation. Given a ℂ-vector space W, ℙW denotes the projective space Proj(⊕_{d≥0}S^{d}(W^{∗})) which parametrizes one-dimensional linear subspaces of W (not one-dimensional quotient spaces of W). Given any integers k ≤ n, G(k, n) denotes the Grassmannian which parametrizes k-dimensional linear subspaces of ℂ^{n}. For a triple of integers k ≤ l ≤ n, F ((k, l), n) denotes the partial flag variety which parametrizes partial flags V_{1} ⊂ V_{2} ⊂ ℂ^{n} of linear subspaces with dim(V_{1}) = k and dim(V_{2}) = l. 1.4. Acknowledgments. The authors would like to thank Tom Graber, Ravi Vakil, Andreas Gathmann, Olivier Debarre, and especially Johan de Jong for many useful discussions.

2. Lines on Hypersurfaces

We denote W = H^{0}(ℙ^{n}, O_{ℙ^{n}}(d)) so that ℙW parametrizes degree d hypersurfaces X ⊂ ℙ^{n}. Let X ⊂ ℙW × ℙ^{n} denote the universal family of degree d hypersurfaces in ℙ^{n}. For each degree d hypersurface X ⊂ ℙ^{n}, denote by F_{1}(X) the subscheme of G(2, n + 1) which parametrizes lines L ⊂ X ⊂ ℙ^{n}. Denote by F ((1, 2), n + 1) the partial flag variety parametrizing pairs (p, L) where L ⊂ ℙ^{n} is a line and p ∈ L is a point. Denote by F_{0,1}(X) the subscheme of F ((1, 2), n + 1) which parametrizes pairs (p, L) with p ∈ L ⊂ X ⊂ ℙ^{n}.

Similarly, let F_{1}(X) ⊂ ℙW × G(2, n + 1) denote the subscheme parametrizing pairs (X, L) with L ∈ F_{1}(X), and let F_{0,1}(X) ⊂ ℙW × F ((1, 2), n + 1) denote the subscheme parametrizing triples (X, p, L) with (p, L) ∈ F_{0,1}(X). There are projection morphisms

π_{0} : ℙW × F ((1, 2), n + 1) → ℙW,  (2)
π_{1} : ℙW × F ((1, 2), n + 1) → ℙ^{n},  (3)
π_{2} : ℙW × F ((1, 2), n + 1) → G(2, n + 1)  (4)

By construction, the morphism (π_{0}, π_{1}) : F_{0,1}(X) → ℙW × ℙ^{n} factors through X ⊂ ℙW ×ℙ^{n}. Denote by ρ : F_{0,1}(X) → X the induced morphism. For a particular hypersurface X ∈ ℙW, denote by ρ_{X} : F_{0,1}(X) → X the fiber of ρ.

The main result of this section is the following theorem:

Theorem 2.1. Let d be a positive integer with d ≤ n−1. For a general hypersurface X ∈ ℙW the morphism ρ_{X} : F_{0,1}(X) → X is flat of relative dimension n − d − 1.

We give the proof in the remainder of this section. From now on we assume that d is given such that d ≤ n − 1.

Denote by O the structure sheaf of ℙW ×F ((1, 2), n+1) and denote by O_{F} the Omodule which is the pushforward of the structure sheaf of F_{0,1}(X). On G(2, n+1) we have a universal rank 2 subbundle of O_{G(2,n+1)}^{⊕n+1}. Denote by S the pullback under π_{2} of this universal subbundle to ℙW × F ((1, 2), n + 1). And denote by U the pullback under π_{0} of the universal rank 1 subbundle O_{ℙW} (−1) ⊂ H^{0}(ℙ^{n}, O_{ℙ^{n}}(d)) ⊗_{ℂ} O_{ℙW}. By restricting a section of H^{0}(ℙ^{n}, O_{ℙ^{n}}(d)) to a line L ⊂ ℙ^{n} parametrized by a point [L] ∈ G(2, n + 1), we get a map U → Sym^{d}(S^{∨}). By adjunction, this gives rise to a map Sym^{d}(S) ⊗_{O} U → O, whose image is exactly the ideal sheaf of F_{0,1}(X). In other words, there is a partial resolution of coherent sheaves:

Sym^{d}(S) ⊗_{O} U ⟶^{σ} O ⟶ O_{F} ⟶ 0.  (5)

In other words, F_{0,1}(X) is the zero scheme of the global section O → Sym^{d}(S^{∨}) ⊗_{O} U^{∨} which is the transpose of σ.

Since Sym^{d}(S) ⊗_{O} U is locally free of rank d + 1, every irreducible component of X has codimension at most d + 1 in ℙW × F ((1, 2), n + 1). Therefore, every (nonempty) fiber of ρ has dimension at least n − d − 1. We define U ⊂ X as a set to be

U = {(X, p) ∈ X | dim(ρ^{−1}(X, p)) ≤ n − d − 1} .  (6)

It follows by upper semicontinuity of the fiber dimension that U is a Zariski open subset of X, and we give it the corresponding structure of open subscheme of X. A priori U might contain points (X, p) ⊂ X for which ρ^{−1}(X, p) is empty. But by [9, exercise V.4.6], it follows that ρ is surjective (also this exercise rederives the statement above about dimensions of fibers).

Notice that the projection morphism π_{0} : X → ℙ^{n} is a projective bundle whose fiber over p ∈ ℙ^{n} is identified, as a subscheme of ℙW, with the hyperplane parametrizing X ∈ ℙW with p ∈ X. In particular, X is a smooth k-scheme. Given the map σ above, we can form the Koszul complex of locally free O-modules in the usual way. By [10, Theorem 17.4 (iii)(4)], this complex is acyclic over U. Therefore the fibers of ρ over U, considered as subschemes of the appropriate fiber of π_{1} : F ((1, 2), n + 1) → ℙ^{n}, all have equal Hilbert polynomial. Since U is smooth, it follows from [7, Theorem III.9.9] that ρ is flat over U.

Let Y ⊂ F_{0,1}(X) denote the complement of U with the induced, reduced scheme structure. Theorem 2 is equivalent to the statement that π_{0}|_{Y} : Y → ℙW is not surjective. Denote by e the codimension of Y in X. Since the fiber dimension of X → ℙW is n − 1, to prove that Y fails to dominate ℙW, it suffices to prove that e > n − 1. In the remainder of this section we will prove that e > n − 1.

On ℙ^{n} let Q denote the locally free quotient sheaf of O_{ℙ^{n}}^{⊕n+1} by O_{ℙV} (−1). The dual injection Q^{∨} ↪ (O_{ℙ^{n}}^{⊕n+1})^{∨} can be considered as a filtration of (O_{ℙ^{n}}^{⊕n+1})^{∨}. The dth symmetric product of this filtration is a filtration of W ⊗_{ℂ} O_{ℙ^{n}}:

W ⊗_{ℂ} O = F^{0,d} ⊃ F^{1,d} ⊃ ⋯ ⊃ F^{d,d} ⊃ F^{d+1,d} = 0.  (7)

Here F^{i,d} is the locally free subsheaf of W ⊗_{ℂ} O which is the image of the multiplication map

Sym^{d−i} (O_{ℙ^{n}}^{⊕n+1})^{∨} ⊗_{O_{ℙ^{n}}} Sym^{i}Q^{∨} → W ⊗_{ℂ} O.  (8)

The associated graded sheaves of this filtration G^{i,d} = F^{i,d}/F^{i+1,d} are canonically isomorphic to the sheaves O_{ℙ^{n}}(d − i) ⊗_{O_{ℙ^{n}}} Sym^{i}Q^{∨}.

In particular, notice that F^{1,d} is simply the kernel of the evaluation map W ⊗_{ℂ} O_{ℙ^{n}} → O_{ℙ^{n}}(d), i.e. the vector bundle parametrizes pairs (Φ, p) where Φ ∈ W is such that Φ(p) = 0. We identify a nonzero section Φ ∈ W, up to nonzero scaling, with the hypersurface it defines X = V (Φ). Then the associated projective bundle ℙF^{1,d} inside ℙW × ℙ^{n} is the closed subscheme parametrizing pairs (X, p) with p ∈ X, i.e. ℙF^{1,d} = X. To prove the inequality e > n − 1 from above, it suffices to prove that for each p ∈ ℙ^{n} (equivalently for any p ∈ ℙ^{n} by homogeneity) the intersection Y ∩ π_{0}^{−1}(p) ⊂ X has codimension greater than n − 1 in π_{0}^{−1}(p). In the remainder of this section we will prove this.

Observe the filtration above is not split on ℙ^{n}. But we can find a covering of ℙ^{n} by open affine subschemes A_{α} ⊂ ℙ^{n} over which we do have a splitting (for example, the standard covering by complements of coordinate hyperplanes). Here by splitting we mean an isomorphism of bundles over A_{α}

s : W ⊗_{ℂ} O_{A_{α}} ⟶ ⊕_{j=0}^{d}O_{ℙ^{n}}(d − j) ⊗_{O_{ℙ^{n}}} Sym^{j}Q^{∨}|_{A_{α}}  (9)

which maps F^{i,d}|_{A_{α}} to the subbundle ⊕_{j=i}^{d}O_{ℙ^{n}}(d−j)⊗_{O_{ℙ^{n}}} Sym^{j}Q^{∨}|_{A_{α}} and such that the induced isomorphism G^{i,d}|_{A_{α}} → O_{ℙ^{n}}(d−i)⊗_{O_{ℙ^{n}}} Sym^{i}Q^{∨}|_{A_{α}} is the isomorphism from above.

Given an open affine A_{α} ⊂ ℙ^{n} we can form the projective bundle ℙ_{A_{α}}(F^{1,d}|_{A_{α}}) over A_{α}. Given a splitting s on A_{α}, denote by

Δ_{j}(s) ⊂ ℙ_{A_{α}}(F^{1,d}|_{A_{α}})  (10)

the closed subscheme which parametrizes pairs (Φ, x), x ∈ A_{α}, Φ ∈ F^{1,d}|x such that the jth component of s(Φ) is zero. Thus Δ_{0}(s) is all of ℙ_{A_{α}}(F^{1,d}|_{A_{α}}). And, considering ℙ_{A_{α}}(F^{1,d}|_{A_{α}}) as an open subscheme of X, Δ_{1}(s) is the intersection of ℙ_{A_{α}}(F^{1,d}|_{A_{α}}) with the singular locus of the projection morphism π_{0} : X → ℙW. Although Δ_{0}(s) and Δ_{1}(s) are independent of the choice of s, the same is not true for Δ_{i}(s) with i > 1. The next result follows immediately from the definition of the Δ_{j}(s).

Lemma 2.2. For j > 0 the codimension of Δ_{j}(s) in ℙ_{A_{α}}(F^{1,d}|_{A_{α}}) equals

rank (O_{ℙ^{n}}(d − j) ⊗_{O_{ℙ^{n}}} Sym^{j}Q^{∨}) = ( \begin{matrix}n − 1 + j \\ n − 1 \end{matrix} ) .  (11)

In particular, for j > 0 the codimension of Δ_{j}(s) in ℙ_{A_{α}}(F^{1,d}|_{A_{α}}) is at least n − 1 + j > n − 1. For each open subscheme A_{α} ⊂ ℙV, each splitting s, and each point p ∈ A_{α}, define the locally closed subscheme

Y_{p,s} := (Y ∩ π_{0}^{−1}(p)) − ∪_{j=1}^{d} (Δ_{j}(s) ∩ π_{0}^{−1}(p)) .  (12)

To establish that e > n − 1, it suffices to prove that the codimension of Y_{p,s} as a subscheme of π_{0}^{−1}(p) has codimension greater than n − 1. In the remainder of this section we prove this inequality.

On the complement of the closed subset Δ(s) := ∪_{j=1}^{d} (Δ_{j}(s)) there is a morphism

ℙ_{A_{α}} (F^{1,d}|_{A_{α}}) − Δ(s) ⟶^{β} ∏_{j=1}^{d} ℙ_{A_{α}} (O_{ℙ^{n}}(d − j) ⊗_{O_{ℙ^{n}}} Sym^{j}Q^{∨}) |_{A_{α}}. (13)

We identify the space

ℙ_{A_{α}} (O_{ℙ^{n}}(d − j) ⊗_{O_{ℙ^{n}}} Sym^{j}Q^{∨}) |_{A_{α}}  (14)

with the scheme parametrizing degree j hypersurfaces in fibers of the projection morphism ℙ_{A_{α}}Q|_{A_{α}} → A_{α}. Thus β assigns to each suitable pair ([Φ], p) a sequence of hypersurfaces in ℙQ|_{p}. Denote this sequence by (X_{1}, …, X_{j}, …, X_{d}). Also, denote by X = X_{1} ∩ ⋯ ∩ X_{d} the intersection of these hypersurfaces.

Lemma 2.3. If we denote by X the hypersurface in ℙV corresponding to Φ, then X_{1} ∩ ⋯ ∩ X_{d} is the fiber of p ∈ X under ρ_{X} : F_{0,1}(X) → X.

Proof. This is most easily seen by passing to local coordinates. Let (x_{0}, …, x_{n}) be a system of homogeneous coordinates on ℙV (i.e. a basis for V^{∨}) and let p be the point with homogeneous coordinates [0, …, 0, 1]. We define a splitting s as follows: for each degree d homogeneous polynomial Φ in (x_{0}, …, x_{n}) we have a unique decomposition

Φ = Φ_{d} + Φ_{d−1}x_{n} + ⋯ + Φ_{d−i}x_{n}^{i} + ⋯ + Φ_{0}x_{n}^{d}  (15)

where each Φ_{i} is a homogeneous polynomial of degree i in (x_{0}, …, x_{n−1}). Then the fiber of F^{1,d} at p consists of those polynomials such that Φ_{0} = 0 and β(Φ) = (Φ_{d}, …, Φ_{1}). For any line L passing through p there is a unique point of the form y = (a_{0}, …, a_{n−1}, 0) contained in L. Let ℙ^{1} → ℙ^{n} be the morphism given by

(t_{0}, t_{1}) ↦ (t_{1}a_{0}, t_{1}a_{1}, …, t_{1}a_{n−1}, t_{0} + t_{1}a_{n}).  (16)

The image of this morphism is just L. Substituting into Φ yields the polynomial on ℙ^{1} given by

t_{0}^{d}Φ_{d}(a_{0}, …, a_{n−1}) + ⋯ + t_{0}^{d−i}t_{1}^{i}Φ_{d−i}(a_{0}, …, a_{n−1}) + …
+t_{0}t_{1}^{d−1}Φ_{1}(a_{0}, …, a_{n−1}).

The line L is contained in X iff this polynomial is identically zero iff each of the terms Φ_{i}(a_{0}, …, a_{n}) is zero. One can show that the homogeneous ideal generated by the terms Φ_{i} is independent of our particular splitting. □

In particular, we conclude that every fiber of β which intersects Y is contained in Y. Therefore the codimension of Y_{p,s} in π_{0}^{−1}(p) equals the codimension of the subvariety

β(Y) ⊂ ∏_{j=1}^{d} ℙ (O_{ℙ^{n}}(d − j) ⊗_{O} Sym^{j}Q^{∨}) |_{p}.  (17)

By construction, β(Y) is the locus parametrizing sequences of hypersurfaces in ℙQ|_{p}, (X_{1}, …, X_{d}), of degrees 1, …, d respectively such that the intersection

X_{(1,...,d)} := X_{1} ∩ ⋯ ∩ X_{d}  (18)

has dimension greater than n − d − 1. So we have reduced Theorem 2.1 to the following theorem:

Theorem 2.4. Let Q be a vector space over k of dimension n and let d be an integer such that 1 ≤ d ≤ n − 1. Let ℙ_{d} denote the scheme ∏_{j=1}^{d} ℙSym^{j}Q^{∨}, which parametrizes d-tuples (X_{1}, …, X_{d}) of hypersurfaces X_{i} ∈ ℙQ of degree i. Denote by D_{d} the closed subscheme of ℙ_{d} which parametrizes sequences (X_{1}, …, X_{d}) such that

dim (X_{(1,...,d)}) > n − d − 1.  (19)

The codimension of D_{d} in ℙ_{d} is greater than n − 1.

Proof. We will prove this by induction on d. Consider first the case d = 1. Since D_{1} = ∅ and the dimension of ℙ_{d} = ℙQ^{∨} is n − 1, the result is true for d = 1.

Let U_{d} denote the open subscheme of ℙ_{d} which is the complement of D_{d}. Then for 1 ≤ d ≤ n − 2, U_{d+1} is contained in U_{d} × ℙSym^{d+1}Q^{∨}. To see this, note that if X_{(1,...,d)} has dimension larger than n − d − 1, then X_{1,...,d+1} is nonempty and has dimension greater than n − d − 2: it is nonempty since X_{d+1} is ample, it has dimension larger than n − d − 1 by the Hauptidealsatz. So we see that the codimension of D_{d+1} in ℙ_{d+1} is the minimum of the codimension of D_{d} in ℙ_{d} and the codimension of D_{d+1}∩(U_{d}×ℙSym^{d+1}Q^{∨}) in U_{d}×ℙSym^{d+1}Q^{∨}. So by induction we are reduced to showing that the codimension of D_{d+1} ∩ (U_{d} × ℙSym^{d+1}Q^{∨}) in U_{d} × ℙSym^{d+1}Q^{∨} is larger than n − 1.

Now suppose that (X_{1}, …, X_{d}, X_{d+1}) is a point in D_{d+1}∩(U_{d}×ℙSym^{d+1}Q^{∨}). By assumption every irreducible component of X_{(1,...,d)} has dimension n − d − 1. Since also X_{(1,...,d+1)} has dimension n − d − 1, we conclude that there is an irreducible component C ⊂ X_{(1,...,d)} such that C ⊂ X_{d+1}. If X_{(1,...,d)} = C_{1} ∪ ⋯ ∪ C_{r} is the irreducible decomposition, then the fiber of D_{d+1} ∩ (U_{d} × ℙSym^{d+1}Q^{∨}) over (X_{1}, …, X_{d}) (which we consider as a subscheme of ℙSym^{d+1}Q^{∨}) is just the union of i = 1, …, r of the set B_{i} ⊂ ℙSym^{d+1}Q^{∨} parametrizing hypersurfaces X_{d+1} such that C_{i} ⊂ X_{d+1}. We are reduced to showing that the codimension of each B_{i} in ℙSym^{d+1}Q^{∨} is greater than n − 1. We prove this in a lemma:

Lemma 2.5. Let Y ⊂ ℙQ be an irreducible subscheme such that dimY = n−d−1. Let B(Y) ⊂ ℙSym^{d+1}Q^{∨} be the locus of hypersurfaces X_{d+1} such that Y ⊂ X_{d+1}. The codimension of B(Y) is greater than n − 1.

Proof. Let Λ ⊂ ℙQ be a (d−1)-plane which is disjoint from Y. Choose coordinates on ℙQ, (x_{0}, …, x_{n−1}) with respect to which Λ = Z(x_{d}, …, x_{n−1}). Let 𝔾_{m} denote the multiplicative group Spec ℂ[t, t^{−1}]. Let μ : 𝔾_{m} ×ℙQ → ℙQ be the group action given by

μ(t, (x_{0}, …, x_{d−1}, x_{d}, …, x_{n−1})) = (t^{−1}x_{0}, …, t^{−1}x_{d−1}, tx_{d}, …, tx_{n−1}). (20)

Since the Hilbert scheme of ℙQ is proper, the valuative criterion implies that the closed subscheme

μ^{−1}(Y) ⊂ 𝔾_{m} × ℙQ  (21)

which is flat over 𝔾_{m}, extends over 0 to yield a closed subscheme

Y ⊂ 𝔸^{1} × ℙQ  (22)

which is flat over 𝔸^{1}. It is easy to see that the fiber of Y over 0 is a scheme whose reduced scheme is just

Z(x_{0}, …, x_{d−1}) ⊂ ℙQ.  (23)

Now we can form the family

B ⊂ 𝔸^{1} × ℙSym^{d+1}Q^{∨}, B_{t} = B(Y_{t}).  (24)

Over 𝔾_{m} the fibers of B are isomorphic. It follows by upper semicontinuity that for t ≠ 0 we have dim(B_{t}) ≤ dim(B_{0}). And of course we have

B_{0} = B(Y_{0}) ⊂ B (Z(x_{0}, …, x_{d−1})) .  (25)

So we are reduced to proving the lemma for the special case Y = Z(x_{0}, …, x_{d−1}). The set B of hypersurfaces X_{d+1} ⊂ ℙQ which contain Z(x_{0}, …, x_{d−1}) is just the projectivization of the kernel of the surjective linear map

H^{0}(ℙQ, O_{ℙQ}(d + 1)) → H^{0}(Y, O_{Y} (d + 1)).

So the codimension of B in ℙQ equals

dim_{ℂ}H^{0}(Y, O_{Y} (d + 1)) = \binom{n}{d + 1}.

For d + 1 ≤ n − 1 (which is one of our hypotheses) we see that \binom{n}{d+1} ≥ n > n − 1. We conclude that the codimension of B(Y) in ℙSym^{d+1}Q^{∨} is greater than n − 1. This proves the lemma. □

By the above lemma, we conclude that the codimension of each B_{i} in ℙSym^{d+1}Q^{∨} is greater than n − 1. So we have proved Theorem 2.4 □

Since we had reduced Theorem 2.1 to Theorem 2.4, we have proved Theorem 2.1. While we are discussing results about lines on hypersurfaces, let us mention two

other results about lines on hypersurfaces. Lemma 2.6. [9, exercise V.4.4.2] For general X and a general line L ⊂ X, the normal bundle N_{L/X} is of the form O_{L}^{⊕d−1} ⊕ O_{L}(1)^{⊕n−1−d}. Theorem 2.7. [9, Theorem V.4.3.2] For general X, the Fano scheme F_{1}(X) is smooth. Therefore F_{0,1}(X) is smooth. By generic smoothness, the general fiber of F_{0,1}(X) → X is smooth.

3. Stable A-graphs and Stable Maps

We follow the notation from [2] regarding stable A-graphs. However, we shall only need to use genus 0 trees. 3.1. Graphs and Trees. Definition 3.1. A graph τ is a 4-tuple (F_{τ} , W_{τ} , j_{τ} , ∂_{τ}) defined as follows:

(1) F_{τ} is a finite set called the set of flags

(2) W_{τ} is a finite set called the set of vertices

(3) j_{τ} : F_{τ} → F_{τ} is an involution

(4) ∂_{τ} : F_{τ} → W_{τ} is a map called the evaluation map. In addition we have the auxiliary definitions

(1) the set of tails S_{τ} ⊂ F_{τ} is the set of fixed points of j_{τ}

(2) the set of edges E_{τ} is the quotient of F_{τ} \ S_{τ} by j_{τ}

(3) for a vertex v ∈ W_{τ}, the valence of v is defined to be val(v) = #(∂^{−1}(v)). We shall often write Flag(τ) in place of F_{τ}, Vertex(τ) in place of W_{τ}, Tail(τ) in place of S_{τ}, Edge(τ) in place of E_{τ}, and \overline{f} in place of j_{τ} (f).

We can associate to a graph its geometric realization |τ| which is a CW-complex defined as follows. The set of 0-cells of |τ| is

|τ|^{0} = Vertex(τ) ⊔ Tail(τ).  (26)

The set of 1-cells of |τ| is

|τ|^{1} = Edge(τ) ⊔ Tail(τ).  (27)

If [0, 1] is a 1-cell associated to an edge \{f, \overline{f}\}, the point 0 is glued to the 0-cell ∂f, and the point 1 is glued to the 0-cell ∂\overline{f}. If [0, 1] is the 1-cell associated to a tail f, the point 0 is glued to the 0-cell ∂f, and the point 1 is glued to the 0-cell f.

Definition 3.2. A tree is a connected graph such that H_{1}(|τ|, ℤ) = 0, i.e. a graph which contains no closed loops.

One important tree is the empty tree λ_{∅}, i.e. the tree such that Vertex(λ_{∅}) = ∅. For each nonnegative integer r define λ_{r} to be the tree with one vertex, Vertex(λ_{r}) = {v}, and with r flags (all of which are tails), Tail(λ_{r}) = {f_{1}, …, f_{r}}. Also, for each pair of nonnegative integers (r_{1}, r_{2}), define λ_{r_{1},r_{2}} to be the connected tree with two vertices v_{1}, v_{2}, with r_{1} tails attached to v_{1} and with r_{2} tails attached to v_{2}.

Definition 3.3. An A-graph is a pair (τ, β_{τ}) where τ is a tree and

β : Vertex(τ) → ℤ_{≥0}  (28)

is a map called the A-structure. We shall often abbreviate (τ, β_{τ}) by just writing τ. We say that an A-graph τ is stable if for each vertex v ∈ Vertex(τ) such that β_{τ} (v) = 0, there are at least 3 distinct flags f ∈ Flag(τ) such that ∂f = v (i.e. the valence of v is at least 3).

One important A-graph is the empty graph, τ_{∅}. This is the unique A-graph whose underlying graph is λ_{∅}. For each pair of nonnegative integers r and e, define τ_{r}(e) to be the unique A-graph whose underlying graph is λ_{r} and such that β(v) = e. Obviously τ_{r}(e) is stable iff either r ≥ 3 or e > 0. For each pair of pairs (r_{1}, r_{2}) and (e_{1}, e_{2}) where r_{1}, r_{2}, e_{1} and e_{2} are nonnegative integers, define τ_{r_{1},r_{2}}(e_{1}, e_{2}) to be the unique A-graph whose underlying graph is λ_{r_{1},r_{2}}, such that β(v_{1}) = e_{1} and such that β(v_{2}) = e_{2}.

There is a category whose objects are the stable A-graphs. The morphisms in this category are each a composition of two basic types of morphisms: contractions and combinatorial morphisms, c.f. [2] for the precise definitions. Essentially a contraction of A-graphs ϕ : τ → σ is a map from the vertices of τ onto the vertices of σ which maps adjacent vertices to adjacent vertices (here two vertices are adjacent if they are equal or if they are connected by an edge). And a combinatorial morphism τ ↩ σ is the inclusion of a subgraph σ into a graph τ. The functor which associates to a stable A-graph the corresponding Behrend-Manin stack is covariant for contractions. But it is contravariant for combinatorial morphisms. Therefore we think of a combinatorial morphism τ ↩ σ as a morphism from τ to σ (which explains our terminology τ ↩ σ for combinatorial morphisms).

Particularly important are morphisms of graphs which remove tails. For each stable A-graph τ we define r_{>0}(τ) to be the stable A-graph obtained by removing every tail f ∈ Tail(τ) such that β(∂f) > 0. We define τ ↩ r_{>0}(τ) to be the canonical combinatorial morphism. For each stable A-graph τ we define r_{0}(τ) to be the stabilization of the A-graph obtained by removing all tails f ∈ Tail(τ) such that β(∂f) = 0. Technically the canonical morphism of graphs from τ to r_{0}(τ) consists of both a combinatorial morphism and a contraction. But we shall denote it by τ ↩ r_{0}(τ) just as if it were a combinatorial morphism. Finally, we define r(τ) := r_{>0}(r_{0}(τ)) = r_{0}(r_{>0}(τ)).

There are numerical invariants associated to an A-graph. Definition 3.4. Given an A-graph τ, define

β(τ) = ∑_{v∈Vertex(τ)} β(v).  (29)

If (X, L) is a polarized variety such that K_{X} =^{num} mL for some integer m, define the expected dimension

dim(X, τ) = −mβ(τ) + #Tail(τ) − #Edge(τ) + (dim(X) − 3).  (30)

3.2. Prestable Curves and Dual Graphs. Definition 3.5. A prestable curve with r marked points (C, (x_{1}, …, x_{r})) is a pair where C is a complete, reduced, at worst nodal curve and x_{i} ∈ C, i = 1, …, r are distinct, nonsingular points of C.

Suppose that (C, (x_{1}, …, x_{r})) is a connected, prestable curve whose arithmetic genus is 0. One associates to (C, (x_{1}, …, x_{n})) a dual graph, Δ: a tree whose vertices {v_{1}, v_{2}, … } correspond to the irreducible components {C_{1}, C_{2}, … } of C, whose edges \{\{f_{1}, \overline{f_{1}}\}, \{f_{2}, \overline{f_{2}}\}, … \} correspond to the nodes \{q_{1}, q_{2}, … \} of C, and whose tails {g_{1}, …, g_{r}} correspond to the marked points {p_{1}, …, p_{r}} of C. Definition 3.6. Let (X, L) be a polarized variety. A prestable map is a pair

((C, (x_{1}, …, x_{n})), C ⟶^{h} X)  (31)

where (C, (x_{1}, …, x_{n})) is a prestable curve, and where C ⟶^{h} X is a morphism of ℂ-schemes.

Just as one associates to a connected prestable curve (C, x_{1}, …, x_{n}) of arithmetic genus 0 a tree Δ(C, x), one can associate an A-graph to a prestable map ((C, x_{1}, …, x_{n}), C ⟶^{h} X) from a connected prestable curve of arithmetic genus 0. The underlying tree of Δ(C, x, h) is simply Δ(C, x). And, given a component C_{i} of C with corresponding vertex v_{i} ∈ Vertex(Δ(C, x)), one defines

β(v_{i}) = ∫_{C_{i}} h_{i}^{∗}(c_{1}(L)).  (32)

The A-graph Δ(C, x, h) is a stable A-graph iff (C, x, h) is a stable map. 3.3. Behrend-Manin stacks.

We refer the reader to [2] for the definition of the stacks \overline{M}(X, τ). These are proper Deligne-Mumford stacks which parametrize stable maps along with some extra data. We shall sometimes deal with these stacks, but more often we shall deal with the open substack M(X, τ) ⊂ \overline{M}(X, τ) of strict maps which we now define. Definition 3.7. Let X be a variety, L a line bundle on X, and let τ be a stable A-graph. A strict τ-map is a datum

((C_{v}), (h_{v} : C_{v} → X), (q_{f}))  (33)

defined as follows:

(1) (C_{v}) is a set parametrized by v ∈ Vertex(τ) of smooth rational curves, i.e. each C_{v} ≅ ℙ^{1}

(2) (h_{v} : C_{v} → X) is a set parametrized by v ∈ Vertex(τ) of morphisms of ℂ-schemes,

(3) (q_{f}) is a set parametrized by f ∈ Flag(τ) of closed points q_{f} ∈ C_{∂f} and satisfying the following conditions

(1) for v ∈ Vertex(τ), the degree of h_{v}^{∗}(L) as a line bundle on C_{v} is β_{τ} (v),

(2) for f_{1}, f_{2} ∈ Flag(τ) distinct flags with ∂f_{1} = ∂f_{2}, q_{f_{1}} ≠ q_{f_{2}},
(3) for f ∈ Flag(τ), we have h_{∂f}(q_{f}) = h_{∂\overline{f}}(q\overline{f}).

Convention: For the empty graph, τ_{∅}, we define a strict τ_{∅}-map to simply be a point in X. Thus the set of strict τ_{∅}-maps is simply X. Definition 3.8. If T is a ℂ-scheme, then a family of strict τ-maps over T is a datum

((π_{v} : C_{v} → T), (h_{v} : C_{v} → X), (q_{f} : T → C_{∂f}))  (34)

defined as follows:

(1) (π_{v} : C_{v} → T) is a set parametrized by v ∈ Vertex(τ) of smooth, proper morphisms whose geometric fibers are rational curves

(2) (h_{v} : C_{v} → X) is a set parametrized by v ∈ Vertex(τ) of morphisms of ℂ-schemes

(3) (q_{f} : T → C_{∂f}) is a set parametrized by f ∈ Flag(τ) of morphisms of

schemes such that π_{∂f} ∘ q_{f} = id_{T} and satisfying the following conditions

(1) for v ∈ Vertex(τ), the degree of h_{v}^{∗}(L) on each geometric fiber of C_{v} → S is β_{τ} (v)

(2) for f_{1}, f_{2} ∈ Flag(τ) distinct flags with ∂f_{1} = ∂f_{2}, q_{f_{1}} and q_{f_{2}} are disjoint sections

(3) for f ∈ Flag(τ), we have h_{∂f} ∘ q_{f} = h_{∂\overline{f}} ∘ q\overline{f}.

Convention: For the empty graph, τ_{∅} we define a family of strict τ_{∅}-maps over T to be a morphism h : T → X.

Suppose given two families of strict τ-maps over S, say

η = ((π_{v} : C_{v} → T), (h_{v} : C_{v} → X), (q_{f} : T → C_{∂f})),  (35)
ζ = ((π′_{v} : C′_{v} → T), (h′_{v} : C′_{v} → X), (q′_{f} : T → C′_{∂f})).  (36)

Definition 3.9. A morphism of families of strict τ-maps over S, ϕ : η → ζ, is a collection of isomorphisms of S-schemes:

ϕ = (ϕ_{v} : C_{v} → C′_{v})  (37)

indexed by v ∈ Vertex(τ) and satisfying (1) for v ∈ Vertex(τ), h′_{v} ∘ ϕ_{v} = h_{v}

(2) for f ∈ Flag(τ), ϕ_{∂f} ∘ q_{f} = q′_{f}.

One defines composition of morphisms in the obvious way. Notice that every morphism is an isomorphism. Thus the category of families of strict τ-maps over S is a groupoid. Given a morphism S′ ⟶^{u} S and a family η of strict τ-maps over S, one has the usual pullback u^{∗}(η) which is a family of strict τ-maps over S′. In this way we have the notion a category fibered in groupoids over the category of ℂschemes along with a clivage normalisée in the sense of [5, Exp. VI]. We denote this category by M(X, τ). We will occasionally also denote by M(X, τ) the associated lax 2-functor from the category of ℂ-schemes to the 2-category of groupoids (with a small skeletal subcategory).

of a functor from the category of ℂ-schemes to the category of groupoids which associates to each S the groupoid of families of strict τ-maps over S. We denote this functor by M(X, τ).

In every case it is easy to see that M(X, τ) is a stack in groupoids over ℂ. In many cases this is even a Deligne-Mumford stack: Theorem 3.10. If X is projective and L is ample, the functor M(X, τ) is a Deligne-Mumford stack which is separated and finite type over ℂ.

Proof. There is a 1-morphism M(X, τ) → \overline{M}(X, τ) where \overline{M}(X, τ) is the refined functor defined in [2]. In [2] it is proved that \overline{M}(X, τ) is a proper Deligne-Mumford stack over ℂ. And it is clear that M(X, τ) → \overline{M}(X, τ) is a representable morphism which is an open immersion. Thus M(X, τ) is a Deligne-Mumford stack which is separated and finite type over ℂ. □ 3.4. Properties and Related Constructions.

Notice that with our notation \overline{M}(X, τ_{r}(e)) is the moduli stack of Kontsevich stable maps \overline{M}_{0,r}(X, e), and M(X, τ_{r}(e)) simply parametrizes those stable maps such that the domain curve is irreducible. Definition 3.11. Suppose that τ is a stable A-graph and f ∈ Flag(τ). Then there is a 1-morphism

ev_{f} : \overline{M}(X, τ) → X  (38)

defined by sending a family of τ-maps, η (with notation as above), to the morphism h_{∂f} ∘ q_{f}.

If α = (α_{F} , α_{V}) : σ → τ is a combinatorial morphism of graphs, τ ↩ σ, there is an associated 1-morphism

\overline{M}(X, α) : \overline{M}(X, τ) → \overline{M}(X, σ).  (39)

If α is the inclusion of σ as a subgraph of τ, then \overline{M}(X, α) is the forgetful morphism which “remembers” only those components of τ-maps whose vertex is contained in σ. The reader is referred to [2, Theorem 3.6] for the precise definition. We will refer to the restriction of \overline{M}(X, α) to M(X, τ) as M(X, α).

If ϕ = (ϕ_{W} , ϕ^{F}) : τ → τ′ is a contraction of stable A-graphs, there is a corresponding 1-morphism of proper Deligne-Mumford stacks

\overline{M}(X, ϕ) : \overline{M}(X, τ) → \overline{M}(X, τ′).  (40)

This morphism “forgets” the labeling of some the individual components of the domain curve. The reader is referred to [2, Theorem 3.6] for the precise definition. We will denote by M(X, ϕ) the restriction of this 1-morphism to the open substack M(X, τ) of \overline{M}(X, τ).

One important case to understand is when β(τ) = 0. We have already defined M(X, τ_{∅}) = \overline{M}(X, τ_{∅}) = X where τ_{∅} is the empty graph. For any stable A-graph τ such that β(τ) = 0 and such that #Tail(τ) = r, we have M(X, τ) = X × M(∗, τ) where M(∗, τ) ⊂ \overline{M}_{0,r} is the obvious substack.

Consider the case that ϕ : τ → τ′ is a contraction of stable A-graphs such that β(τ) = β(τ′) = 0. Then M(X, τ) → \overline{M}(X, τ′) is simply the product of id_{X} : X → X with the 1-morphism M(∗, ϕ) : M(∗, τ) → \overline{M}(∗, τ′). In particular, using the notation of [2], consider the case that ϕ is an isogeny, i.e. ϕ is the morphism which removes some subset of the set of tails from τ and then stabilizes the resulting (possibly unstable) graph. Lemma 3.12. Let τ, τ′ be stable A-graphs such that β(τ) = β(τ′) = 0 and let ϕ : τ → τ′ be an isogeny. Then M(X, ϕ) : M(X, τ) → \overline{M}(X, τ′) is smooth of relative dimension dim(X, τ) − dim(X, τ′) with geometrically connected fibers.

Proof. Of course it is equivalent to prove that

M(∗, ϕ) : M(∗, τ) → \overline{M}(∗, τ′)  (41)

is smooth of relative dimension dim(X, τ)−dim(X, τ′) with geometrically connected fibers. Now it follows by Proposition 7.4 of [2] that M(∗, τ) and \overline{M}(∗, τ′) have the expected dimension. Thus all we really need to show is that M(∗, τ) → \overline{M}(∗, τ′) is smooth with geometrically irreducible fibers. Moreover, since every isogeny is a composition of morphisms obtained by stably removing one tail, we may suppose that ϕ : τ → τ′ corresponds to stably removing one tail f ∈ Tail(τ).

There are two cases. Suppose first of all that when we remove f from τ, the resulting graph is unstable. But then ∂f = v is a vertex with valence 3. Since a rational curve with 3 marked points has no moduli, we conclude that M(∗, τ) → \overline{M}(∗, τ′) is an open immersion.

The second case is that when we remove f from τ, the resulting graph is stable, i.e. the resulting graph is just τ′. But then if v = ϕ(∂f), we conclude that ϕ : M(∗, τ) → \overline{M}(∗, τ′) is simply an open subset of the universal curve over \overline{M}(∗, τ′) corresponding to the vertex v. In both cases we conclude that \overline{M}(∗, τ) → \overline{M}(∗, τ′) is smooth with geometrically connected fibers. □

For each stable A-graph τ define e = β(τ) and define r = #Tail(τ). Then there is a contraction ϕ : τ → τ_{r}(e) which is unique up to a labeling of the tails of τ. Definition 3.13. The contraction ϕ above is the canonical contraction. The corresponding 1-morphism (well-defined up to relabeling the tails)

M(X, ϕ) : M(X, τ) → \overline{M}_{0,r}(X, e)  (42)

will be referred to as the canonical contraction morphism.

Notice that the image of M(X, ϕ) as a subset of the set |\overline{M}_{0,r}(X, e)| is welldefined. Proposition 3.14. Let ϕ : τ → τ′ be a contraction of stable A-graphs. The image of the 1-morphism M(X, ϕ) is a locally closed subset of the topological space |\overline{M}(X, τ)|.

Proof. For notation’s sake let’s denote the continuous map of topological spaces

|\overline{M}(X, ϕ)| : |\overline{M}(X, τ)| → |\overline{M}(X, τ′)|  (43)

by f : M → M′ and let’s denote the open substack M(X, τ) of \overline{M}(X, τ) by M^{o}. Then f : M → M′ is a closed map. And it is easy to see that f^{−1}(f(M^{o})) = M^{o}. Therefore f(M^{o}) = f(M)−f(M −M^{o}) is a difference of closed sets and so is locally closed. □

We now fix n and α and consider the set S of all images

{Image(M(X, ϕ))}  (44)

as ϕ ranges over all contractions of stable A-graphs to τ_{n}(α). The set of isomorphism classes of such contractions is clearly finite. The previous lemma shows that S forms a locally closed decomposition of the topological space |\overline{M}_{0,n}(X, α)|, i.e. a partition of |\overline{M}_{0,n}(X, α)| into locally closed subsets. This partition is what we call the Behrend-Manin decomposition.

4. Flatness and Dimension Results

In this section we consider the dimensions of the stacks M(X, τ) and the evaluation morphisms. The main property we are interested in is the following: Definition 4.1. Given a stable A-graph τ, we say that D(X, τ) holds if the dimension of every irreducible component of M(X, τ) equals the expected dimension dim(X, τ).

By deformation theory there is an a priori lower bound on the dimension of any irreducible component of M(X, τ): Lemma 4.2. Every irreducible component of \overline{M}(X, τ) has dimension at least dim(X, τ). In particular, every irreducible component of M(X, τ) has dimension at least dim(X, τ).

Proof. This is a standard result. In the case that τ = τ_{r}(e) it follows from [3, section 5.2]. In the general case the theorem follows from [1]. The theorem isn’t actually stated in [1], so we show how it follows from results there.

Let 𝔐(τ) denote the stack of τ-marked prestable curves as in [1]. There is a forgetful 1-morphism of algebraic (Artin) stacks

\overline{M}(X, τ) → 𝔐(τ).  (45)

By [1, Lemma 1], 𝔐(τ) is smooth of dimension

#Tail(τ) − #Edge(τ) − 3.  (46)

Let ℭ(τ) → 𝔐(τ) be the universal curve. By [1, Proposition 4], \overline{M}(X, τ) is an open substack of the relative morphism-scheme,

Mor_{𝔐(τ)} (ℭ(τ), X × 𝔐(τ)) .  (47)

By [9, Theorem 2.17.1], it follows that every irreducible component of \overline{M}(X, τ) has dimension at least

h^{0}(C, f^{∗}T_{X}) − h^{1}(C, f^{∗}T_{X}) + #Tail(τ) − #Edge(τ).  (48)

By Riemann-Roch, we have (h^{0} − h^{1})(C, f^{∗}T_{X}) = −K_{X}.f_{∗}[C] + dim(X). We conclude that every irreducible component of \overline{M}(X, τ) has dimension at least

−K_{X}.f_{∗}[C] + dim(X) − 3 + #Flag(τ) − #Edge(τ) = dim(X, τ).  (49)

□

When Vertex(τ) has more than one element, we can try to reduce D(X, τ) to D(X, τ_{i}) for some proper subgraphs τ_{i} of τ, thus giving an inductive proof that D(X, τ) holds. To carry out such a proof, we need to know that the evaluation morphisms have constant fiber dimension. So we introduce the following property: Definition 4.3. Given a stable A-graph τ and a flag f ∈ Flag(τ), we say that E(X, τ, f) holds if

ev_{f} : M(X, τ) → X  (50)

is dominant and has constant fiber dimension dim(X, τ) − dim(X).

Notice that if there is any flag f ∈ Flag(τ) such that E(X, τ, f) holds, then it follows that D(X, τ) holds.

In the case that X ⊂ ℙ^{N} is a complete intersection, then the properties D and E are equivalent to stronger properties. Definition 4.4. Given a stable A-graph τ, we say that LCI(X, τ) holds if M(X, τ) is a local complete intersection and if D(X, τ) holds. Given a stable A-graph τ and a flag f ∈ Flag(τ), we say that FE(X, τ, f) holds if

ev_{f} : M(X, τ) → X  (51)

is flat of relative dimension dim(X, τ) − dim(X).

Lemma 4.5. If X ⊂ ℙ^{N} is a complete intersection, then D(X, τ) holds iff LCI(X, τ) holds. Also E(X, τ, f) holds iff FE(X, τ, f) holds. The same result holds with M(X, τ) replaced by \overline{M}(X, τ).

Proof. Suppose that X is a complete intersection of r = N − n hypersurfaces of degrees d_{1}, …, d_{r}. Consider M(ℙ^{N}, τ) and denote the universal curve by

π : C → M(ℙ^{N}, τ).  (52)

Let h : C → ℙ^{N} denote the universal map. Since

O_{ℙ^{N}} (d) := O_{ℙ^{N}} (d_{1}) ⊕ ⋯ ⊕ O_{ℙ^{N}} (d_{r})  (53)

is generated by global sections, also h^{∗}O_{ℙ^{N}} (d) is generated by global sections. On a genus 0 tree, if F is a sheaf generated by global sections then H^{1}(C, F) = 0, so R^{1}π_{∗} (h^{∗}O_{ℙ^{N}} (d)) = 0.

Now by [2, Proposition 7.4], M(ℙ^{N}, τ) is smooth of dimension

(N + 1)β(τ) + (N − 3) + #Flag(τ) − #Edge(τ).  (54)

So by [11, Corollary II.2] the pushforward E := π_{∗} (h^{∗}O_{ℙ^{N}} (d)) is a locally free sheaf of rank

∑_{i=1}^{r} χ(C, h^{∗}O_{ℙ^{N}} (d_{i})) = ∑_{i=1}^{r} (d_{i}β(τ) + 1) = (∑_{i} d_{i}) β(τ) + r.  (55)

Now the defining equations of the hypersurfaces in ℙ^{N} give a global section σ of E, and M(X, τ) is precisely the zero scheme of σ. Finally notice that the expected codimension, dim(ℙ^{n}, τ) − dim(X, τ), of M(X, τ) in M(ℙ^{N}, τ) is just

−K_{ℙ^{N}} .f_{∗}[C] + K_{X}.f_{∗}[C] + dim(ℙ^{N}) − dimX = (∑_{i} d_{i}) β(τ) + r.  (56)

Thus, if M(X, τ), then it follows that M(X, τ) is a local complete intersection. So if D(X, τ) holds, then also LCI(X, τ) holds. The opposite inclusion is obvious.

Now suppose that E(X, τ, f) holds. In particular D(X, τ) holds, so LCI(X, τ) holds. But now by [10, Theorem 23.1], ev_{f} is a dominant morphism from a Cohen-Macaulay scheme to a smooth scheme with constant fiber dimension, therefore it is flat. So FE(X, τ, f) holds. The opposite inclusion is obvious.

The same proof works when we replace M(X, τ) by \overline{M}(X, τ). □

Consider the following diagram:

Diagram 1

Here τ is an A-graph which contains the two subgraphs

a_{1} : τ ↩ τ_{1}, a_{2} : τ ↩ τ_{2}.  (57)

The edge {f_{1}, f_{2}} of τ is made up of the two tails f_{1} ∈ Tail(τ_{1}), f_{2} ∈ Tail(τ_{2}). Let f ∈ Flag(τ_{2}) be any flag (possibly f = f_{2}). Lemma 4.6. If FE(X, τ_{1}, f_{1}) and FE(X, τ_{2}, f) hold, then FE(X, τ, f) holds.

Proof. The combinatorial morphisms a_{1} and a_{2} give rise to a 1-morphism

M(X, a_{1}, a_{2}) : M(X, τ) → M(X, τ_{1}) ×_{ev_{f_{1}} ,X,ev_{f_{2}}} M(X, τ_{2}).  (58)

It is clear from the definition of strict τ-maps that M(X, a_{1}, a_{2}) is an open immersion.

Since ev_{f_{1}} : M(X, τ_{1}) → X is flat of relative dimension dim(X, τ_{1}) − dim(X), it follows that the projection morphism

pr_{2} : M(X, τ_{1}) ×_{ev_{f_{1}} ,X,ev_{f_{2}}} M(X, τ_{2}) → M(X, τ_{2})  (59)

is flat of relative dimension dim(X, τ_{1}) − dim(X). And ev_{f} : M(X, τ_{2}) → X is flat of relative dimension dim(X, τ_{2}) − dim(X). Thus the composite morphism

M(X, τ_{1}) ×_{ev_{f_{1}} ,X,ev_{f_{2}}} M(X, τ_{2}) ⟶^{pr_{2}} M(X, τ_{2}) ⟶^{ev_{f}} X  (60)

is flat of relative dimension dim(X, τ_{1}) + dim(X, τ_{2}) − 2dim(X). Of course ev_{f} : M(X, τ) → X is simply the restriction of the composite morphism, so it is flat of the same relative dimension. But notice that

dim(X, τ_{1}) + dim(X, τ_{2}) − dim(X) =

(−K_{X}.β(τ_{1}) + −K_{X}.β(τ_{2})) + 2 (dim(X) − 3) +
(#Flag(τ_{1}) + #Flag(τ_{2})) + (#Edge(τ_{1}) + #Edge(τ_{2})) − dim(X) =
−K_{X}.β(τ) + 2 (dim(X) − 3) + (#Tail(τ) + 2) −
(#Edge(τ) − 1) − dim(X) = dim(X, τ).

(61)

From this it follows that

ev_{f} : M(X, τ) → X  (62)

is flat of constant fiber dimension dim(X, τ)−dim(X). Thus FE(X, τ, f) holds. □

Definition 4.7. Suppose that τ is a stable A-graph and define the maximum component degree of τ to be

E(τ) = sup_{v∈Vertex(τ)} β(v).  (63)

Proposition 4.8. Suppose that τ is a stable A-graph with E(τ) = E. If for each e = 0, …, E we have FE(X, τ_{1}(e), f) holds, then for each flag f ∈ Flag(τ), FE(X, τ, f) holds.

Proof. We prove this by induction on #Vertex(τ). Suppose τ has a single vertex. Then τ = τ_{r}(e) for some r and e. If e = 0, then \overline{M}(X, τ_{r}(0)) = \overline{M}(∗, τ_{r}(0)) × X, and the evaluation morphism is just projection. So ev_{f} is flat of relative dimension

dim(\overline{M}(∗, τ_{r}(0))) = dim(X, τ_{r}(0)) − dim(X),  (64)

i.e. FE(X, τ_{r}(0), f) holds.

Next consider the case τ = τ_{r}(e) with e > 0. For any flag f ∈ Flag(τ_{r}(e)) there is a combinatorial morphism a : τ_{r}(e) ↩ τ_{1}(e) which maps the unique flag f_{1} ∈ Flag(τ_{1}(e)) to f. The associated 1-morphism

M(X, a) : M(X, τ_{r}(e)) → M(X, τ_{1}(e))  (65)

is isomorphic to an open subset of the (e − 1)-fold fiber product of the universal curve over M(X, τ_{1}(e)). Since the universal curve is smooth of relative dimension 1 over M(X, τ_{1}(e)), we conclude that M(X, a) is smooth of relative dimension e − 1. The evaluation morphism ev_{f} : M(X, τ_{r}(e)) → X factors as the composition

M(X, τ_{r}(e)) ⟶^{M(X,a)} M(X, τ_{1}(e)) ⟶^{ev_{f_{1}}} X.  (66)

Therefore if FE(X, τ_{1}(e), f_{1}) holds, then ev_{f} is flat of relative dimension

(e − 1) + (dim(X, τ_{1}(e)) − dim(X)) = dim(X, τ_{r}(e)) − dim(X),  (67)

in other words FE(X, τ_{r}(e), f) holds. Thus the Proposition is proved when τ has a single vertex.

Now suppose that τ has more than one vertex and suppose that for all e ≤ E = E(τ), FE(τ_{1}(e), f) holds. By way of induction, assume that the proposition is true for all graphs τ′ such that #Vertex(τ′) < #Vertex(τ). Let {f_{1}, f_{2}} be any edge. Define τ_{1} and τ_{2} to be the graphs obtained by breaking the edge into two tails (see diagram 1). Let f ∈ Flag(τ) be a flag, and without loss of generality suppose that f ∈ Flag(τ_{2}). Now E(τ_{2}) ≤ E(τ) and #Vertex(τ_{2}) < #Vertex(τ), so by the induction assumption FE(X, τ_{2}, f) holds. Also E(τ_{1}) ≤ E(τ) and #Vertex(τ_{1}) < #Vertex(τ), so by the induction assumption FE(X, τ_{1}, f_{1}) holds. Then by Lemma 4.6, we conclude that FE(X, τ, f) holds. So the proposition is proved by induction. □

5. Specializations

In the previous section we reduced the flatness and dimension results for a general stable A-graph τ to flatness and dimension results for the the stable A-graphs τ_{1}(e) with 0 ≤ e ≤ E(τ). In this section we will use specializations to reduce the flatness and dimension results for all τ_{1}(e), e > 1 to flatness and dimension results for a finite number of cases τ_{1}(e), e = 1, …, E(X) where E(X) is the threshold degree of X. We define a stable A-graph σ to be basic if for each vertex v ∈ Vertex(σ), we have β(v) ≤ E(X). The specializations we produce will show that every irreducible component of \overline{M}(X, τ) contains a basic locally closed subset M(X, σ). Thus to understand the irreducible components of \overline{M}(X, τ) it suffices to understand the irreducible components which pass through the general point of a basic locus M(X, σ).

Convention Suppose that we have a contraction ϕ : σ → τ. There is an induced morphism

M(X, ϕ) : M(X, σ) → \overline{M}(X, τ)  (68)

which is unramified with locally closed image. We will speak of M(X, σ) as though it is a substack of \overline{M}(X, τ). Thus given an irreducible component M ⊂ M(X, τ) and an irreducible component N ⊂ M(X, σ) we will say N ⊂ \overline{M} to mean that the image of N is contained in \overline{M}.

The basic lemma is the following easy version of Mori’s bend-and-break lemma. Lemma 5.1. Let e > 0. There is no complete curve contained in a fiber of the evaluation morphism

ev_{f_{1},f_{2}} : M(X, τ_{2}(e)) → X × X.  (69)

Proof. Suppose that C is a complete curve and ζ : C → M(X, τ_{2}(e)) has image in a fiber of ev_{f_{1},f_{2}}. Denote the family ζ of strict τ_{2}(e)-maps by (π : Σ → C, h, (q_{1}, q_{2})). Denote ev_{f_{1},f_{2}}(C) = (p_{1}, p_{2}). Let H ⊂ X be a hyperplane section containing neither p_{1} nor p_{2}. Define C′ = Σ ×_{X} H, so C′ is a finite ramified cover of C. Let B be the normalization of an irreducible component of C′ which dominates C. The basechange Σ×_{C} B now admits the two sections q_{1}, q_{2} as well as a third section q_{3} which is everywhere disjoint from both q_{1} and q_{2}. Any ℙ^{1}-bundle with three everywhere disjoint sections is isomorphic to ℙ^{1} ×B and the three sections are constant sections {0} × B, {1} × B, {∞} × B. But now the morphism h : Σ ×_{C} B → X contracts the sections q_{1} and q_{2}. By the rigidity lemma, [11, p. 43], we conclude that h factors through the projection Σ×_{C} B ≅ ℙ^{1} ×B → ℙ^{1}. So we conclude that C → M(X, τ) is a constant map. □ Corollary 5.2. Let M ⊂ \overline{M}(X, τ_{2}(e)) be an irreducible, closed substack and suppose that the fibers of ev_{f_{1},f_{2}} : M → X × X have dimension at least 1. Then M ∩ (\overline{M}(X, τ_{2}(e)) − M(X, τ_{2}(e))) is either all of M or contains an irreducible component with codimension 1 in M.

Proof. Suppose that M intersects M(X, τ_{2}(e)). Define I ⊂ X × X to be the image I = ev_{f_{1},f_{2}}(M). In order to prove that

∂M := M ∩ (\overline{M}(X, τ_{2}(e)) − M(X, τ_{2}(e)))  (70)

has an irreducible component of codimension 1 in M, it suffices to prove that for every (p_{1}, p_{2}) ∈ I, ev^{−1}(p_{1}, p_{2})∩∂M ⊂ ev^{−1}(p_{1}, p_{2})∩M has codimension 1. Suppose that it has codimension at least 2. Then the coarse moduli space |ev^{−1}(p_{1}, p_{2}) ∩ ∂M| ⊂ |ev^{−1}(p_{1}, p_{2})∩M| has codimension at least 2. Since the coarse moduli spaces are proper varieties, we can find a complete curve C in |ev^{−1}(p_{1}, p_{2})∩M| which does not intersect |ev^{−1}(p_{1}, p_{2}) ∩ ∂M|. Since \overline{M}(X, τ_{2}(e)) is a Deligne-Mumford stack, there exists a finite ramified cover C′ → C such that C′ → |\overline{M}(X, τ_{2}(e))| factors through \overline{M}(X, τ_{2}(e)) → |\overline{M}(X, τ_{2}(e))|. But then by Lemma 5.1, we conclude that C′ → |\overline{M}(X, τ_{2}(e))| is constant, which contradicts the construction of C. Therefore ∂M ⊂ M has codimension 1. □

The main application is the following: Proposition 5.3. Suppose that X ⊂ ℙ^{N} is a complete intersection. Suppose that FE(X, τ_{1}(e), f_{1}) holds for every e < E and suppose that every irreducible component of M(X, τ_{1}(E)) has dimension at least 2dim(X). Then FE(X, τ_{1}(E), f_{1}) holds as well.

Proof. By Lemma 4.5, to prove that FE(X, τ_{1}(E), f_{1}) holds, it suffices to prove that E(X, τ_{1}(E), f_{1}) holds. Let ζ ∈ M(X, τ_{1}(e)) be a point, denote p = ev_{f_{1}}(ζ), and let M ⊂ ev_{f_{1}}^{−1}(p) be an irreducible component which contains p. We need to prove that dim(M) = dim(X, τ_{1}(e)) − dim(X).

Now consider the forgetful morphism

M(X, a) : M(X, τ_{2}(e)) → M(X, τ_{1}(e)).  (71)

This is a smooth surjective morphism: let N ⊂ M(X, τ_{2}(e)) be the preimage of M. Then dim(N) = dim(M) + 1, thus we have to prove that

dim(N) = dim(X, τ_{1}(e)) + 1 − dim(X).  (72)

It suffices to prove that the general fiber of ev_{f_{2}} : N → X has dimension at most dim(X, τ_{1}(e)) + 1 − 2dim(X) (since we already know the dimension is at least this large).

Choose any point q ≠ p in ev_{f_{2}}(N) and consider N_{q} := ev_{f_{2}}^{−1}(q)∩N. By assumption, dim(X, τ_{1}(e))+1−2dim(X) ≥ 1, so dim(N_{q}) ≥ 1. Define \overline{N} ⊂ \overline{M}(X, τ_{2}(e)) to be the closure of N. By Corollary 5.2 we conclude that ∂\overline{N} ⊂ \overline{N} has codimension 1. In other words, there is a stable A-graph σ′ ≅ τ̸_{2}(e) whose canonical contraction is ϕ′ : σ′ → τ_{2}(e) and such that \overline{N} ∩ Image(M(X, σ′)) ⊂ \overline{N} has an irreducible component of codimension 1.

Now there is precisely one stable A-graph σ_{0} ≅ τ̸_{2}(e) whose stabilization after removing f_{2} equals τ_{1}(e), namely:

Diagram 2

By the assumption that q ≠ p, the point (q, p) ∉ ev_{f_{1},f_{2}}(M(X, σ_{0})). So σ′ is not σ_{0}. We conclude that the image of M(X, σ′) in \overline{M}(X, τ_{1}(e)) (under the map which stably removes f_{2}) is again a boundary component \overline{M}(X, σ) for some ϕ : σ → τ_{1}(e) (not the identity). Therefore \overline{M} ∩Image(M(X, σ)) ⊂ \overline{M} is a locally closed substack such that some irreducible component has codimension one in \overline{M}.

Since σ ≅ τ̸_{1}(E), we have E(σ) < E. By our assumption and by Proposition 4.8, we conclude that FE(X, σ, f_{1}) holds. In particular, \overline{M} ∩ Image(M(X, σ)) has dimension at most dim(X, σ) − dim(X). So the dimension of \overline{M} is at most dim(X, σ) + 1 − dim(X). Since dim(X, σ) + 1 ≤ dim(X, τ_{1}(e)), we conclude that dim(M) ≤ dim(X, τ_{1}(e)) − dim(X). So FE(X, τ_{1}(e), f_{1}) holds. □

Remark The condition that X ⊂ ℙ^{N} be a complete intersection is essentially superfluous. If instead we had worked throughout with the property E(X, τ, f) rather than FE(X, τ, f) (which is a little trickier), then the argument above proves the analogous result without this condition on X. Definition 5.4. If X ⊂ ℙ^{N} is a complete intersection of hypersurfaces of degrees d_{1}, d_{2}, …, d_{r}, define the threshold degree to be

E(X) = E(N, (d_{1}, …, d_{r})) = ⌊\frac{N + 2 − r}{N + 1 − (d_{1} + ⋯ + d_{r})}⌋ .  (73)

In particular, if X ⊂ ℙ^{N} is a hypersurface of degree d < \frac{N+1}{2} then E(X) = 1. Corollary 5.5. Suppose that X ⊂ ℙ^{N} is a complete intersection of hypersurfaces of degrees d_{1}, d_{2}, …, d_{r}. If FE(X, τ_{1}(e), f_{1}) holds for each 1 ≤ e ≤ E(X), then for every stable A-graph τ and flag f ∈ Flag(τ), FE(X, τ, f) holds. In particular, if X ⊂ ℙ^{N} is a hypersurface of degree d < \frac{N+1}{2} then it suffices to prove FE(X, τ_{1}(1), f_{1}).

Proof. By Proposition 4.8, to prove that FE(X, τ, f) always holds it suffices to prove that FE(X, τ_{1}(e), f_{1}) always holds (for e > 0). Now for e > E(X) we have by Lemma 4.2 that every irreducible component of M(X, τ_{1}(e)) has dimension at least

\begin{matrix}(N + 1 − (d_{1} + ⋯ + d_{r})) e + (N − r − 3) + 1 ≥ \\ (N + 1 − (d_{1} + ⋯ + d_{r})) \frac{N+2−r}{N+1−(d_{1}+···+d_{r})} + (N − r − 3) + 1, \end{matrix}  (74)

which, of course, is just 2dim(X). So by Proposition 5.3 and induction, to prove FE(X, τ_{1}(e), f_{1}) for all e, it suffices to prove FE(X, τ_{1}(e), f_{1}) in the cases e = 1, …, E(X). □ Corollary 5.6. With the same hypotheses as in Corollary 5.5, suppose that FE(X, τ_{1}(e), f_{1}) holds for each 1 ≤ e ≤ E(X). Then for every stable A-graph τ, M(X, τ) has pure dimension dim(X, τ).

Proof. If E(τ) = 0, then M(X, τ) = M(∗, τ) × X and the result follows from [2, Proposition 7.4]. Suppose that E(τ) > 0 and let v ∈ Vertex(τ) be such that β(v) > 0. Define a : τ′ ↩ τ to be the combinatorial morphism which adds a new tail f to v. Then M(X, a) is smooth and surjective of relative dimension 1. By Corollary 5.5, FE(X, τ′, f) holds. In particular, M(X, τ′) has pure dimension dim(X, τ′). It follows that M(X, τ) has pure dimension dim(X, τ′) − 1 = dim(X, τ). □

A second application of the proof of Proposition 5.3 is the following: Proposition 5.7. Suppose that X ⊂ ℙ^{N} is a complete intersection. Suppose that FE(X, τ_{1}(e), f_{1}) holds for every e < E and suppose that every irreducible component of M(X, τ_{1}(E)) has dimension at least 2dim(X). Then for every irreducible component M ⊂ M(X, τ_{0}(E)) there is a graph σ = τ_{0,0}(i, j), 0 < i, j and i+j = E, and an irreducible component N ⊂ M(X, τ_{0,0}(i, j)) such that N ⊂ \overline{M} is a codimension 1 subvariety.

Proof. Let M′ ⊂ M(X, τ_{1}(e)) be the irreducible component which dominates M. By the proof of Proposition 5.3 there is a graph σ ≅ τ̸_{1}(e) with canonical contraction ϕ : σ → τ_{1}(e) such that \overline{M′} ∩ M(X, σ) ⊂ \overline{M} has codimension 1. Now M(X, σ) has dimension dim(X, σ) and by Lemma 4.2 \overline{M′} has dimension at least dim(X, τ_{1}(e)). But dim(X, σ) = dim(X, τ_{1}(e)) − #Edge(σ). So σ has exactly one edge, i.e. σ = τ_{1,0}(i, j) for some i, j with i+j = E. By stability i, j > 0. Moreover, \overline{M′} ∩M(X, σ) has dimension dim(X, σ) so there is an irreducible component N′ ⊂ M(X, σ) such that N′ ⊂ \overline{M′}.

Since M(X, τ_{1,0}(i, j)) → M(X, τ_{0,0}(i, j)) is smooth of relative dimension 1, there is an irreducible component N ⊂ M(X, τ_{0,0}(i, j)) such that N′ is the preimage of N. Thus N ⊂ \overline{M} is a codimension 1 subvariety. □ Definition 5.8. Let X ⊂ ℙ^{N} be a complete intersection with threshold degree E(X) = E. A stable A-graph τ is basic for X if its maximal component degree E(τ) satisfies E(τ) ≤ E(X). Definition 5.9. For an A-graph τ, define its degree 0 subgraph to be the maximal subgraph τ ↩ τ^{0} such that E(τ^{0}) = 0, i.e. τ^{0} is the (possibly disconnected) subgraph of τ with

Vertex(τ^{0}) = {v ∈ Vertex(τ)|β(v) = 0}  (75)

and

Flag(τ^{0}) = {f ∈ Flag(τ)|∂f ∈ Vertex(τ^{0})}.  (76)

A contraction of A-graphs ϕ : σ → τ is nice if ϕ induces an isomorphism σ^{0} ≅ τ^{0}. Theorem 5.10. Let X ⊂ ℙ^{N} be a complete intersection, τ a stable A-graph and M ⊂ M(X, τ) an irreducible component. Suppose that FE(X, τ_{1}(e), f_{1}) holds for each 1 ≤ e ≤ E(X). Then there exists a nice contraction ϕ : σ → τ and an irreducible component N ⊂ M(X, σ) such that σ is basic and such that N ⊂ \overline{M}.

Proof. We will prove this by induction on the maximal component degree E(τ). If E(τ) ≤ E(X), then we can take ϕ to be the identity τ → τ and N = M.

Suppose that E > E(X). By way of induction, assume the theorem is proved for all graphs τ with E(τ) < E. We will deduce the theorem for graphs with E(τ) = E by induction on #Vertex(τ), and thus establish the theorem by induction on E(τ).

First we consider the case that #Vertex(τ) = 1, i.e. τ = τ_{r}(E) for some r. Define a : τ_{r}(E) ↩ τ_{0}(E) be the combinatorial morphism which strips the tails from τ_{r}(E). Then M(X, a) : M(X, τ_{r}(E)) → M(X, τ_{0}(E)) is smooth, surjective with connected fibers of dimension r. So we conclude that M is the preimage of an irreducible component M′ ⊂ M(X, τ_{0}(E)). Now by Proposition 5.7 there is a nice contraction ψ : ρ → τ_{0}(E) and an irreducible component L ⊂ M(X, ρ) such that L ⊂ \overline{M}′. As E(ρ) < E, by assumption there exists a nice contraction ϕ′ : σ′ → ρ such that σ′ is basic and there exists an irreducible component N′ ⊂ M(X, σ) such that N′ ⊂ \overline{L}. Therefore ψ ∘ ϕ′ : σ′ → τ_{0}(E) is a nice contraction and N′ ⊂ \overline{M′}. Now let v ∈ Vertex(σ′) be any vertex and let b : σ ↩ σ′ be the graph obtained by attaching r tails to v. Note that M(X, b) is smooth, surjective with connected fibers of dimension r. Let ϕ : σ → τ_{r}(E) be the contraction obtained from ψ ∘ ϕ′ by sending the r tails of σ to the r tails of τ_{r}(E). Then ϕ is a nice contraction and σ is basic. Define N ⊂ M(X, σ) to be the preimage of N′ under M(X, a). The morphism \overline{M}(X, a) is compatible with M(X, b), i.e. \overline{M}(X, a) ∘ M(X, ϕ) = M(X, ψ ∘ϕ′)∘M(X, b), and \overline{M}(X, a) is smooth along the image of M(X, ϕ). Thus we conclude that N ⊂ \overline{M}, the theorem is proved for M.

Now we consider the general case. For each graph τ, define

l(τ) := #Vertex(τ).  (77)

When l = 1, we have the case in the last paragraph. Suppose l > 1 and, by way of induction on l, suppose that for all stable A-graphs τ with E(τ) = E and with l(τ) < l, the theorem is proved. Suppose that τ is a stable A-graph with E(τ) = E and l(τ) = l. Let {f_{1}, f_{2}} be an edge of τ. Define a_{1} : τ ↩ τ_{1} and a_{2} : τ ↩ τ_{2} be the two subgraphs obtained by breaking the edge (see diagram 1).

Now the morphism M(X, a_{1}) : M(X, τ) → M(X, τ_{1}) is the composition of an open immersion and the projection of the fiber product

M(X, τ_{1}) ×_{ev_{f_{1}} ,X,ev_{f_{2}}} M(X, τ_{2}) → M(X, τ_{1}).  (78)

By Proposition 5.3 the morphism ev_{f_{2}} is flat. Therefore M(X, a_{1}) is flat. So M dominates an irreducible component M_{1} of M(X, a_{1}). Since l(τ_{1}) = l(τ) − l(τ_{2}) and l(τ_{2}) > 0 by the assumption that E(τ_{2}) = E, we have l(τ_{1}) < l(τ). So by the induction assumption, there exists a nice contraction ϕ_{1} : ρ_{1} → τ_{1} and an irreducible component L_{1} ⊂ M(X, ρ_{1}) such that ρ_{1} is basic and such that L_{1} ⊂ \overline{M_{1}}.

Since \overline{M}(X, a_{1}) is proper, there exists an irreducible subvariety L ⊂ \overline{M} such that \overline{M}(X, a_{1})(L) = L_{1} and such that the fiber dimension of L → L_{1} is at least the fiber dimension of M(X, a_{1}). Up to replacing L by an open subset, we may suppose L is contained in one of the locally closed substacks M(X, ρ). Since \overline{M}(X, a_{1}) maps L into M(X, ρ_{1}), we must have that ρ is glued from ρ_{1} and a graph ρ_{2} by making an edge out of f_{1} and f_{2}. Moreover ρ_{2} must contract to τ_{2}. But then the dimension of L is at most

dim(X, ρ_{1}) + dim(X, ρ_{2}) − dim(X) =
dim(X, ρ_{1}) + (dim(X, τ) − dim(X, τ_{1})) − (#Edge(ρ_{2}) − #Edge(τ_{2})) .

(79)

Thus we conclude that #Edge(ρ_{2}) = #Edge(τ_{2}), i.e. ρ_{2} = τ_{2}. So ψ : ρ → τ is a nice contraction and L ⊂ M(X, ρ) is an irreducible component such that L ⊂ \overline{M}. Moreover, l(ρ) = l(ρ_{2}) = l(τ_{2}) < l(τ). By the induction assumption, there exists a nice contraction ϕ : σ → ρ and an irreducible component N ⊂ M(X, σ) such that σ is basic and such that N ⊂ \overline{L}. But then ψ ∘ϕ : σ → τ is nice and N ⊂ \overline{M}, i.e. the theorem is proved for M. So the theorem is proved by induction on E and l. □

The previous theorem suggests a strategy for proving that any given M(X, τ) is irreducible:

(1) Determine all nice contractions ϕ : σ → τ such that σ is basic.

(2) Determine all irreducible components N ⊂ M(X, σ).

(3) Show that for each N, there is a unique irreducible component M(N) ⊂ M(X, σ) which contains N.

(4) Prove that all of the putative irreducible components M(N) are actually

equal. The first step (1) is a combinatorial problem. The simplest case for (2) is when M(X, σ) is itself irreducible for each basic σ. One can try to prove (3) by a deformation theory argument; if one proves that the general point of M is a smooth point of the stack M(X, τ), then it follows that there is a unique irreducible component M(N) which contains N. We will prove (4) by linking up basic graphs using almost basic graphs (we will explain this further below). Although one should be able to carry out this strategy in the case of complete intersections (and perhaps even more general varieties), in the remainder of this paper we will restrict ourselves to hypersurfaces X ⊂ ℙ^{N} with d < \frac{N+1}{2}. Then the steps above all reduce to questions regarding lines on X.

6. Properties of evaluation morphisms

In the last two sections we investigated when an evaluation morphism ev_{f} : M(X, τ) → X is flat of the expected dimension. In this section we also investigate when the general fiber is irreducible, and when the morphism ev_{f} is unobstructed at a general point of M(X, τ). By the same techniques in the last two sections, we reduce these properties for a general basic A-graph τ to the property for A-graphs of the form τ_{1}(e) with e = 1, …, E(X). Using this result, we carry out Steps (2) and (3) of the strategy of proof in the previous sections.

The new property of evaluation morphisms we want to consider is the following.

Definition 6.1. Suppose X ⊂ ℙ^{N} is a smooth subvariety, τ is a stable A-graph and f ∈ Flag(τ). We say that B(X, τ, f) holds if we have

(1) FE(X, τ, f) holds,

(2) the general fiber of ev_{f} is geometrically irreducible,

(3) in M(X, τ) there is a point [h : C → X] which is free, i.e. h^{∗}T_{X} is generated by global sections.

The difficult item to check is still (1). We return to this point at the end of this section. First we give a reformulation of item (3) of the definition above. Lemma 6.2. Suppose that (C, q_{f_{i}}) is a genus 0 prestable curve with dual graph τ, and suppose that E is a locally free sheaf on C. The following are equivalent:

(1) E is generated by global sections.

(2) For each irreducible component C_{v} of C, the restriction E_{v} of E to C_{v} is generated by global sections.

(3) For each irreducible component C_{v} of C and each point q ∈ C_{v}, we have H^{1}(C_{v}, E_{v}(−q)) = 0.

Proof. Clearly (1) implies (2). By Grothendieck’s Lemma [7, exercise V.2.4], E_{v} splits as a direct sum of line bundles L_{1} ⊕⋯⊕L_{r}. If (2) is satisfied, then each L_{v} is generated by global sections, i.e. if we identify C_{v} with ℙ^{1}, then L_{i} = O_{ℙ^{1}}(a_{i}) with a_{i} ≥ 0. Since H^{1}(ℙ^{1}, O_{ℙ^{1}}(a−1)) = 0 for a ≥ 0, we conclude that H^{1}(C_{v}, E_{v}(−q)) = 0. So (2) implies (3).

Finally suppose that (3) is satisfied. We shall prove that E is generated by global sections by induction on the number of vertices of τ, i.e. on the number of irreducible components of C. If C has a single irreducible component, then C is isomorphic to ℙ^{1}. By Grothendieck’s Lemma we know E = O_{ℙ^{1}}(a_{1})⊕⋯⊕O_{ℙ^{1}}(a_{r}) for some integers a_{i}. Since H^{1}(ℙ^{1}, E(−q)) = 0, we conclude that each a − 1 ≥ −1, i.e a ≥ 0. So E is generated by global sections.

Now suppose that τ has more than one vertex and let v_{1} be any leaf of τ, i.e. v_{1} is adjacent to exactly one other vertex. Let i_{1} : C_{1} → C be the irreducible component associated to v_{1}, let i_{2} : C_{2} → C be the union of all the other irreducible components of C and let q be the unique point of intersection of C_{1} and C_{2}. Let E_{1} denote the restriction of E to C_{1} and let E_{2} denote the restriction of E to C_{2}. By the induction assumption, we may assume that E_{2} is generated by global sections. But now we have an exact sequence of sheaves on C:

0 → (i_{1})_{∗}(E_{1}(−q)) → E → (i_{2})_{∗}(E_{2}) → 0.  (80)

The obstruction to lifting the global sections of E_{2} to global sections of E is an element of H^{1}(C_{1}, E_{1}(−q)), which is zero by assumption. So every global section of E_{2} is the restriction of a global section of E. Thus the locus where E isn’t generated by global sections (i.e. the cokernel of the morphism H^{0}(C, E) ⊗_{ℂ} O_{E} → E) is a closed subset of C_{1} − C_{2}. Since τ has more than one vertex, we can find a second leaf v_{2} of τ. Repeating the argument with v_{2} we conclude that E is generated by global sections. □ Lemma 6.3. With the same notation as in Lemma 6.2, if E satisfies any of the three equivalent conditions above, and if p ∈ C is any smooth point, then H^{1}(C, E(−p)) = 0.

Proof. If C has a single irreducible component, this follows from the equivalent condition (3) in Lemma 6.2. Suppose that C has l > 1 irreducible components. By way of induction, suppose that the lemma has been proved for all curves with fewer than l irreducible components. We can find a leaf v_{1} of C such that p is not contained in the corresponding irreducible component C_{2}. Let C_{2}, i_{1}, i_{2}, and q be as in the proof of Lemma 6.2. Then we have a short exact sequence:

0 → (i_{1})_{∗}(E_{1}(−q)) → E(−p) → (i_{2})_{∗}(E_{2}(−p)) → 0.  (81)

By the induction assumption, both H^{1}(C_{1}, E_{1}(−q)) = 0 and H^{1}(C_{2}, E_{2}(−p)) = 0. So by the long exact sequence in cohomology associated to the short exact sequence above, we conclude that H^{1}(C, E(−p)) = 0. So the lemma is proved by induction on l. □ Lemma 6.4. Suppose n > 2 and X ⊂ ℙ^{n} is a general hypersurface of degree d < \frac{n+1}{2}. Then B(X, τ_{1}(1), f_{1}) holds.

Proof. By Theorem 2.1, for general X we have that FE(X, τ_{1}(1), f_{1}) holds, i.e. we have (1). By Lemma 2.6, for general X there is a free line on X, i.e. we have (3).

By Theorem 2.7, for general X, F_{0,1}(X) is smooth. Thus by generic smoothness, the general fiber of ev_{f_{1}} is smooth. By Lemma 2.3, the general fiber of ev_{f_{1}} is a complete intersection in ℙ^{n−1} of dimension n − d − 1 > 1. Thus the general fiber is geometrically connected by repeated application of [4, corollaire 3.5, exp. XII]. Since a smooth, geometrically connected scheme is geometrically irreducible, we have (2). □

The main theorem of this section is the following: Proposition 6.5. Suppose X ⊂ ℙ^{N} is a smooth subvariety which satisfies B(X, τ_{1}(e), f_{1}) for e = 1, …, E. Let τ be an A-graph such that E(τ) ≤ E. Then we have the following:

(1) For each f ∈ Flag(τ), we have B(X, τ, f).

(2) M(X, τ) is an irreducible stack.

Proof. Both statements are trivial in case τ is empty, so assume τ is nonempty. Observe that (1) implies (2): given v ∈ Vertex(τ), define a new A-graph τ′ and a combinatorial morphism α : τ′ ↩ τ which attaches a new flag f′ to τ at v. Then M(X, α) : M(X, τ′) → M(X, τ) is smooth, surjective with geometrically irreducible fibers. So M(X, τ′) is irreducible iff M(X, τ) is irreducible. By (1), ev_{f′} : M(X, τ′) → X is flat and the general fiber is geometrically irreducible. Since X is irreducible, it follows that M(X, τ′) is irreducible. So it remains to prove (1).

First of all, suppose that β(τ) = 0. Let α : τ ↩ ∅ be the unique morphism. Then ev_{f} coincides with M(X, α) : M(X, τ) → M(X, ∅) = X. Thus (1) follows from Lemma 3.12. So we are reduced to the case that β(τ) > 0.

We prove (1) by induction on the number of vertices of τ. Suppose that τ has a single vertex, i.e. τ = τ_{r}(e) for some r > 0. Let α : τ_{r}(e) ↩ τ_{1}(e) be the unique combinatorial morphism which maps f_{1} ∈ Flag(τ_{1}(e)) to f ∈ Flag(τ_{r}(e)). Then ev_{f} factors as the composition

M(X, τ_{r}(e)) ⟶^{M(X,α)} M(X, τ_{1}(e)) ⟶^{ev_{f_{1}}} X.  (82)

Of course M(X, α) is an open immersion into the (r − 1)-fold fiber product of the universal curve, so M(X, α) is smooth with geometrically irreducible fibers. And by B, ev_{f_{1}} is flat and the general fiber is geometrically irreducible. Therefore the composition is flat and the general fiber is geometrically irreducible, i.e. items (1) and (2) of condition B(X, τ_{r}(e), f) hold. The condition that h^{∗}T_{X} is generated by global sections is independent of the number of vertices, so item (3) of B(X, τ_{r}(e), f) holds as well, i.e. B(X, τ_{r}(e), f) holds.

Now suppose that there is more than one vertex, say #Vertex(τ) = l > 1. By way of assumption, suppose that B(X, σ, f) has been proved for all A-graphs σ such that #Vertex(σ) < l. Let {f_{1}, f_{2}} be any edge and consider the subgraphs α_{1} : τ ↩ τ_{1} and α_{2} : τ ↩ τ_{2} as in diagram 1. Without loss of generality, suppose that f is in τ_{1}. Then ev_{f} factors as the composition:

M(X, τ) ⟶^{M(X,α_{1})} M(X, τ_{1}) ⟶^{ev_{f}} X.  (83)

Now M(X, α_{1}) factors as the composition of an open immersion and the projection

π_{1} : M(X, τ_{1}) ×_{ev_{f_{1}} ,X,ev_{f_{2}}} M(X, τ_{2}) → M(X, τ_{1}).  (84)

Since #Vertex(τ_{i}) < l for i = 1, 2, the induction assumption says that (2) holds for ev_{f_{i}} : M(X, τ_{i}) → X. Since ev_{f_{1}} is open, the general fiber of π_{1} dominates the general fiber of ev_{f_{2}}. So π_{1} is flat and the general fiber is geometrically irreducible. Thus the same is true of M(X, α_{1}). Since #Vertex(τ_{1}) < l, ev_{f} : M(X, τ_{1}) → X is flat and the general fiber is geometrically irreducible. Thus the composition is flat and the general fiber is geometrically irreducible, i.e. items (1) and (2) of B(X, τ, f) hold.

Finally we consider item (3) of B(X, τ, f). Each of the two projections M(X, α_{1}) and M(X, α_{2}) are dominant. By the induction assumption, for i = 1, 2 the set of points in M(X, τ_{1}) which parametrize stable maps with h ∗ T_{X} generated by global sections is an open, dense set U_{i}. The preimage of each U_{i} in M(X, τ) is an open dense set, and the intersection of these two open dense sets is an open dense set. For a point in this intersection – using the equivalent condition (2) of Lemma 6.2 – we have that the restriction of h^{∗}T_{X} to each irreducible component with vertex v ∈ τ_{1} is generated by global sections, and also the restriction of h^{∗}T_{X} to each irreducible component with vertex v ∈ τ_{2} is generated by global sections. So by the equivalent condition (2) of Lemma 6.2, we conclude that h^{∗}T_{X} is generated by global sections. Thus item (3) of B(X, τ, f) is satisfied. This completes the proof that B(X, τ, f) holds, and the proposition is proved by induction. □

Propsition 6.5 simplifies Step 2 in the strategy of the last section to checking that B(X, τ_{1}(e), f_{1}) holds for all e = 1, …, E(X). Next we reduce Step 3 to checking that B(X, τ_{1}(e), f_{1}) holds for all e = 1, …, E(X). Proposition 6.6. Suppose that τ is a stable A-graph, f ∈ Tail(τ) and suppose that B(X, τ, f) holds. Suppose that α : τ → σ is a contraction. The morphism M(X, α) maps a general point of M(X, τ) to a point in the smooth locus of the morphism ev_{f} : \overline{M}(X, σ) → X.

Proof. As in the proof of Proposition 6.5, the case that β(σ) = 0 follows from Lemma 3.12. So we are reduced to the case β(σ) > 0.

Let 𝔐(σ) denote the (non-separated) Artin stack of prestable σ-curves as in [2, definition 2.6]. There is a 1-morphism \overline{M}(X, σ) → 𝔐(σ) given by forgetting the map to X. “Remembering” the map to X gives an isomorphism of \overline{M}(X, σ) with the relative scheme of morphisms Mor_{𝔐}(σ)(σ, X), by [1, prop. 4]. Since 𝔐(σ) is smooth by [1, prop. 2], to prove that ev_{f} is smooth at a point, it suffices to prove the following morphism is smooth at this point:

(π, ev_{f}) : \overline{M}(X, σ) → 𝔐 × X.  (85)

By [9, Theorem II.1.7], to check that a point ζ = ((C_{v}), (q_{f′}), (h_{v})) of \overline{M}(X, σ) is in the smooth locus of (π, ev_{f}), it suffices to check that H^{1}(C, h^{∗}T_{X}(−q_{f})) = 0. For a general point in the image of M(X, α), this follows from item (3) of B(X, τ, f) along with Lemma 6.3. □ Corollary 6.7. Suppose X ⊂ ℙ^{N} is a smooth subvariety which satisfies B(X, τ_{1}(e), f_{1}) for all e = 1, …, E. Let τ be an A-graph with E(τ) ≤ E and suppose that α : τ → σ is a contraction. The morphism M(X, α) maps a general point of M(X, τ) to a smooth point of \overline{M}(X, σ).

Proof. This is trivial if τ is empty. Suppose τ is not empty and let v be a vertex of τ. Let τ ↩ τ′ be the combinatorial morphism which attaches a new tail, f, at the vertex v. Let σ ↩ σ′ be the combinatorial morphism which attaches a new tail, f, at the vertex of σ which is the image of v. Let α′ : τ′ → σ′ be the contraction which restricts to α and which maps f to f. By Proposition 6.5, B(X, τ′, f) holds. By Proposition 6.6, M(X, α′) maps a general point of M(X, τ′) to a point in the smooth locus of ev_{f} : \overline{M}(X, σ′) → X. A point in the smooth locus of ev_{f} is a smooth point of \overline{M}(X, σ′). The image of this point in \overline{M}(X, σ) is also a smooth point. Since M(X, τ′) surjects onto M(X, τ), a general point of M(X, τ′) maps to a general point of M(X, τ). Thus M(X, α) maps a general point of M(X, τ) to a smooth point of \overline{M}(X, σ). □

Remark: Now suppose B(X, τ_{1}(e), f_{1}) holds for all e = 1, …, E, suppose that τ is a stable A-graph with E(τ) ≤ E and suppose that α : τ → σ is a contraction. By Corollary 6.7 and (2) of Proposition 6.5, we conclude that there is a unique irreducible component M(α) of \overline{M}(X, σ) which contains the image of M(X, α), and M(α) is smooth of the expected dimension at a general point. So Steps (2) and (3) of the strategy in the last section are successful.

Finally we give a simpler criterion for when B(X, τ, f) holds for all τ with E(τ) ≤ E, where E is some fixed integer, and also reduce the number of components M(α) we have to deal with in Step (4) of our strategy. Proposition 6.8. Suppose that X ⊂ ℙ^{N} is a smooth subvariety satisfying

(1) B(X, τ_{1}(1), f_{1}) holds,
(2) FE(X, τ_{1}(e), f_{1}) holds for e = 1, …, E, and
(3) M(X, τ_{0}(e)) is irreducible for e = 1, …, E.

Then for each stable A-graph τ with E(τ) ≤ E and each flag f ∈ Flag(τ), B(X, τ, f) holds and there is a nice contraction α : σ → τ such that E(σ) ≤ 1 and such that M(X, α) maps the general point of M(X, σ) to a smooth point of \overline{M}(X, τ).

Proof. It is easy to see that there is always a nice contraction α : σ → τ such that E(σ) ≤ 1. By Proposition 6.6, we know that M(X, α) maps a general point of M(X, σ) to a smooth point of \overline{M}(X, τ).

By Proposition 6.5, to prove that B(X, τ, f) holds for all τ with E(τ) ≤ E, it suffices to prove that B(X, τ_{1}(e), f_{1}) holds for all e = 1, …, E. Let \overline{M}(X, τ_{1}(e))′ denote the normalization of \overline{M}(X, τ_{1}(e)). Consider the Stein factorization \overline{M}(X, τ_{1}(e))′ → Z → X of ev_{f_{1}}. By Proposition 6.6, there is a open, dense subset U ⊂ M(X, σ) such that M(X, α) maps U into the smooth locus of ev_{f_{1}}. So M(X, α)|_{U} : U → \overline{M}(X, τ_{1}(e)) factors through \overline{M}(X, τ_{1}(e))′. Now consider the image V of U in Z. By Proposition 6.5, the general fiber of ev_{f_{1}}|_{U} : U → X is geometrically irreducible. Therefore V → X is generically injective.

Let \overline{V} ⊂ Z be the Zariski closure of V with the induced, reduced scheme structure. Then \overline{V} is an irreducible stack and \overline{V} → X is surjective and generically injective. In particular, \overline{V} is a nonempty irreducible component of Z. So the preimage of V in \overline{M}(X, τ_{1}(e)) is a nonempty irreducible component of \overline{M}(X, τ_{1}(e)). By Corollary 5.6, every stratum in the Behrend-Manin decomposition of \overline{M}(X, τ_{1}(e)) has the expected dimension. Thus M(X, τ_{1}(e)) is Zariski dense in \overline{M}(X, τ_{1}(e)). By assumption, M(X, τ_{0}(e)) is irreducible. Since M(X, τ_{1}(e)) → M(X, τ_{1}(e)) is smooth with geometrically connected fibers, also M(X, τ_{1}(e)) is irreducible. Therefore \overline{M}(X, τ_{1}(e)) is irreducible. Therefore \overline{V} = Z and we conclude that the general fiber of \overline{M}(X, τ_{1}(e))′ → X is normal and geometrically connected, thus geometrically irreducible. It follows that the general fiber of M(X, τ_{1}(e)) → X is geometrically connected, i.e. we have established item (2) of the definition of B(X, τ_{1}(e), f_{1}).

To establish item (3) of the definition of B(X, τ_{1}(e), f_{1}), observe that the locus of points in \overline{M}(X, τ_{1}(e)) parametrizing stable maps for which h^{∗}T_{X} is generated by global sections is an open locus. By Lemma 6.2 and the assumption B(X, τ_{1}(1), f_{1}), for a general point of M(X, σ), we have that h^{∗}T_{X} is generated by global sections. So this open set intersects the general point of the image of M(X, α), so it is nonempty. Therefore it intersects M(X, τ) and item (3) follows. □

We summarize the results of this section for the case of complete intersections in the following corollary. Corollary 6.9. Suppose that X ⊂ ℙ^{N} is a smooth complete intersection of threshold degree E(X) which satisfies:

(1) B(X, τ_{1}(1), f_{1}) holds,

(2) FE(X, τ_{1}(e), f_{1}) holds for e = 1, …, E(X), and
(3) M(X, τ_{0}(e)) holds for e = 1, …, E(X).

Then we have (1) For each basic A-graph τ and each flag f ∈ Flag(τ), B(X, τ, f) holds.

(2) For each stable A-graph τ and each contraction α : σ → τ of a basic Agraph σ to τ, there is a unique irreducible component M(α) of \overline{M}(X, τ) which contains the image of M(X, α). Moreover M(α) is smooth of the expected dimension at a general point of the image of M(X, α).

(3) \overline{M}(X, τ) is the union of the irreducible components M(α) as α : σ → τ ranges over nice contractions such that E(σ) ≤ 1.

Proof. The only new statement is (3). By Theorem 5.10, we know that each irreducible component of \overline{M}(X, τ) is one of the irreducible components M(α′) for a nice contraction α′ : σ′ → τ with σ a basic A-graph. Of course we can find a nice contraction β : σ → σ′ such that E(σ) ≤ 1. Let α : σ → τ be the composition of β and α′. Then M(α′) is an irreducible component which contains the image of M(X, α). So M(α′) = M(α), i.e. we have proved (3). □

7. Equating Irreducible Components

Suppose that X ⊂ ℙ^{N} is a complete intersection which satisfies the hypotheses of Corollary 6.9. Then for each stable A-graph τ, we know that \overline{M}(X, τ) has the expected dimension and is a union of irreducible components M(α) as α : σ → τ ranges over nice contractions with E(σ) ≤ 1. To prove that \overline{M}(X, τ) (and hence M(X, τ)) is irreducible, we are reduced to proving that the irreducible components M(α) are all equal.

Suppose that B(X, τ_{1}(e), f_{1}) holds for all e = 1, …, E where E is some integer with E ≥ E(X). Fix a stable A-graph τ and let S_{E}(τ) be the set of (isomorphism classes of) nice contractions α : σ → τ with E(σ) ≤ E. Define a relation α ≤ α′ if there exists a contraction ϵ : σ → σ′ such that α = α′ ∘ ϵ. If α ≤ α′, then observe M(α) = M(α′). Form the equivalence relation ≅ on S_{E}(τ) generated by ≤. Notice conclusion (3) of Corollary 6.9 implies that every equivalence class contains a contraction α : σ → τ such that E(σ) ≤ 1. Since M(α) = M(α′) if α ≅ α′, we see that the number of irreducible components of \overline{M}(X, τ) is bounded by the number of equivalence classes of ≅ on S_{E}(τ). So to prove that \overline{M}(X, τ) is irreducible, it suffices to prove that every two elements of S_{E}(τ) are equivalent. Definition 7.1. Given X ⊂ ℙ^{N} a smooth complete intersection, define the modified threshold degree of X to be E′(X) = max(E(X), 2). Proposition 7.2. Suppose that X ⊂ ℙ^{N} is a smooth complete intersection such that for e = 1, …, E′(X), we have B(X, τ_{1}(e), f_{1}) holds. Then for each positive integer e, every two elements of S_{E′(X)}(τ_{0}(e)) are equivalent. In particular \overline{M}(X, τ_{0}(e)) is irreducible.

Proof. Recall a connected tree τ is called an path if τ has precisely one or two vertices (so no vertex has valence greater than 2). The number of vertices in a path is the diameter of the path. Given any connected tree τ, the diameter of τ, diam(τ), is defined to be the maximum diameter of a subgraph which is a path. If α : σ → τ_{0}(e) is a nice contraction, then there are at most e vertices in σ. So the diameter of σ is at most e. Moreover, there is a unique contraction α_{e} : σ_{e} → τ_{0}(e) with diam(σ_{e}) = e. Here σ_{e} is the A-graph whose underlying graph is the path of length e, and for each vertex v ∈ σ_{e} we have β(v) = 1.

To prove that any two elements in S_{E′(X)}(τ_{0}(e)) are equivalent, it suffices to prove that any two nice contractions α : σ → τ with E(σ) = 1 are equivalent. We will prove that for each such α : σ → τ with diam(σ) < e, there is a nice contraction α′ : σ′ → τ such that σ ≅ σ′, E(σ′) = 1 and diam(σ′) ≥ diam(σ). From this it follows by induction that all such contractions are equivalent to α_{e} : σ_{e} → τ.

Suppose that α : σ → τ is a nice contraction with E(σ) = 1 and diam(σ) < e. Let γ ↪ σ be a subgraph which is a path such that diam(γ) = diam(σ). Since γ does not equal σ, there exists a vertex v_{1} of γ such that the valence of v_{1} is at least 3. Let f_{1}, f_{2} be an edge of σ not contained in γ such that ∂f_{1} = v_{1}. Let v_{2} = ∂f_{2}. Form the nice contraction ϵ : σ → ρ which contracts v_{1} and v_{2} to a common vertex v of σ with β(v) = 2. The nice contraction α : σ → τ_{0}(e) factors through a nice contraction α_{ρ} : ρ → τ_{0}(e).

The image of γ in ρ is a path γ_{ρ} which contains v. Now let γ′ → γ_{ρ} be a contraction of a path of length diam(γ) + 1 which contracts two adjacent vertices w_{1} and w_{2} to v (where β(w_{1}) = β(w_{2}) = 1). There is a unique nice contraction ϵ′ : σ′ → ρ such that γ′ is a path in σ′, such that the restriction of ϵ′ to γ′ is just γ′ → ρ, such that every flag of v not contained in γ_{ρ} is the image of a flag of w_{1}, and which is an isomorphism from σ′ − γ′ → ρ − γ_{ρ}. Define α′ = α_{ρ} ∘ ϵ. Then α′ : σ′ → τ_{0}(e) is a nice contraction, E(σ′) = 1, α ≅ α′ and diam(σ′) = diam(σ)+1. This proves the claim.

So by induction on the diameter of σ, every element of S_{E′(X)}(τ_{0}(e)) is equivalent to α_{e} : σ_{e} → τ_{0}(e). In particular \overline{M}(X, τ_{0}(e)) = \overline{M}_{0,0}(X, e) is irreducible. □ Corollary 7.3. With the same hypotheses as in Proposition 7.2, for each stable A-graph τ we have

(1) \overline{M}(X, τ) is an integral, local complete intersection stack of the expected dimension dim(X, τ), and M(X, τ) is the unique dense stratum in the Behrend-Manin decomposition.

(2) For each flag f ∈ Flag(τ), B(X, τ, f) holds.

(3) For each contraction α : σ → τ, \overline{M}(X, τ) is smooth at the general point of the image of M(X, α) : M(X, σ) → \overline{M}(X, τ).

Proof. By Corollary 5.5, FE(X, τ_{1}(e), f_{1}) holds for all integers e > 0. By assumption, B(X, τ_{1}(1), f_{1}) holds. And by Proposition 7.2, M(X, τ_{0}(e)) is irreducible for each integer e > 0. Thus by Proposition 6.8, for every stable A-graph τ and every flag f ∈ Flag(τ), we have that B(X, τ, f) holds. This establishes (2).

As in the proof of Proposition 6.5, (2) implies that for every stable A-graph τ, M(X, τ) is irreducible of the expected dimension. By a parameter count, we conclude that M(X, τ) is the unique dense stratum of the Behrend-Manin decomposition of \overline{M}(X, τ). So \overline{M}(X, τ) is also irreducible of the expected dimension, and generically smooth. So D(X, τ) holds. By Lemma 4.5, we conclude that LCI(X, τ) holds, i.e. \overline{M}(X, τ) is a local complete intersection stack. Since it is generically smooth, and thus generically reduced, it is reduced. So \overline{M}(X, τ) is an integral, local complete intersection stack of the expected dimension dim(X, τ) and M(X, τ) is the unique dense stratum in the Behrend-Manin decomposition. This establishes (1).

Finally (3) follows from (1) and Corollary 6.7. □

Finally, we prove that a general hypersurface X ⊂ ℙ^{n} of degree d < \frac{n+1}{2} satisfies the hypotheses of Proposition 7.2. Proposition 7.4. Suppose n > 2, d ≤ \frac{n+1}{2} and suppose X ⊂ ℙ^{n} is a hypersurface of degree d, so E′(X) = 2. If B(X, τ_{1}(1), f_{1}) holds for X (recall from Lemma 6.4 that B(X, τ_{1}(1), f_{1}) holds for a general X) then also B(X, τ_{1}(2), f_{1}) holds. For such an X, the results of Corollary 7.3 hold.

Proof. By Corollary 5.5, LCI(X, τ_{1}(2), f_{1}) holds. Since M(X, τ_{0}(2)) is the unique dense stratum in the Behrend-Manin decomposition, to prove that M(X, τ_{0}(2)) is irreducible, it is equivalent to prove that \overline{M}(X, τ_{0}(2)) is irreducible. To see that \overline{M}(X, τ_{0}(2)) is irreducible, observe by Theorem 5.10 that every irreducible component of \overline{M}(X, τ) is of the form M(α) for a nice contraction α : σ → τ_{0}(2) with E(α) = 1. But there is a unique such contraction, namely α_{2} : σ_{2} → τ_{0}(2). So M(X, τ_{0}(2)) is irreducible.

Now by property 6.8, B(X, τ_{1}(2), f_{1}) holds. □

References

[1] K. Behrend. Gromov-Witten invariants in algebraic geometry. Inventiones Mathematica, 127:601–617, 1997. arXiv: math.AG/9601011.

[2] K. Behrend and Y. Manin. Stacks of stable maps and Gromov-Witten invariants. Duke Math Journal, 85:1–60, 1996. arXiv: math.AG/9506023.

[3] W. Fulton and R. Pandaharipande. Notes on stable maps and quantum cohomology. In Algebraic geometry – Santa Cruz 1995, pages 45–96. American Mathematical Society, 1995. arXiv: math.AG/9608011.

[4] A. Grothendieck. Cohomologie locale des faisceaux cohérents ... (SGA 2), volume 2 of Advanced Studies in Pure Mathematics. North-Holland Publishing Co., 1968.

[5] A. Grothendieck. Revêtements etales et groupe fondamental (SGA 1), volume 224 of Lecture notes in mathematics. Springer-Verlag, 1971.

[6] J. Harris and J. Starr. Rational curves on hypersurfaces of low degree, II. preprint arXiv: math.AG/0207257.

[7] R. Hartshorne. Algebraic Geometry, volume 52 of Graduate Texts in Mathematics. Springer-Verlag, 1977.

[8] B. Kim and R. Pandharipande. The connectedness of the moduli space of maps to homogeneous spaces. In Symplectic geometry and mirror symmetry, chapter 5. World Scientific, 2001. arXiv: math.AG/0003168.

[9] J. Kollár. Rational Curves on Algebraic Varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. Springer-Verlag, 1996.

[10] H. Matsumura. Commutative ring theory, volume 8 of Cambridge studies in advanced mathematics. Cambridge University Press, 1986.

[11] D. Mumford. Abelian varieties, volume 5 of Tata Institute of Fundamental Research Studies in Mathematics. Oxford University Press, 1970.

Department of Mathematics, Harvard University, Cambridge, MA 02138, USA

E-mail address: harris@math.harvard.edu

Department of Mathematics and Statistics, Queens University, Kingston, Ontario, K7L3N6, Canada

E-mail address: mikeroth@mast.queensu.ca

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

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