CUBIC FOURFOLDS AND SPACES OF RATIONAL CURVES

A.J DE JONG AND JASON STARR

Abstract. For a general nonsingular cubic fourfold X ⊂ ℙ^{5} and e ≥ 5 an odd integer, we show that the space M_{e} parametrizing rational curves of degree e on X is non-uniruled. For e ≥ 6 an even integer, we prove that the generic fiber dimension of the maximally rationally connected fibration of M_{e} is at most one, i.e. passing through a very general point of M_{e} there is at most one rational curve. For e < 5 the spaces M_{e} are fairly well understood and we review what is known.

1. Introduction

Let k be an algebraically closed field of characteristic 0. Let X be a nonsingular cubic hypersurface in ℙ_{k}^{5}. For each integer e ≥ 1 denote by M_{e} the variety parametrizing smooth, geometrically connected curves in X of degree e and arithmetic genus 0, i.e., M_{e} is the scheme of rational curves of degree e in X. The scheme M_{e} is an irreducible variety of dimension 3e + 1. This is nontrivial and is discussed in Section 2. In this paper we consider the birational geometry of M_{e}, specifically the following questions:

(1) What is the Kodaira dimension of M_{e}?

(2) In case the Kodaira dimension is negative, what is the dimension of the general fiber of the maximally rationally connected fibration of M_{e} (cf. [16])?

These questions were originally raised by Joe Harris with regard to the well-known problem of rationality/irrationality of cubic fourfolds (we do not solve this problem). It is a pleasure to acknowledge useful conversations with Joe Harris.

In Section 2 we discuss different compactifications of M_{e} and how they are related. Let \overline{M}_{e} be a desingularization of a compactification of M_{e}. Question 2 can be rephrased: For a very general point p ∈ \overline{M}_{e}, what is the maximal dimension of a closed, rationally connected subvariety Z ⊂ \overline{M}_{e} containing p? Equivalently, denoting by \overline{M}_{e} → Q the MRC fibration (in the sense of [16, Def. IV.5.3]), what is the difference dim(M_{e}) − dim(Q)? If this is 0, then for a very general point p ∈ \overline{M}_{e} there is no non-constant morphism ℙ^{1} → \overline{M}_{e} whose image contains p, i.e., \overline{M}_{e} is not uniruled. We note that the invariant dim Z is a birational invariant of M_{e} (so it is independent of the choice of desingularized compactification).

Discussions with Joe Harris have led to the list of maximal dimensions for small values of e:

\begin{matrix}e & 1 & 2 & 3 & 4 \\ dim M_{e} & 4 & 7 & 10 & 13 \\ dim Z & 0 & 3 & 2 & 3 \end{matrix}

Date: February 5, 2004.

We pause to explain this table. The case of lines is well known, namely M_{1} is a 4-dimensional hyperKähler manifold [3, Prop. 1]. In the case of conics, the set of all conics residual to a fixed line is a 3 dimensional rationally connected variety Z. In the case of cubic rational curves, note that a general cubic curve lies on a unique cubic surface and moves in a 2-dimensional linear system on it. So Z has dimension at least 2. A general quartic rational curve lies on a unique cubic threefold, and moves in a 3-dimensional rationally connected family on it (c.f. [10, Theorem 8.2]), so Z has dimension at least 3. This gives a lower bound for the numbers in the bottom row of the diagram, which is easily seen to be the actual dimension of Z when e = 1 or 2. For e = 3 and e = 4, we have not verified these numbers give the actual dimensions, but we would be surprised if they turn out to be larger. Ana-Maria Castravet conjectured that for e = 4, the actual dimension of Z is 3 and the target of the MRC fibration of \overline{M}_{4} is birational to the relative intermediate Jacobian of the family of hyperplane sections of X – in other words, the relative intermediate Jacobian of the family of hyperplane sections of X is not uniruled.

Theorem 1.1. Let X ⊂ ℙ^{5} be a very general cubic fourfold. For every odd degree e ≥ 5, the variety M_{e} is not uniruled. For every even degree e ≥ 6 the variety M_{e} has dim(Z) ≤ 1.

Actually our method gives something a little better than Theorem 1.1.

Theorem 1.2. Let X ⊂ ℙ^{5} a smooth cubic hypersurface, and let \overline{M_{e}} be a nonsingular projective model of M_{e}. There is a canonical section ω_{e} ∈ H^{0}(\overline{M}_{e}, Ω_{\overline{M}_{e}}^{2}) with the following property:

(a) If e is odd, e ≥ 5, if X is general, and if p is a general point of \overline{M}_{e}, then ω_{e} induces a nondegenerate pairing on T_{p}(\overline{M}_{e}).

(b) If e is even, e ≥ 6, if X is general, and if p ∈ \overline{M}_{e} a general point, then the linear map T_{p}(\overline{M}_{e}) → T_{p}^{∨}(\overline{M}_{e}) induced by ω_{e} has a 1-dimensional kernel.

Corollary 1.3. If e is odd, e ≥ 5, and if X is general, then the Kodaira dimension κ(M_{e}) ≥ 0.

The corollary follows as the form ω_{e}^{(3e+1)/2} is a nonzero section of the canonical line bundle.

In Section 2 we recall three different moduli spaces and how they are related. In Section 4 we give a general method to produce ω_{e} on the Kontsevich moduli stack M_{e} of stable maps for any e ≥ 1. This is different than producing the form ω_{e} on \overline{M}_{e}. In a preliminary section, Section 3, we prove that every p-form on any tame, finite type Deligne-Mumford stack over a field k (not necessarily algebraically closed, nor of characteristic 0) gives rise to a p-form on every desingularization of the coarse moduli space, cf. Lemma 3.6. Thus producing the 2-form on M_{e} is stronger than producing the 2-form on \overline{M}_{e}. In Section 5 we describe how to compute the associated alternating pairing on Zariski tangent spaces of M_{e}. In Section 6 we show that this pairing is nondegenerate for a general point of M_{5}. The case e = 5 is particularly simple: almost no explicit calculation is necessary. In Section 7 we prove that ω_{e} is generically non-degenerate for every odd degree e ≥ 5, and the kernel of the pairing is generically 1-dimensional for every even degree e ≥ 6. In Section 8 we sketch a proof that M_{6} is not uniruled and pose some questions about the spaces M_{e}.

Finally, Theorem 1.2 implies Theorem 1.1 thanks to the following lemma.

Lemma 1.4. Let M be a smooth, projective scheme, let ω be a 2-form on M, and suppose that at a general point p ∈ \overline{M}_{e} the rank of the 2-form ω is r. Then dim(Z) ≤ dim(M) − r, i.e., the codimension of the maximal rationally connected subvariety Z passing through a very general point of M is at least r.

Proof. Denote d = dim(Z). If d = 0, there is nothing to prove. Suppose that d is positive. Then, by [16, Theorem IV.5.8], for a very general point p ∈ M there is a morphism g : ℙ^{1} → M whose image contains p and such that g^{∗}T_{M} contains a locally free subsheaf E ⊂ g^{∗}T_{M} with E an ample locally free sheaf of rank d and whose cokernel is a trivial locally free sheaf of rank n − d (this is in the proof of [16, Theorem IV.5.8], not in the statement). Consider the sheaf map induced by ω, i.e., g^{∗}T_{M} → g^{∗}Ω_{M}. Since g^{∗}T_{M} is semipositive, the sheaf g^{∗}Ω_{M} is seminegative. There is no nonzero map from an ample locally free sheaf to a seminegative locally free sheaf. So E is contained in the kernel of the sheaf map. Therefore d ≤ dim(M) − r.

□

2. Discussion of moduli spaces

In this section we discuss three related functors, each of which gives a compactification of the space of smooth rational curves. The spaces representing these functors are birational. Since we are studying birational properties of these spaces the distinction between them is not crucial to the rest of the paper. We find it useful to pause, compare these three spaces, and point out what is and is not known about them.

Let k be a field, not necessarily algebraically closed, nor of characteristic 0. Let X ⊂ ℙ^{N} be a quasi-projective scheme over k. Denote by M_{e} the scheme parametrizing families of smooth, proper, geometrically connected curves C ⊂ X of arithmetic genus 0 and degree e. Even before compactifying M_{e}, there are several versions of M_{e} and we concentrate on two of these: M_{e}^{h} and M_{e}^{c}. The scheme M_{e}^{h} is the open subscheme of the Hilbert scheme Hilb^{et+1}(X) (cf. [9]) parametrizing smooth curves. And M_{e}^{c} is the open subvariety of the Chow variety Chow_{1,e}(X) (cf. [16, Def. I.3.20]) parametrizing cycles of smooth curves. There is not universal acceptance of the definition of the Chow variety (e.g. there is also the definition in [2]), but Kollár’s definition is best suited to our needs. In particular, there is the fundamental class morphism, also called the Hilbert-Chow morphism, from M_{e}^{h} to M_{e}^{c}.

Lemma 2.1. [16, Thm. 6.3] There exists a fundamental class morphism FC : (M_{e}^{h})^{sn} → M_{e}^{c}, where (M_{e}^{h})^{sn} is the semi-normalization of M_{e}^{h} (cf. [16, Def. I.7.2.1]). The morphism F C is an isomorphism. Therefore there is a morphism (FC)^{−1} : M_{e}^{c} → M_{e}^{h} that is equivalent to the semi-normalization of M_{e}^{h}. In particular it is bijective on points.

Proof. This follows from [16, Thm. 6.3] and the semi-normal analogue of Zariski’s Main Theorem. □

It does happen that M_{e}^{h} is not semi-normal so that M_{e}^{c} and M_{e}^{h} are not isomorphic, e.g. whenever M_{e}^{h} is not reduced. A simple example of this is given by any pair (X, L) where L ⊂ ℙ^{3} is a line and X ⊂ ℙ^{3} is a smooth hypersurface of degree d ≥ 4 containing L. In this case there is a unique connected component of M_{1}^{h} whose reduced scheme consists just of the point [L] ∈ M_{1}^{h}, but M_{1}^{h} is non-reduced.

For the special case that X ⊂ ℙ^{n} is a smooth cubic hypersurface – the case of interest in this paper – we expect that M_{e}^{h} is always semi-normal.

Question 2.2. If X ⊂ ℙ^{n} is a smooth cubic hypersurface, and if char(k) = 0, is M_{e}^{h} semi-normal? Is M_{e}^{h} normal?

There are some partial answers. For n arbitrary and e = 1, M_{1}^{h} is smooth by [5, Thm. 7.8]. For n = 3 and e arbitrary, M_{e}^{h} is an open subset of a projective space and so it is smooth. For n = 4 and e = 2, 3, M_{e}^{h} is smooth by [11, Lemma 3.2, Lemma 4.6]. For n = 4 and e arbitrary, M_{e}^{h} is an irreducible, reduced, local complete intersection scheme by [12]. So, by Serre’s criterion, to prove that M_{e}^{h} is normal it suffices to prove that M_{e}^{h} is nonsingular in codimension one. We do not know whether this is true.

Let X ⊂ ℙ^{N} be a projective scheme over a field k. Denote by \overline{M}_{e}^{h} the closure of M_{e}^{h} in Hilb^{et+1}(X) and denote by \overline{M}_{e}^{c} the closure of M_{e}^{c} in Chow_{1,e}(X). These are the first two compactifications of M_{e} which we consider.

Many results about the Hilbert scheme and the Chow variety are known. For instance, by [16, Thm. I.6.3], the morphism FC extends to a morphism FC : (\overline{M}_{e}^{h})^{sn} → \overline{M}_{e}^{c}. Both \overline{M}_{e}^{c} and \overline{M}_{e}^{h} have certain drawbacks. For example the morphism (FC)^{−1} does not extend to a regular morphism \overline{M}_{e}^{c} → \overline{M}_{e}^{h} (this fails even in the case X = ℙ^{N}). Moreover, the closed subsets \overline{M}_{e}^{h} ⊂ Hilb^{et+1}(X) and \overline{M}_{e}^{c} ⊂ Chow_{1,e}(X) are typically not open (i.e., they are typically not a union of connected components of the full Hilbert scheme, resp. Chow variety). Because of this, it is difficult to carry out an infinitesimal analysis of \overline{M}_{e}^{h} and \overline{M}_{e}^{c} as in [16, Section I.2].

If char(k) = 0, there is a third compactification of M_{e} that is very useful: the Kontsevich moduli space of stable maps (this compactification also exists in positive characteristic, but it is not as well-behaved). A prestable map from an r-pointed curve of genus g to X of degree e defined over a field L/k is a triple (C, (p_{1}, …, p_{r}), f) where C is a geometrically connected, reduced, at-worst-nodal curve of arithmetic genus g defined over L, where p_{1}, …, p_{r} is an ordered set of distinct L-rational points in the nonsingular locus of C, and where f : C → X is a morphism of k-schemes such that the degree of f^{∗}O(1) is e. The triple is called a stable map if there are no infinitesimal automorphisms of the triple. There is a notion of families of stable maps and morphisms between stable maps. There is a Deligne-Mumford stack that is proper over k, \overline{M}_{g,n}(X, e), parametrizing flat families of stable maps from r-pointed curves of genus g to X of genus g of degree e. The coarse moduli space \overline{M}_{g,n}(X, e) of the stack \overline{M}_{g,n}(X, e) is a projective k-scheme. The Deligne-Mumford stack and its coarse moduli space are described in detail in [4, 7].

In the special case that X ⊂ ℙ^{5} is a smooth cubic hypersurface, we denote by \overline{M}_{e} the Kontsevich moduli space of stable maps to X of genus 0 with no marked points and degree e.

Lemma 2.3. The scheme M_{e}^{h} is isomorphic to an open substack of \overline{M}_{e}.

Proof. This follows from the definitions of \overline{M}_{e} and M_{e}^{h}. □

There is an analogue of the morphism FC, i.e., a 1-morphism FC : (\overline{M}_{e})^{sn} → \overline{M}_{e}^{c}.

One drawback of \overline{M}_{e} as compared to \overline{M}_{e}^{h} and \overline{M}_{e}^{c} is that it is a stack rather than a scheme, which makes some arguments more technical. On the other hand, the deformation and obstruction theory of \overline{M}_{e} and the “boundary” are understood quite well. These are the key components in the proof of the following proposition.

Proposition 2.4. [13, Prop. 7.4] Let n ≥ 5 be an integer and let X ⊂ ℙ^{n} a cubic hypersurface. If X is general, the stack \overline{M}_{e} is irreducible and reduced of the expected dimension (n − 2)e + (n − 4) and has only local complete intersection singularities.

Proof. Every case except n = 5 follows from [13, Prop. 7.4]. Thus suppose that n = 5.

By [13, Cor. 7.3], to prove the proposition it suffices to check that for e = 1 and e = 2, the following three conditions hold:

(i) the evaluation morphism ev : \overline{M}_{0,0}(X, e) → X is surjective and has constant fiber dimension,

(ii) a general fiber of ev is irreducible, and (iii) there exists a free stable map of degree e, i.e., a stable map [C, f] such that

f^{∗}T_{X} is generated by global sections.

Case I, e = 1: First consider (iii). For every every smooth cubic hypersurface X ⊂ ℙ^{5}, and for every point p ∈ X, there exists a line L ⊂ X containing p. By [6, Prop. 4.14], for every smooth variety X and for a very general point p ∈ X, every rational curve in X containing p is free. So for a very general point p in X, and for any line L containing p, (iii) holds.

By [16, Cor. II.3.5.4.2], for a very general point p ∈ X, the evaluation morphism ev : \overline{M}_{0,1}(X, 1) → X is smooth over p. The fiber F is canonically a complete intersection of hypersurfaces in ℙ^{n−1} of dimension n−4. Whenever n ≥ 5, this complete intersection is connected. Since F is smooth and connected, it is irreducible. Thus (ii) holds.

Finally, if X is a general hypersurface, then by [11, Thm. 2.1], (i) holds.

Case II, e = 2: Let p ∈ X be a general point and let L ⊂ X be any line containing p. Then L is free. Thus any degree 2 cover of L by a rational curve is a stable map that is free. Thus (iii) holds.

There is an a priori lower bound on the dimension of every irreducible component of every fiber of ev_{f} : \overline{M}_{0,1}(X, 2) → X, namely the difference of the expected dimension of \overline{M}_{0,1}(X, 2) and dim(X), which is 4. Condition (i) is the condition that every fiber of ev_{f} has dimension exactly 4. Condition (ii) is the condition that at least one fiber is irreducible and reduced of dimension 4.

Suppose that X contains no linear ℙ^{2} – this holds for a general cubic hypersurface in ℙ^{5}. Then every stable map f : C → X of degree 2 that is not a double cover of a line is an embedded plane conic. The span of the conic C, say Λ ⊂ ℙ^{n}, intersects X in a plane cubic curve C′ ⊂ Λ. Of course C ⊂ C′, and the residual curve is a line L ⊂ X.

Conversely, for a general pair of a line L ⊂ X and a linear ℙ^{2} Λ containing L, the residual to L in Λ ∩ X is a plane conic. Thus the set of embedded plane conics in X passing through a general point p, is isomorphic to an open subset of the space of lines \overline{M}_{1}. This space is smooth of dimension 4. So to finish the proof of (i) and (ii), it suffices to show that this set is Zariski dense in ev^{−1}(p) for every p ∈ X. In other words, for every p ∈ X, the subset of ev^{−1}(p) consisting of double covers of lines is not dense in any irreducible component of ev^{−1}(p).

Since X is general, the morphism ev : \overline{M}_{0,1}(X, 1) → X is flat. Thus the variety parametrizing lines in X containing p has dimension 1. Thus the variety parametrizing double covers of lines containing p has dimension 3 (there is a 2-dimensional family of double covers of a given line by rational curves). Combined with the lower bound of 4, it follows that the variety parametrizing double covers of lines containing p is not dense in any irreducible component of ev^{−1}(p). Therefore (i) and (ii) hold. □

Remark 2.5. (1) The proposition is false for n = 3 and n = 4. For n = 3, M_{e} is disconnected (the Picard number of X is not 1). For n = 4 and e ≥ 2, there is an irreducible component Y_{e} ⊂ \overline{M}_{e} parametrizing degree e covers of lines in X. The open subset \overline{M}_{e} − Y_{e} is irreducible, reduced of the expected dimension and has only local complete intersections (cf. [12]).

(2) Even though the proof above works only for a general hypersurface X, we suspect the proposition holds for every smooth cubic hypersurface X ⊂ ℙ^{n}.

(3) In fact the argument above proves much more than the proposition, namely for every stable genus 0 A-graph τ and every flag f of τ, a certain condition B(X, τ, f) holds (cf. [13, Cor. 7.3]). In particular, \overline{M}(X, τ) is irreducible.

Corollary 2.6. For X ⊂ ℙ^{5} a general cubic hypersurface, the schemes M_{e}^{c} and M_{e}^{h} are irreducible and reduced of dimension 3e + 1. They are birational to each other and to \overline{M}_{e}.

3. Trace maps and descent for p-forms

Let X be a quasi-projective variety over a field k with char(k) = 0. If X is smooth and projective and if k = ℂ, Hodge theory gives linear maps from H^{p+1,q+1}(X) to H^{p,q}(M̃_{e}), where M̃_{e} is a desingularization of \overline{M}_{0,0}(X, e). The map pulls back forms to the universal curve over \overline{M}_{0,0}(X, e), and then uses “integration along fibers”. In the proof of the main theorem, we need a version of this that holds when X is neither smooth nor projective. In the next two sections we prove the following version (the proof is algebraic, not Hodge-theoretic): Let X be a quasi-projective variety. Let g, r, p and q be nonnegative integers. There are linear maps,

H^{q+1} (X, Ω^{p+1}) → H^{q} (\overline{M}_{g,r}(X, e), Ω^{p}) .

When q = 0 this map gives p-forms on the Kontsevich moduli stack. This, in turn, gives p-forms on a desingularization of the coarse moduli space of the stack. This follows by a more general result, Proposition 3.6, which is the main result of this section: Let k be a field (not necessarily algebraically closed, nor of characteristic 0), and let M be a finite type Deligne-Mumford stack over k. If M is tame and if the coarse moduli space M is smooth, then every p-form on M is the pullback of a unique p-form on M (up to torsion).

The proof uses trace maps for proper morphisms, f : Y → Z, where Z is normal and f is étale on a dense open subset of Y,

Tr_{f}^{p} : f_{∗}(Ω_{Y}^{p}) → (Ω_{Z}^{p})^{∨∨}.

3.1. If f is finite étale. Let f : Y → Z be a morphism of schemes and let F be a coherent sheaf on Z. There is a morphism of O_{Z}-modules, f_{∗}O_{Y} ⊗_{O_{Z}} F → f_{∗}f^{∗}F. If f is finite and f_{∗}O_{Y} is a locally free O_{Z}-module, this morphism is an isomorphism. Also, there is a trace map Tr_{f} : f_{∗}O_{Y} → O_{Z} defined in the usual way. Therefore, there is a trace map Tr_{f} : f_{∗}f^{∗}F → F.

Let f : Y → Z be a finite étale k-morphism of finite type k-schemes. For each integer p ≥ 0, the pullback map (df)^{†} : f^{∗}Ω_{Z/k}^{p} → Ω_{Y/k}^{p} is an isomorphism. Combined with the trace map from the last paragraph, we get a map satisfying the following properties.

Lemma 3.1. Let f : Y → Z be a finite étale k-morphism of finite type k-schemes. Denote by n the degree of f. For each integer p ≥ 0 there exists a unique morphism of O_{Z}-modules, Tr_{f}^{p} : f_{∗}Ω_{Y/k}^{p} → Ω_{Z/k}^{p} satisfying the following properties.

(i) For p = 0, Tr_{f}^{0} : f_{∗}O_{Y} → O_{Z} is the usual trace map.

(ii) For every open subset U ⊂ Z, for every pair of integers, p, p′ ≥ 0, for every section σ ∈ H^{0}(U, Ω_{Z/k}^{p}), and for every section τ ∈ H^{0}(f^{−1}(U), Ω_{Y/k}^{p′}),

Tr_{f}^{p+p′}f_{∗}(f^{∗}σ ∧ τ) = σ ∧ Tr_{f}^{p′}(f_{∗}τ).

Moreover, the following properties hold.

(iii) For every integer p and every section τ ∈ H^{0}(Y, Ω_{Y/k}^{p}),

Tr_{f}^{p+1}(f_{∗}dτ) = d(Tr_{f}^{p}(f_{∗}τ)).

(iv) Let Z′ be a finite type k-scheme and let g : Z′ → Z be a morphism of kschemes. Denote by Y′ the fiber product Z′ ×_{Z} Y, and denote by f′ : Y′ → Z′ and g′ : Y′ → Y the projection morphisms. For each integer p ≥ 0, there is a commutative diagram of O_{Z′}-modules.

g^{∗}f_{∗}Ω_{Y/k}^{p}  ⟶^{g^{∗}Tr_{f}^{p}}  g^{∗}Ω_{Z/k}^{p} ⟶^{(dg)^{†}} Ω_{Z′/k}^{p}
≅↓  ↓^{=}
(f′)_{∗}(g′)^{∗}Ω_{Y/k}^{p} ⟶^{(f′)_{∗}(dg′)^{†}} (f′)_{∗}Ω_{Y′/k}^{p} ⟶^{Tr_{f′}^{p}} Ω_{Z′/k}^{p}

(v) Let h : X → Y be a finite étale morphism. Then for each integer p ≥ 0, Tr_{f∘h}^{p} = Tr_{f}^{p} ∘ f_{∗}Tr_{h}^{p}.

3.2. If f is proper and generically étale. Let f : Y → Z be a proper kmorphism of finite type k-schemes. Suppose that Z is connected and geometrically normal, and suppose that there is a dense open subset of Y on which f is étale. Denote by Z_{smooth} ⊂ Z the maximal open subscheme that is smooth over k. Then there exists an open subset U ⊂ Z_{smooth}, dense in Z, such that V = f^{−1}(U) is dense in Y and such that f|_{V} : V → U is finite étale. By Lemma 3.1, for each p ≥ 0, there is a trace map Tr_{f|_{V}}^{p} : (f|_{V})_{∗}Ω_{V/k}^{p} → Ω_{U/k}^{p}.

Definition 3.2. A trace map is a morphism of O_{Z}-modules, Tr_{f}^{p} : f_{∗}Ω_{Y}^{p} → (Ω_{Z}^{p})^{∨∨}, whose restriction to U equals Tr_{f|_{V}}^{p}.

It is straightforward to check that, if a trace map exists, it is unique and it is independent of the choice of U.

Proposition 3.3. Let Z be a finite type k-scheme that is connected and geometrically normal. Let f : Y → Z be a proper morphism that is étale on a dense open subset of Y. For each integer p ≥ 0, there exists a trace map Tr_{f}^{p} : f_{∗}Ω_{Y/k}^{p} → (Ω_{Z/k}^{p})^{∨∨}.

Proof. Denote by i : U → Z the open immersion. The morphism Tr_{f|_{V}} determines a morphism of O_{Z}-modules, i_{∗}Tr_{f|_{V}}^{p} : f_{∗}Ω_{Y/k}^{p} → i_{∗}i^{∗}Ω_{Z/k}^{p}. Because U ⊂ Z_{smooth}, i^{∗}Ω_{Z/k}^{p} = i^{∗}(Ω_{Z/k}^{p})^{∨∨}. Therefore there is a injective morphism of O_{Z}-modules, (Ω_{Z/k}^{p})^{∨∨} → i_{∗}i^{∗}Ω_{Z/k}^{p}. The proposition exactly says that the image of i_{∗}Tr_{f|_{V}}^{p} is contained in (Ω_{Z/k}^{p})^{∨∨}. By hypothesis, Z is normal. And (Ω_{Z/k}^{p})^{∨∨} is reflexive. Therefore, (Ω_{Z/k}^{p})^{∨∨} is the intersection (in i_{∗}i^{∗}Ω_{Z/k}^{p}) of its localization at every codimension 1 point. Thus it suffices to check that for every codimension 1 point η ∈ Z − U, the image of i_{∗}Tr_{f|_{V}}^{p} is contained in the localization of (Ω_{Z/k}^{p})^{∨∨} at η.

The image of i_{∗}Tr_{f|_{V}}^{p} does not change if we replace Y by the disjoint union of the irreducible components of Y, with the induced reduced scheme structure. Therefore assume that Y is reduced and every connected component of Y is irreducible. The morphism f is finite on an open subset of Z whose complement has codimension ≥ 2. Thus there exists an open affine W ⊂ Z_{smooth} such that η ∈ W and such that f|_{W} : f^{−1}(W) → W is finite. Denote A = H^{0}(W, O_{Z}) and denote B = H^{0}(f^{−1}(W), O_{Y}). By [19], the image of Ω_{B/k}^{p} under i_{∗}Tr_{f|_{V}}^{p} is contained in Ω_{A/k}^{p} (Zannier only proves this when Y is connected, but the generalization here follows trivially). In particular, the image of i_{∗}Tr_{f|_{V}}^{p} is contained in the localization of (Ω_{Z/k}^{p})^{∨∨} at η.

□

Remark 3.4. For each integer p ≥ 0, there is a generic trace map

(Tr_{f}^{⊗p})_{gen} : f_{∗}(Ω_{Y/k}^{1})^{⊗p} → (Ω_{Z/k}^{1})^{⊗p} ⊗_{O_{Z}} K(Z).

By the proposition, when restricted to the submodule corresponding to an exterior power, the generic trace map factors through (Ω_{Z}^{p})^{∨∨}. One might hope that the entire generic trace map factors through the reflexive hull of (Ω_{Z}^{1})^{⊗p}. This is true, for instance, if f : Y → Z is étale away from codimension 2. But typically this is not the case: Consider the morphism f : 𝔸_{k}^{1} → 𝔸_{k}^{1} which pulls back a coordinate t on the target to u^{2}, the square of a coordinate u on the domain. Then the generic trace of du ⊗ du is \frac{1}{4t}dt ⊗ dt.

Lemma 3.5. Let Z be a finite type k-scheme that is connected and geometrically normal. Let f : Y → Z be a proper morphism that is étale on a dense open subset of Y. The trace maps Tr_{f}^{p} satisfy the following properties.

(i) For p = 0, Tr_{f}^{0} : f_{∗}O_{Y} → O_{Z} is the usual trace map.

(ii) For every pair of integers, p, p′ ≥ 0, for every section σ ∈ H^{0}(Z, Ω_{Z}^{p}), and for every section τ ∈ H^{0}(Y, Ω_{Y}^{q}),

Tr_{f}^{p+p′}f_{∗}(f^{∗}σ ∧ τ) = σ ∧ Tr_{f}^{p′}(f_{∗}τ).

(iii) For every integer p and every section τ ∈ H^{0}(Y, Ω_{Y/k}^{p}),

Tr_{f}^{p+1}(f_{∗}dτ) = d(Tr_{f}^{p}(f_{∗}τ)).

(iv) Let Z′ be a finite type k-scheme that is connected and geometrically normal. Let g : Z′ → Z be a morphism of k-schemes such that g^{−1}(U) is dense in Z′. Denote by Y′ the closure in Z′ ×_{Z} Y of the inverse image of g^{−1}(U). And denote by f′ : Y′ → Z′ and g′ : Y′ → Y the projection morphisms. For each integer p ≥ 0, there is a commutative diagram of O_{Z′}-modules.

g^{∗}f_{∗}Ω_{Y/k}^{p}  ⟶^{g^{∗}Tr_{f}^{p}}  g^{∗}Ω_{Z/k}^{p} ⟶^{(dg)^{†}} Ω_{Z′/k}^{p}
↓  ↓^{=}
(f′)_{∗}(g′)^{∗}Ω_{Y/k}^{p} ⟶^{(f′)_{∗}(dg′)^{†}} (f′)_{∗}Ω_{Y′/k}^{p} ⟶^{Tr_{f′}^{p}} Ω_{Z′/k}^{p}

(v) Let h : X → Y be a proper morphism. Suppose that Y is normal and h is étale on a dense open subset of X. Then f ∘ h is étale on a dense open subset of X, and for each integer p ≥ 0, Tr_{f∘h}^{p} = Tr_{f}^{p} ∘ f_{∗}Tr_{h}^{p}.

3.3. Descent for p-forms on a stack. Let k be a field (not necessarily algebraically closed nor of characteristic 0). Let M be a finite type Deligne-Mumford stack over k. Recall from [1], that M is tame if the stabilizer group of every geometric point of M has order prime to char(k). Recall from [15] that there exists a coarse moduli space, M, for M, and M is an algebraic space of finite type over k. Suppose that M is tame, irreducible, and generically reduced and that M is a geometrically normal k-scheme.

Denote by c : M → M the morphism of M to the coarse moduli space. Let U ⊂ M be the maximal open subscheme over which c is smooth. Then for every p, the pullback map H^{0}(U, Ω_{M/k}^{p}) → H^{0}(c^{−1}(U), Ω_{M/k}^{p}) is an isomorphism. It is not true that the pullback map over all of M is an isomorphism, even modulo torsion. For instance, let char(k) ≠ 2, consider 𝔸_{k}^{2} with coordinates x, y, let Γ be the cyclic group of order 2, and let Γ act on 𝔸_{k}^{2} by x ↦ −x, y ↦ −y. Let M be the quotient stack [𝔸_{k}^{2}/Γ]. Then the 2-form dx ∧ dy is Γ-invariant and thus gives rise to a global section of Ω_{M/k}^{2}. But this 2-form is not the pullback of any global section of Ω_{M/k}^{2}.

A slightly weaker result is true, and will be proved in Proposition 3.6. This result is good enough for our application. First we explain the result, then we prove it. If M is tame, irreducible and generically reduced and if M is a geometrically normal k-scheme, then for each integer p ≥ 0 there is a k-linear map,

c_{∗} : H^{0}(M, (Ω_{M/k}^{p})^{free}) → H^{0}(M, (Ω_{M/k}^{p})^{∨∨}).

Here, for a coherent sheaf F, the notation (F)^{free} denotes the maximal torsion-free quotient, i.e., the image of the sheaf map F → F ⊗_{O_{M}} K(M). There is also a generic pullback map,

c_{gen}^{∗} : H^{0}(M, (Ω_{M/k}^{p})^{∨∨}) → H^{0}(M, (Ω_{M/k}^{p})^{free} ⊗_{O_{M}} K(M)).

And the composition c_{gen}^{∗} ∘ c_{∗} equals the obvious inclusion map.

Let N be a finite type k-scheme that is geometrically normal. Let g : N → M be a k-morphism such that g^{−1}(U) is dense. There is a generic pullback map,

g_{gen}^{∗} : H^{0}(M, (Ω_{M/k}^{p})^{∨∨}) → H^{0}(N, (Ω_{N/k}^{p})^{∨∨} ⊗_{O_{N}} K(N)).

In fact the image of g_{gen}^{∗}c_{∗} is contained in the image of H^{0}(N, (Ω_{N/k}^{p})^{∨∨}).

Proposition 3.6. Let k be a field (not necessarily algebraically closed nor of characteristic 0), and let M be a finite type Deligne-Mumford stack. Suppose that M is tame, irreducible, and generically reduced, and that the coarse moduli space M is a k-scheme that is geometrically normal. For each integer p ≥ 0, there is a k-linear map,

c_{∗} : H^{0}(M, (Ω_{M/k}^{p})^{free}) → H^{0}(M, (Ω_{M/k}^{p})^{∨∨}),

whose composition with the generic pullback map, c_{gen}^{∗}, is the obvious inclusion map. Moreover, for every finite type k-scheme N, and for every k-morphism g : N → M, if N is geometrically normal and if g^{−1}(U) ⊂ N is dense, then the image of g_{gen}^{∗}c_{∗} is contained in the image of H^{0}(N, (Ω_{N/k}^{p})^{∨∨}).

Proof. If there exists a map c_{∗} such that c_{gen}^{∗}c_{∗} is the inclusion map, then c_{∗} is unique. Thus we may prove that c_{∗} exists after étale base change of M: the uniqueness of c_{∗} guarantees the cocycle condition for étale descent.

By [1, Lem. 2.2.3], there exists an étale covering {M_{i} → M} such that, (i) each base change M_{i} ×_{M} M is a finite quotient stack [U_{i}/Γ_{i}], (ii) each U_{i} is a scheme finite over M_{i}, (iii) each Γ_{i} is a finite group whose order is prime to char(k), (iv) Γ_{i} acts on U_{i} by M_{i}-morphisms, and

(v) the quotient M_{i}-scheme U_{i}//Γ_{i} equals M_{i}. Thus, without loss of generality, assume that M = [U/Γ] where U is a scheme finite over M and where Γ is a group whose order is prime to char(k) acting on U by M-morphisms.

Denote by h : U → M the morphism above. By Proposition 3.3, there is a morphism

Tr_{h}^{p} : H^{0}(U, (Ω_{U/k}^{p})^{free}) → H^{0}(M, (Ω_{M/k}^{p})^{∨∨}).

The global sections of (Ω_{M/k}^{p})^{free} are precisely the Γ-invariant global sections of (Ω_{U/k}^{p})^{free}. So there is an induced morphism

(Tr_{h}^{p})^{Γ} : H^{0}(M, (Ω_{M/k}^{p})^{free}) → H^{0}(M, (Ω_{M/k}^{p})^{∨∨}).

It is straightforward to check that \frac{1}{|Γ|}(Tr_{h}^{p})^{Γ} satisfies the condition for c_{∗}.

Consider g_{gen}^{∗}c_{∗}. Denote by N the fiber product N ×_{M} M. Denote by c′ : N → N and g′ : N → M the projection morphisms. Then (c′)_{gen}^{∗}g_{gen}^{∗}c_{∗} equals (g′)^{∗}c_{gen}^{∗}c_{∗}. And this equals (c′)_{gen}^{∗}(c′)_{∗}(g′)^{∗}. Since (c′)_{gen}^{∗} is injective, g_{gen}^{∗}c_{∗} = (c′)_{∗}(g′)^{∗}. In particular, the image is contained in H^{0}(N, (Ω_{N/k}^{p})^{∨∨}).

□

4. Construction of the 2-form

Let k be a field (not necessarily algebraically closed, nor of characteristic 0). Let M be a finite type Deligne-Mumford stack over k and let p : C → M be a representable 1-morphism of Deligne-Mumford stacks such that

(i) p is proper and flat of relative dimension 1, and (ii) every geometric fiber of p is a reduced, at-worst-nodal curve,

i.e., p : C → M is a semi-stable family of curves. There is a canonical morphism from the sheaf of relative Kähler differentials to the dualizing sheaf Ω_{p}^{1} → ω_{p}. This is an isomorphism on the open substack U ⊂ C which is the smooth locus of p. For each integer i ≥ 0, this isomorphism induces a morphism of O_{U}-modules,

ϕ_{U,i}: Ω_{C/k}^{i+1}|_{U} → (Ω_{C/k}^{i+1}/p^{∗}Ω_{M/k}^{i+1})|_{U} ≅ p^{∗}Ω_{M/k}^{i} ⊗ ω_{p}|_{U}.

This morphism has the property that for every section α ∈ Ω_{M/k}^{i} and β ∈ Ω_{C/k}^{j}, ϕ_{U,i+j}(p^{∗}α ∧ β) = p^{∗}α ∧ ϕ_{U,j}(β).

Lemma 4.1. For each integer i ≥ 0 there exists a unique morphism of O_{C}-modules, ϕ_{i} : Ω_{C/k}^{i+1} → p^{∗}Ω_{M/k}^{i} ⊗_{O_{C}} ω_{p}, such that ϕ_{i}|_{U} = ϕ_{U,i} and such that for every section α ∈ Ω_{M/k}^{i} and β ∈ Ω_{C/k}^{j}, ϕ_{i+j}(p^{∗}α ∧ β) = p^{∗}α ∧ ϕ_{j}(β).

Proof. If ϕ_{i} exists, then by construction it annihilates p^{∗}Ω_{M}^{i+1}, i.e., it factors through the quotient. The quotient has a canonical subsheaf isomorphic to p^{∗}Ω_{M}^{i} ⊗Ω_{p}^{1} with an obvious map to p^{∗}Ω_{M}^{i} ⊗ ω_{p}. The lemma claims this map extends to the entire quotient. It also claims the extension is unique. Uniqueness is straightforward: the cokernel of p^{∗}Ω_{M}^{i} ⊗ Ω_{p}^{1} is a sheaf that is torsion on fibers, whereas the sheaf p^{∗}Ω_{M}^{i} ⊗ ω_{p} is torsion-free on fibers. So there is no nonzero map from the cokernel to p^{∗}Ω_{M}^{i} ⊗ ω_{p}, i.e., if the map extends, then the extension is unique. The extension problem is equivalent to the vanishing of a section of a sheaf Ext. This vanishing can be checked after passing to the completion of the local ring at each geometric closed point of C, i.e., it suffices to check that the sheaf map extends formally locally at each geometric closed point of C.

Since the property can be checked formally locally, without loss of generality assume that M is a scheme. Let z ∈ C be a closed point. Denoting A = { \atop O_{M,p(z)}} and B = Ô_{C,z}, there is an isomorphism

B ≅ A[[x, y]]/⟨xy − a⟩.

for some element a ∈ A. By Remark 4.4, the base change of ϕ_{U,i} does extend to a map ϕ_{i} ⊗_{O_{ℂ}} B as required. □

Corollary 4.2. For each pair of integers, i, j ≥ 0, there is a k-linear map,

H^{j+1}(C, Ω_{C/k}^{i+1}) → H^{j}(M, Ω_{M/k}^{i}).

Proof. The morphism ϕ_{i} induces a k-linear map,

H^{j+1}(C, Ω_{C/k}^{i+1}) → H^{j+1}(C, p^{∗}Ω_{M/k}^{i} ⊗ ω_{p}).

Associated to the morphism p, there is a Leray spectral sequence for the target vector space. Because R^{l}p_{∗}(p^{∗}Ω_{M/k}^{i} ⊗ω_{p}) = Ω_{M/k}^{i} ⊗R^{l}p_{∗}ω_{p} is zero for l ≥ 2, there is an abutment map,

H^{j+1}(C, p^{∗}Ω_{M/k}^{i} ⊗ ω_{p}) → H^{j}(M, Ω_{M/k}^{i} ⊗ R^{1}p_{∗}ω_{p}).

And there is a trace isomorphism R^{1}p_{∗}ω_{p} ⟶^{≅} O_{M}. Composing these maps gives the k-linear map,

H^{j+1}(C, Ω_{C/k}^{i+1}) → H^{j}(M, Ω_{M/k}^{i}).
□

Assume that char(k) = 0. Let X be a quasi-projective k-scheme and let \overline{M}_{g,r}(X, e) be the Kontsevich moduli space of stable maps from r-pointed curves of arithmetic genus g to X of degree e. There is a universal curve p : C → \overline{M}_{g,r}(X, e) satisfying the hypotheses above. And there is an evaluation morphism ev : C → X. For each pair of integers, i, j ≥ 0, there is a pullback morphism

ev^{∗} : H^{j+1}(X, Ω_{X/k}^{i+1}) → H^{j+1}(C, Ω_{C/k}^{i+1}).

Composing with the k-linear map from Corollary 4.2 gives the following.

Corollary 4.3. For each pair of integers i, j ≥ 0, there is an “integration along fibers” morphism,

H^{j+1}(X, Ω_{X/k}^{i+1}) → H^{j}(\overline{M}_{g,r}(X, e), Ω_{\overline{M}_{g,r}(X,e)/k}^{i}).

In particular, suppose X is the smooth locus of a cubic hypersurface in ℙ^{4}. As will be recalled in the next section, the Griffiths residue calculus gives a canonical map from a 1-dimensional k-vector space to H^{1}(X, Ω_{X/k}^{3}). If X is projective, this map is an isomorphism of k-vector spaces. Using the map above, for each integer e > 0, this gives a global section ω_{e} of Ω^{2} on the stack \overline{M}_{e} parametrizing stable maps from curves of arithmetic genus 0 to X of degree e. The section ω_{e} is well-defined up to non-zero scalar. This is the object of study in the rest of the article.

Remark 4.4. Let A be a ring and let B = A[x, y]/(xy − a) for some a ∈ A. Consider the canonical exact sequence

0 → Ω_{A}^{1} ⊗ B → Ω_{B}^{1} → Ω_{B/A}^{1} → 0.

Exactness on the left follows as B is a complete intersection flat over A whose cotangent complex L_{B/A} is quasi-isomorphic to Ω_{B/A}^{1}. Moreover, the relative dualizing sheaf is the determinant of L_{B/A} (which is perfect of amplitude [−1, 0]). So, the relative dualizing module ω_{B/A} is free with generator

θ = \frac{dx ∧ dy}{xy − a} .

and there is a canonical B-module homomorphism

Ω_{B/A}^{1} ⟶ ω_{B/A}

which is determined by the rules dx ↦ xθ and dy ↦ −yθ. From this we will define maps

Ω_{B}^{i} → Ω_{A}^{i−1} ⊗_{A} ω_{B/A}.

Namely, any element in Ω_{B}^{i} can be written as a B-linear combination of forms of the type η, η ∧ dx, η ∧ dy and η ∧ dx ∧ dy, where η is in Ω_{A}^{j}, with j = i, i − 1, or i − 2. We claim there exists a map as above such that

η ↦ 0, η ∧ dx ↦ η ⊗ xθ, η ∧ dy ↦ −η ⊗ yθ, η ∧ dx ∧ dy ↦ −η ∧ da ⊗ θ.

The reader easily verifies that this is well defined (the main concern being that forms of the type η ∧ (ydx + xdy − da) and η ∧ (ydx + xdy − da) ∧ dx get mapped to zero).

5. Explicit description of the 2-form

Let k be a field of characteristic 0 and let X be a quasi-projective k-scheme. As in the last section, for each pair of integers, i, j ≥ 0, there is k-linear map H^{j+1}(X, Ω_{X/k}^{i+1}) → H^{j}(\overline{M}_{g,r}(X, e), Ω_{\overline{M}_{g,r}(X,e)/k}^{i}). When j = 0, this gives global

sections of Ω^{i}. Let z ∈ M be a geometric closed point, and consider the fiber of this section at z. The goal of this section is to describe the fiber of the section in terms of the local geometry of the associated stable map f : C → X, i.e., in terms of the pullback of the tangent bundle of X, etc. In the special case that X is the smooth locus of a cubic hypersurface in ℙ^{4} and g = 0, we give an explicit description of the fiber of this section.

5.1. Explicit description of H^{1}(X, Ω_{X}^{3}). First we recall a small part of the Griffiths residue calculus [8, Section 8]. This is also discussed very briefly in [18, Section 0]. Let X ⊂ ℙ^{n} be a hypersurface of degree d, and let U ⊂ X be the smooth locus. The cotangent sequence is,

0 ⟶ O_{U}(−d) ⟶ Ω_{ℙ^{n}}^{1}|_{U} ⟶ Ω_{U}^{1} ⟶ 0

Taking the exterior power of this sequence, and twisting by O_{X}(d)|_{U}, gives an exact sequence,

0 ⟶ Ω_{U}^{n−2} ⟶ Ω_{ℙ^{n}}^{n−1}|_{U} ⊗ O_{U}(d) ⟶ Ω_{ℙ^{n}}^{n}|_{U} ⊗ O_{U}(2d) ⟶ 0

(This also follows by taking the dual of the first exact sequence and twisting by Ω_{ℙ^{n}}^{n}|_{U} ⊗ O_{U}(d).) The connecting homomorphism in cohomology gives a map,

H^{0}(ℙ^{n}, Ω_{ℙ^{n}}^{n} ⊗ O_{ℙ^{n}}(2d)) → H^{1}(U, Ω_{U}^{n−2}).

In the special case of a cubic fourfold, there is an exact sequence,

0 ⟶ Ω_{U}^{3} ⟶ Ω_{ℙ^{5}}^{4}|_{U} ⊗ O_{U}(3) ⟶ Ω_{ℙ^{5}}^{5}|_{U} ⊗ O_{U}(6) ⟶ 0. (1)

Of course Ω_{ℙ^{5}}^{5} ⊗ O_{ℙ^{5}}(6) ≅ O_{ℙ^{5}}. Thus the connecting homomorphism,

H^{0}(ℙ^{5}, Ω_{ℙ^{5}}^{5} ⊗ O_{ℙ^{5}}(6)) → H^{1}(U, Ω_{U}^{3}),

is a map from a 1-dimensional vector space to H^{1}(U, Ω_{U}^{3}). If U = X, this map is an isomorphism. Choose a nonzero element in H^{0}(ℙ^{5}, Ω_{ℙ^{5}}^{5} ⊗ O_{ℙ^{5}}(6)), and define ω_{e}^{pre} to be the image of this element in H^{1}(U, Ω_{U}^{3}). Define ω_{e} ∈ H^{0}(\overline{M}_{e}, Ω_{\overline{M}_{e}/k}^{2}) to be the global section associated to ω_{e}^{pre}.

5.2. The explicit description. Let f : C → X be a point of M_{e} ⊂ \overline{M}_{e}. Assume that C ≅ ℙ^{1} is smooth and that f is a regular embedding into the smooth locus U ⊂ X. Consider the sequence of vector bundles over C given by the normal bundle N_{C/X} of C in X mapping to the normal bundle N_{C/ℙ^{5}} of C in ℙ^{5},

0 ⟶ N_{C/X} ⟶ N_{C/ℙ^{5}} ⟶ f^{∗}N_{X/ℙ^{5}} ⟶ 0.  (2)

Of course N_{X/ℙ^{5}} ≅ O_{X}(3), so that f^{∗}N_{X/ℙ^{5}} ≅ O_{C}(3e), where the notation O_{C}(a) indicates any invertible sheaf of degree a on C. In particular, observe that ⋀^{3} N_{C/X} = O_{C}(3e − 2) and that ⋀^{4} N_{C/ℙ^{5}} = O_{C}(6e − 2). The Zariski tangent space T_{[f]}(M_{e}), which is the same thing as the dual vector space of the fiber Ω_{M_{e}}^{1}|_{[f]}, is given by the space of global sections H^{0}(C, N_{C/X}) (c.f. [16, Theorem I.2.8]). So the fiber Ω_{M_{e}}^{2}|_{[f]} is just the vector space dual of ⋀^{2} H^{0}(C, N_{C/X}). And the 2-form ω_{e} gives a procedure to associate to any two sections of N_{C/X} a complex number.

Next consider the exact sequence

0 ⟶ ⋀^{3} N_{C/X} ⊗ O_{C}(−3e) ⟶ ⋀^{3} N_{C/ℙ^{5}} ⊗ O_{C}(−3e) ⟶ ⋀^{2} N_{C/X} ⟶ 0. (3)

This sequence is obtained from Equation 2 by taking exterior powers and twisting by O_{C}(−3e). In any case, the sheaf on the left is O_{C}(−2) by what was said above. Choose an isomorphism H^{1}(C, O_{C}(−2)) = ℂ, and let

δ : H^{0}(C, ⋀^{2} N_{C/X}) → H^{1}(C, ⋀^{3} N_{C/X} ⊗ O_{C}(−3)) = H^{1}(C, O(−2)) = ℂ,

be the boundary map on cohomology coming from the exact sequence above. This is another procedure which associates to any two sections of N_{C/X} a complex number. In the following theorem we prove that the two procedures agree. The best argument for this is the usual one: What else could it be? The actual proof is even more annoying.

Theorem 5.1. Up to a nonzero scalar factor the pairing associated to ω_{e} on T_{[f]}(M_{e}) = H^{0} (C, N_{C/X}) is equal to the pairing (s_{1}, s_{2}) ↦ δ(s_{1} ∧ s_{2}).

Proof. Observe that the construction of Section 4 is compatible with arbitrary base change of the stack M. To prove the theorem, perform a base change to the Artin local ring Z = SpecA which is the base of the universal first order deformation of C ⊂ X, say C ⊂ Z × X. The construction of Section 4 restricts the exact sequence from Equation 1 to C and then pushes-out the sequence by the map

f^{∗} (Ω_{X}^{3}) → Ω_{C}^{3} → p^{∗} (Ω_{Z}^{2}) ⊗ ω_{C/Z}.

Then the construction takes the cohomology of the resulting sequence to obtain the 2-form ω_{e}. By a diagram chase, the resulting sequence is simply the “Serre dual” of the sequence from Equation 3 from which the theorem follows.

First consider the universal first order deformation of C ⊂ X. By Serre duality the vector space V = H^{1} (C, I/I^{2} ⊗ ω_{C}) is dual to H^{0} (C, N_{C/X}). Here I is the ideal sheaf of C in X. Consider the local Artin k-algebra, A = k ⊕ V, where V ⊂ A is an ideal of square zero. Set Z = Spec A. Denote by C → Z the universal first order deformation of C. Let s_{1}, …, s_{A} be an ordered basis for H^{0} (C, N_{C/X}) and let t_{1}, …, t_{A} in V be the dual ordered basis. The elements s_{1}, …, s_{A} are canonically identified with O_{C}-linear maps I/I^{2} → O_{C}. Let p ∈ C be a point, let U ⊂ X be an open affine subset containing p, and let g_{1}, g_{2}, g_{3} be generators for H^{0}(U, I) as an H^{0}(U, O_{X})-module. Then the ideal of C is locally generated by the equations

g̃_{j} := g_{j} + ∑_{i=1}^{A} t_{i} · s_{i}(f_{j}), j = 1, 2, 3

g̃_{j} ∈ O_{X}[t_{1}, …, t_{A}]/⟨t_{i}t_{i′}, t_{i}g_{j}, g_{j}g_{j′}|i, i′ = 1, …, A, j, j′ = 1, 2, 3⟩.

Denote by p: C → Z and f̃: C → X the two projections.

To prove the theorem, we compute the 2-form on Z obtained from the construction of Section 4 applied to (p: C → Z, f̃: C → X). This is not as crazy as it sounds; namely Ω_{A/k}^{2} ⊗_{A} k = ∧^{2}V, so this computation will provide the necessary information.

To compute f̃^{∗}η, form the pullback by f̃^{∗} of the exact sequence from Equation 1. Considered as an element of the Yoneda-Ext group Ext_{C}^{1}(O_{C}, Ω_{C}^{3}), the element f̃^{∗}η is simply the push-out of this exact sequence by the canonical map f̃^{∗} (Ω_{X}^{3}) → Ω_{C}^{3}. According to Section 4, take the image of f^{∗}η under the map

Ext_{C}^{1}(O_{C}, Ω_{C}^{3}) → Ext_{C}^{1}(O_{C}, p^{∗} (Ω_{Z}^{2}) ⊗ ω_{C/Z})

In terms of Yoneda-Ext, take an additional push-out of the exact sequence by Ω_{C}^{3} → p^{∗} (Ω_{Z}^{2}) ⊗ ω_{C/Z}. So, in terms of Yoneda-Ext, the exact sequence is obtained as the push-out of the pullback of Equation 1 by the map f̃^{∗}Ω_{X}^{3} → p^{∗} (Ω_{Z}^{2}) ⊗ ω_{C/Z}.

Of course it is only necessary to compute the restriction of this exact sequence to the closed fiber, so restrict the push-out exact sequence to the closed fiber. In particular, the restriction to the closed fiber of p^{∗} (Ω_{Z}^{2}) ⊗ ω_{C/Z} is ⋀^{2} V ⊗_{k} Ω_{C}^{1}. The next step is an explicit local description of the map

ψ: Ω_{X}^{3}|_{C} → ⋀^{2} V ⊗_{k} Ω_{C}^{1}.

Let t be a regular function on X restricting to a local coordinate on C. Any local 3-form on X is an O_{X}-linear combination of the forms ϵ_{jj′} = df_{j} ∧df_{j′} ∧dt, 1 ≤ j < j′ ≤ 3 and the form df_{1} ∧ df_{2} ∧ df_{3}. So it suffices to evaluate ψ on these 3-forms. The result is

ψ(η_{jj′}) = ∑_{i,i′=1}^{A} s_{i}(f_{j})s_{i′}(f_{j′})t_{i} ∧ t_{i′} ⊗ dt, 1 ≤ j < j′ ≤ 3,  (4)
ψ(df_{1} ∧ df_{2} ∧ df_{3}) = 0.  (5)

Of course there is a more “global” way of thinking about ψ. The exact sequence

0 ⟶ I/I^{2} ⟶ Ω_{X}^{1}|_{C} ⟶ Ω_{C}^{1} ⟶ 0,  (6)

determines a canonical map α : Ω_{X}^{3}|_{C} → ⋀^{2} I/I^{2} ⊗_{O_{C}} Ω_{C}^{1}. And there is a map of O_{C}-modules β : I/I^{2} → V ⊗_{k} O_{C} defined as the transpose of the map H^{0} (C, N_{C/X}) ⊗_{k} O_{C} → N_{C/X}. The global description of ψ is as the composition of α with ⋀^{2} β ⊗ Id_{Ω_{C}^{1}} .

Just as the exact sequence in Equation 6 induces the map α, also the exact sequence

0 ⟶ Ĩ/Ĩ^{2} ⟶ Ω_{ℙ^{5}}^{1}|_{C} ⟶ Ω_{C}^{1} ⟶ 0.  (7)

induces a map α′ : Ω_{ℙ^{5}}^{4}|_{C} → ⋀^{3} Ĩ/Ĩ^{2} ⊗ Ω_{C}^{1} where Ĩ is the ideal sheaf of C in ℙ^{5}. By adjunction, there are isomorphisms Ω_{ℙ^{5}}^{5}|_{C} ⊗ O_{C}(3e) ≅ Ω_{X}^{4}|_{C} and Ω_{X}^{4}|_{C} ≅ ⋀^{3} I/I^{2} ⊗ Ω_{C}^{1}. Combining these adjunction isomorphisms gives an isomorphism,

α′′: Ω_{ℙ^{5}}^{5}|_{C} ⊗ O_{C}(6e) → ⋀^{3} I/I^{2} ⊗ O_{C}(3e) ⊗ Ω_{C}^{1}.

Both terms in this map are isomorphic to O_{C}. Choosing such isomorphisms, α′′ is just an isomorphism of O_{C} to itself.

We leave it to the reader to verify that the following diagram commutes,

\begin{aligned}0 & ⟶ & Ω_{X}^{3}|_{C} & ⟶ & Ω_{ℙ^{5}}^{4}|_{C}(3e) & ⟶ & O_{C} & ⟶ & 0 \\ & & ^{α}↓ & & _{α′}↓ & _{α′′}↓ \\ 0 & ⟶ & ⋀^{2} I/I^{2} ⊗ Ω_{C}^{1} & ⟶ & ⋀^{3} Ĩ/Ĩ^{2}(3e) ⊗ Ω_{C}^{1} & ⟶ & O_{C} & ⟶ & 0. \end{aligned} (8)

The top exact sequence is the restriction to C of Equation 1, and the bottom exact sequence is the dual of Equation 3 tensored with Ω_{C}^{1}. More canonically, the last term in the top sequence is Ω_{ℙ^{5}}^{5}|_{C}(6e) and the last term in the bottom sequence is ⋀^{3} I/I^{2}(3e) ⊗ Ω_{C}^{1}. The diagram follows using the isomorphisms of these sheaves with O_{C} from the last paragraph.

The conclusion is that the extension of O_{C} by ∧^{2}V ⊗_{k} Ω_{C}^{1} obtained from f̃^{∗}η is precisely the Serre dual exact sequence of Equation 3 used to define the coboundary map δ. Hence the coboundary map on cohomology H^{0}(C, O_{C}) → H^{1}(C, ⋀^{2} I/I^{2} ⊗ Ω_{C}^{1}) is the dual of δ. □

6. Proof of Theorem 1.2: degree five case

The strategy of the proof of Theorem 1.2 is the following. Form the ℙ^{55} parametrizing all cubic hypersurfaces in ℙ^{5}. Let U_{e} → ℙ^{55} be the Deligne-Mumford stack over ℙ^{55} parametrizing pairs ([X], [C]) of a cubic hypersurface X ⊂ ℙ^{5} and a smooth rational curve C ⊂ X of degree e such that X is smooth along C and such that H^{1}(C, N_{C/X}) is zero (i.e. C ⊂ X is unobstructed). The last condition guarantees that U_{e} → ℙ^{55} is a smooth morphism. Also, by Proposition 2.4, a general fiber of U_{e} → ℙ^{55} is irreducible. In particular, U_{e} is also irreducible.

There is a straightforward generalization of the construction of Section 4 to the relative setting. This produces (locally over ℙ^{55}) a 2-form ω_{e} that is a global section of Ω_{U_{e}/ℙ^{55}}^{2} such that the restriction of ω_{e} to any fiber is the 2-form of the fiber constructed in Section 4. The rank of ω_{e} on fibers is a lower semicontinuous function on U_{e}, so to prove that the rank of ω_{e} is the maximum possible for a general pair ([X], [C]), it suffices to find a single pair ([X], [C]) ∈ U_{e} where the rank of ω_{e} is the maximum possible.

Let

0 ⟶ O(a_{1}) ⊕ O(a_{2}) ⊕ O(a_{3}) ⟶ N_{C/ℙ^{5}} ⟶ O(3e) ⟶ 0

be the usual exact sequence, where a_{1} + a_{2} + a_{3} = 3e − 2. In other words N_{C/X} = ⨁ O(a_{i}). The extension class of this sequence is an element ψ of H^{1}(ℙ^{1}, O(a_{1} − 3e) ⊕ O(a_{2} − 3e) ⊕ O(a_{3} − 3e)). Write ℙ^{1} = Proj (S), where S = ℂ[X_{0}, X_{1}]. Then, using Serre duality, ψ equals ψ_{1} ⊕ ψ_{2} ⊕ ψ_{3} for ψ_{i} ∈ Hom (S_{3e−a_{i}−2}, ℂ). Writing elements of H^{0}(C, N_{C/X}) in the form (g_{1}, g_{2}, g_{3}) for g_{i} ∈ H^{0}(C, O(a_{i})), then the pairing takes the following form

⟨( \begin{matrix}g_{1} \\ g_{2} \\ g_{3} \end{matrix} ) , ( \begin{matrix}h_{1} \\ h_{2} \\ h_{3} \end{matrix} )⟩ = ψ_{3}(g_{1}h_{2} − g_{2}h_{1}) + ψ_{2}(g_{1}h_{3} − g_{3}h_{1}) + ψ_{1}(g_{2}h_{3} − g_{3}h_{2}).

To compute the pairing for a given curve, we have to find the linear functionals ψ_{1}, ψ_{2}, ψ_{3} above. For large e this reduces to a rather involved computation. We will present this computation later, but first we show that in the special case e = 5 there is a short solution (which will hopefully motivate the reader to brave the computations of the next two sections).

Theorem 6.1. Let f : C → X be a general quintic rational curve on a general cubic fourfold X. Then N_{C/X} = O(4) ⊕ O(4) ⊕ O(5) and the extension class ψ of the sequence 0 → N_{C/X} → N_{C/ℙ^{5}} → O(15) → 0 is a general point of the space Hom (S_{9} ⊕ S_{9} ⊕ S_{8}, ℂ).

Proof. Fix a rational normal curve C ⊂ ℙ^{5} of degree 5. Its normal bundle N_{C/ℙ^{5}} is O(7)^{⊕4}. Thus any (not necessarily smooth) cubic fourfold X containing C determines a homomorphism of O_{C}-modules

φ_{X} : O(7)^{⊕4} → O(15).

Note that φ_{X} = 0 if and only if X is singular along C, which happens if and only if the defining equation of X is a section of I^{2}(3). The following computations are left to the reader,

dim H^{0}(ℙ^{5}, I(3)) = 40, dim H^{0}(ℙ^{5}, I^{2}(3)) = 4, dim Hom_{C}(O(7)^{4}, O(15)) = 36.

Thus the rule X ↦ φ_{X} is onto. Hence a general exact sequence of the form 0 → Ker (α) → O(7)^{4} − α → O(15) → 0 occurs as the normal bundle sequence for a general (nonsingular) X. The theorem follows. □

To finish, choose ψ_{i} as follows,

ψ_{1}(∑_{i=0}^{9} a_{i}X_{0}^{9−i}X_{1}^{i}) = ∑_{i=0}^{9} ν_{i}a_{i}, ψ_{1}(∑_{i=0}^{9} a_{i}X_{0}^{9−i}X_{1}^{i}) = ∑_{i=0}^{9} μ_{i}a_{i},

and

ψ_{1}(∑_{i=0}^{8} a_{i}X_{0}^{8−i}X_{1}^{i}) = ∑_{i=0}^{8} λ_{i}a_{i}.

Choose ν_{i}, μ_{i} and λ_{i} general. Form the matrix of the pairing with respect to the obvious basis of H^{0}(ℙ^{1}, O(4) ⊕ O(4) ⊕ O(5)). The computation gives,

\begin{matrix}0 & 0 & 0 & 0 & 0 & λ_{0} & λ_{1} & λ_{2} & λ_{3} & λ_{4} & μ_{0} & μ_{1} & μ_{2} & μ_{3} & μ_{4} & μ_{5} \\ 0 & 0 & 0 & 0 & 0 & λ_{1} & λ_{2} & λ_{3} & λ_{4} & λ_{5} & μ_{1} & μ_{2} & μ_{3} & μ_{4} & μ_{5} & μ_{6} \\ 0 & 0 & 0 & 0 & 0 & λ_{2} & λ_{3} & λ_{4} & λ_{5} & λ_{6} & μ_{2} & μ_{3} & μ_{4} & μ_{5} & μ_{6} & μ_{7} \\ 0 & 0 & 0 & 0 & 0 & λ_{3} & λ_{4} & λ_{5} & λ_{6} & λ_{7} & μ_{3} & μ_{4} & μ_{5} & μ_{6} & μ_{7} & μ_{8} \\ 0 & 0 & 0 & 0 & 0 & λ_{4} & λ_{5} & λ_{6} & λ_{7} & λ_{8} & μ_{4} & μ_{5} & μ_{6} & μ_{7} & μ_{8} & μ_{9} \\ −λ_{0} & −λ_{1} & −λ_{2} & −λ_{3} & −λ_{4} & 0 & 0 & 0 & 0 & 0 & ν_{0} & ν_{1} & ν_{2} & ν_{3} & ν_{4} & ν_{5} \\ −λ_{1} & −λ_{2} & −λ_{3} & −λ_{4} & −λ_{5} & 0 & 0 & 0 & 0 & 0 & ν_{1} & ν_{2} & ν_{3} & ν_{4} & ν_{5} & ν_{6} \\ −λ_{2} & −λ_{3} & −λ_{4} & −λ_{5} & −λ_{6} & 0 & 0 & 0 & 0 & 0 & ν_{2} & ν_{3} & ν_{4} & ν_{5} & ν_{6} & ν_{7} \\ −λ_{3} & −λ_{4} & −λ_{5} & −λ_{6} & −λ_{7} & 0 & 0 & 0 & 0 & 0 & ν_{3} & ν_{4} & ν_{5} & ν_{6} & ν_{7} & ν_{8} \\ −λ_{4} & −λ_{5} & −λ_{6} & −λ_{7} & −λ_{8} & 0 & 0 & 0 & 0 & 0 & ν_{4} & ν_{5} & ν_{6} & ν_{7} & ν_{8} & ν_{9} \\ −μ_{0} & −μ_{1} & −μ_{2} & −μ_{3} & −μ_{4} & −ν_{0} & −ν_{1} & −ν_{2} & −ν_{3} & −ν_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ −μ_{1} & −μ_{2} & −μ_{3} & −μ_{4} & −μ_{5} & −ν_{1} & −ν_{2} & −ν_{3} & −ν_{4} & −ν_{5} & 0 & 0 & 0 & 0 & 0 & 0 \\ −μ_{2} & −μ_{3} & −μ_{4} & −μ_{5} & −μ_{6} & −ν_{2} & −ν_{3} & −ν_{4} & −ν_{5} & −ν_{6} & 0 & 0 & 0 & 0 & 0 & 0 \\ −μ_{3} & −μ_{4} & −μ_{5} & −μ_{6} & −μ_{7} & −ν_{3} & −ν_{4} & −ν_{5} & −ν_{6} & −ν_{7} & 0 & 0 & 0 & 0 & 0 & 0 \\ −μ_{4} & −μ_{5} & −μ_{6} & −μ_{7} & −μ_{8} & −ν_{4} & −ν_{5} & −ν_{6} & −ν_{7} & −ν_{8} & 0 & 0 & 0 & 0 & 0 & 0 \\ −μ_{5} & −μ_{6} & −μ_{7} & −μ_{8} & −μ_{9} & −ν_{5} & −ν_{6} & −ν_{7} & −ν_{8} & −ν_{9} & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} .

Finally, to complete the proof of Theorem 1.2 for e = 5, consider the determinant of this matrix. For λ_{0} = 1, λ_{1} = 2, λ_{2} = −1, λ_{3} = 1, λ_{4} = 1, λ_{5} = 1, λ_{6} = −1, λ_{7} = −4, λ_{8} = 2, μ_{0} = 1, μ_{1} = 2, μ_{2} = −1, μ_{3} = 2, μ_{4} = 5, μ_{5} = −1, μ_{6} = 13, μ_{7} = −1, μ_{8} = 1, μ_{9} = 1, ν_{0} = 1, ν_{1} = 2, ν_{2} = 3, ν_{3} = 5, ν_{4} = 4, ν_{5} = −5, ν_{6} = −6, ν_{7} = −7, ν_{8} = −5, ν_{9} = 1 the determinant equals 445717799641. Since this is nonzero, Theorem 1.2 is true for e = 5.

7. Proof of Theorem 1.2

By Section 6, to prove Theorem 1.2 it suffices to determine a certain extension class ψ. The proof for e = 5 was short because ψ can be chosen to be a general element of the Ext group. Comparing the dimension of the parameter space U_{e} of pairs ([X], [C]) (cf. Section 6) and the dimension of the relevant Ext group, the Ext group grows more quickly. So, for large e, the extension class ψ will not be a general element of the Ext group.

Instead we work with a specific pair ([X], [C]) ∈ U_{e} for which we can prove the rank of ω_{e} is maximal and h^{1}(C, N_{C/X}) = 0. The reader is warned that for this pair, X is not smooth! But X is smooth on an open set containing C, and this is all that matters.

The proof of Theorem 1.2 in the case that e is odd is almost identical to the proof in the case that e is even. For this reason, most of the argument is carried out for both cases simultaneously. For each construction, the even case is specified by a subscript “ϵ” and the odd case is specified by a subscript “o”. Arguments that apply verbatim to both cases will not have a subscript (i.e., if there is no subscript, a true statement is obtained by either applying the subscript “o” throughout, or by applying the subscript “ϵ” throughout). In the odd case, the degree is e_{o} = 2r_{o} + 1 for some integer r_{o} ≥ 2. In the even case, the degree is e_{ϵ} = 2r_{ϵ} for some integer r_{ϵ} ≥ 3.

7.1. Computation of N_{C/ℙ^{5}}. We begin by specifying C and computing N_{C/ℙ^{5}}. As in the last section, choose homogeneous coordinates X_{0}, X_{1} on ℙ^{1}. Choose homogeneous coordinates Y_{0}, Y_{1}, Y_{2}, Y_{3}, Y_{4}, Y_{5} on ℙ^{5}. Consider the maps f_{o} : ℙ^{1} → ℙ^{5}, resp. f_{ϵ} : ℙ^{1} → ℙ^{5} given by

f_{o}([X_{0} : X_{1}]) = [X_{0}^{2r_{o}+1} : X_{0}^{2r_{o}}X_{1} : X_{0}^{r_{o}+1}X_{1}^{r_{o}} : X_{0}^{r_{o}}X_{1}^{r_{o}+1} : X_{0}X_{1}^{2r_{o}} : X_{1}^{2r_{o}+1}],

resp.

f_{ϵ}([X_{0} : X_{1}]) = [X_{0}^{2r_{ϵ}} : X_{0}^{2r_{ϵ}−1}X_{1} : X_{0}^{r_{ϵ}+1}X_{1}^{r_{ϵ}−1} : X_{0}^{r_{ϵ}−1}X_{1}^{r_{ϵ}+1} : X_{0}X_{1}^{2r_{ϵ}−1} : X_{1}^{2r_{ϵ}}].

This is a closed immersion, and local inverses are given by [Y_{0} : ⋯: Y_{5}] ↦ [Y_{0} : Y_{1}] and [Y_{0} : ⋯: Y_{5}] ↦ [Y_{4} : Y_{5}] on ℙ^{5} − 𝕍(Y_{0}, Y_{1}) and ℙ^{5} − 𝕍(Y_{4}, Y_{5}) respectively (the image of C does not intersect 𝕍(Y_{0}, Y_{1}, Y_{4}, Y_{5})). To compute the normal bundle of C in ℙ^{5}, we use the Euler sequence for T_{ℙ^{1}} and for T_{ℙ^{5}}. There is a map between these Euler sequences induced by f_{o}, resp. f_{ϵ}, and the important term is

d̃f_{o} : O_{ℙ^{1}}(1)^{⊕2} → f_{o}^{∗} (O_{ℙ^{5}}(1)^{⊕6}) = O_{ℙ^{1}}(2r_{o} + 1)^{⊕6},

resp.

d̃f_{ϵ} : O_{ℙ^{1}}(1)^{⊕2} → f_{ϵ}^{∗} (O_{ℙ^{5}}(1)^{⊕6}) = O_{ℙ^{1}}(2r_{ϵ})^{⊕6}.

These maps are given by the matrices

d̃f_{o} = [ \begin{matrix}(2r_{o} + 1)X_{0}^{2r_{o}} & 0 \\ 2r_{o}X_{0}^{2r_{o}−1}X_{1} & X_{0}^{2r_{o}} \\ (r_{o} + 1)X_{0}^{r_{o}}X_{1}^{r_{o}} & r_{o}X_{0}^{r_{o}+1}X_{1}^{r_{o}−1} \\ r_{o}X_{0}^{r_{o}−1}X_{1}^{r_{o}+1} & (r_{o} + 1)X_{0}^{r_{o}}X_{1}^{r_{o}} \\ X_{1}^{2r_{o}} & 2r_{o}X_{0}X_{1}^{2r_{o}−1} \\ 0 & (2r_{o} + 1)X_{1}^{2r_{o}} \end{matrix} ] ,

resp.

d̃f_{ϵ} = [ \begin{matrix}2r_{ϵ}X_{0}^{2r_{ϵ}−1} & 0 \\ (2r_{ϵ} − 1)X_{0}^{2r_{ϵ}−2}X_{1} & X_{0}^{2r_{ϵ}−1} \\ (r_{ϵ} + 1)X_{0}^{r_{ϵ}}X_{1}^{r_{ϵ}−1} & (r_{ϵ} − 1)X_{0}^{r_{ϵ}+1}X_{1}^{r_{ϵ}−2} \\ (r_{ϵ} − 1)X_{0}^{r_{ϵ}−2}X_{1}^{r_{ϵ}+1} & (r_{ϵ} + 1)X_{0}^{r_{ϵ}−1}X_{1}^{r_{ϵ}} \\ X_{1}^{2r_{ϵ}−1} & (2r_{ϵ} − 1)X_{0}X_{1}^{2r_{ϵ}−2} \\ 0 & 2r_{ϵ}X_{1}^{2r_{ϵ}−1} \end{matrix} ] .

Observe that both matrices have rank 2 at every point of ℙ^{1}. The normal bundle of C in ℙ^{5} is the cokernel of d̃f_{o}, resp. d̃f_{ϵ}. To compute this, consider the sheaf morphism T_{o} : O_{ℙ^{1}}(2r_{o}+1)^{⊕6} → O_{ℙ^{1}}(3r_{o}+1)^{⊕4}, resp. T_{ϵ} : O_{ℙ^{1}}(2r_{ϵ})^{⊕6} → O_{ℙ^{1}}(3r_{ϵ}− 1) ⊕ O_{ℙ^{1}}(3r_{ϵ}) ⊕ O_{ℙ^{1}}(3r_{ϵ}) ⊕ O_{ℙ^{1}}(3r_{ϵ} − 1) given by the matrices

\begin{matrix}T_{o} = [ \begin{matrix}(r_{o} − 1)X_{1}^{r_{o}} & −r_{o}X_{0}X_{1}^{r_{o}−1} & X_{0}^{r_{o}} \\ 0 & X_{1}^{r_{o}} & −r_{o}X_{0}^{r_{o}−1}X_{1} \\ 0 & 0 & (r_{o} − 1)X_{1}^{r_{o}} \\ 0 & 0 & 0 \end{matrix} | … \\ | \begin{matrix}0 & 0 & 0 \\ (r_{o} − 1)X_{0}^{r_{o}} & 0 & 0 \\ −r_{o}X_{0}X_{1}^{r_{o}−1} & X_{0}^{r_{o}} & 0 \\ X_{1}^{r_{o}} & −r_{o}X_{0}^{r_{o}−1}X_{1} & (r_{o} − 1)X_{0}^{r_{o}} \end{matrix} ] , \end{matrix}

resp.

\begin{matrix}T_{ϵ} = [ \begin{matrix}(r_{ϵ} − 2)X_{1}^{r_{ϵ}−1} & −(r_{ϵ} − 1)X_{0}X_{1}^{r_{ϵ}−1} & X_{0}^{r_{ϵ}−1} \\ 0 & 2X_{1}^{r_{ϵ}} & −r_{ϵ}X_{0}^{r_{ϵ}−2}X_{1}^{2} \\ 0 & 0 & (r_{ϵ} − 2)X)1^{r_{ϵ}} \\ 0 & 0 & 0 \end{matrix} | … \\ | \begin{matrix}0 & 0 & 0 \\ (r_{ϵ} − 2)X_{0}^{r_{ϵ}} & 0 & 0 \\ −r_{ϵ}X_{0}^{2}X_{1}^{r_{ϵ}−2} & 2X_{0}^{r_{ϵ}} & 0 \\ X_{1}^{r_{ϵ}−1} & −(r_{ϵ} − 1)X_{0}^{r_{ϵ}−2}X_{1} & (r_{ϵ} − 2)X_{0}^{r_{ϵ}−1} \end{matrix} ] . \end{matrix}

It is straightforward to verify that T_{o} ∘ d̃f_{o} is zero, resp. T_{ϵ} ∘ d̃f_{ϵ} is zero. And T_{o}, resp. T_{ϵ}, has rank 4 everywhere. Thus T_{o}, resp T_{ϵ}, gives an isomorphism of N_{C/ℙ^{5}} with O_{ℙ^{1}}(3r_{o} + 1)^{⊕4}, resp. O_{ℙ^{1}}(3r_{ϵ} − 1) ⊕ O_{ℙ^{1}}(3r_{ϵ}) ⊕ O_{ℙ^{1}}(3r_{ϵ}) ⊕ O_{ℙ^{1}}(3r_{ϵ} − 1). Each of these isomorphisms is taken to be an identification of locally free sheaves.

7.2. Computation of N_{C/X}. Next we specify X and compute the normal bundle N_{C/X}. Observe that the quadric equations Q_{a} = Y_{1}Y_{4} − Y_{0}Y_{5} and Q_{b} = Y_{2}Y_{3} − Y_{0}Y_{5} both vanish on the image of f_{o}, resp. f_{ϵ}. Let L_{a} and L_{b} be any linear homogeneous polynomials in Y_{0}, …, Y_{5} which are linearly independent and consider the homogeneous cubic polynomial F = L_{a}Q_{a} + L_{b}Q_{b} (later we will specialize to the case that L_{a} and L_{b} are general linear homogeneous polynomials in Y_{0} and Y_{5} alone). For our purposes it is convenient to make a “change of variables” and define M = L_{a} + L_{b} and N_{o} = L_{a} + r_{o}L_{b}, resp. N_{ϵ} = L_{a} + (r_{ϵ} − 1)L_{b} (here we are using that r_{o} ≠ 1, resp. r_{ϵ} ≠ 2, to see that L_{a} and L_{b} are uniquely determined by M and N_{o}, resp. N_{ϵ}). Consider X = {[Y_{0} : ⋯: Y_{5}] ∈ ℙ^{5}|F (Y_{0}, …, Y_{5}) = 0}. Observe that X is singular along the common zero locus of L_{a}, L_{b}, Q_{a} and Q_{b} – which will typically be a geometrically connected degree 4 curve of arithmetic genus 1.

To determine whether X is smooth along the image of f_{o}, resp. f_{ϵ}, we need to compute the pullback of the “gradient vector” [\frac{∂F}{∂Y_{i}}]_{i=0,...,5}. Define L̃_{a} = f^{∗}L_{a}, L̃_{b} = f^{∗}L_{b}, M̃ = f^{∗}M and Ñ = f^{∗}N, considered as sections of of H^{0}(ℙ^{1}, f^{∗}O_{ℙ^{5}}(1)) = H^{0}(ℙ^{1}, O_{ℙ^{1}}(5)). The pullback of the gradient vector of F is the sheaf morphism U_{o} : O_{ℙ^{1}}(2r_{o} + 1)^{⊕6} → O_{ℙ^{1}}(6r_{o} + 3), resp. U_{ϵ} : O_{ℙ^{1}}(2r_{ϵ})^{⊕6} → O_{ℙ^{1}}(6r_{ϵ}) given by

U_{o} = \begin{matrix}[ −X_{1}^{2r_{o}+1}(L̃_{a} + L̃_{b}) X_{0}X_{1}^{2r_{o}} L̃_{a} X_{0}^{r_{o}}X_{1}^{r_{o}+1}L̃_{b} | & … \\ | X_{0}^{r_{o}+1}X_{1}^{r_{o}} L̃_{b} X_{0}^{2r_{o}}X_{1}L̃_{a} −X_{0}^{2r_{o}+1}(L̃_{a} + L̃_{b}) ] , \end{matrix}

resp.

U_{ϵ} = \begin{matrix}[ −X_{1}^{2r_{ϵ}}(L̃_{a} + L̃_{b}) X_{0}X_{1}^{2r_{ϵ}−1}L̃_{a} X_{0}^{r_{ϵ}−1}X_{1}^{r_{ϵ}+1}L̃_{b} | & … \\ | X_{0}^{r_{ϵ}+1}X_{1}^{r_{ϵ}−1}L̃_{b} X_{0}^{2r_{ϵ}−1}X_{1}L̃_{a} −X_{0}^{2r_{ϵ}}(L̃_{a} + L̃_{b}) ] . \end{matrix}

If L̃_{a} and L̃_{b} have no common zeroes and if L̃_{a} + L̃_{b} is nonzero at the points [1 : 0] and [0 : 1], then these matrices are everywhere nonzero, i.e., X is smooth along C. From now on, assume this is the case. The matrix U factors as U = S ∘ T where S_{o} : N_{C/ℙ^{5}} → O_{ℙ^{1}}(6r_{o} + 3), resp. S_{ϵ} : N_{C/ℙ^{5}} → O_{ℙ^{1}}(6r_{ϵ}), are given by the matrices

S_{o} = \frac{−1}{r_{o} − 1} [ X_{1}^{r_{o}+1}M̃ X_{0}X_{1}^{r_{o}} Ñ_{o} X_{0}^{r_{o}}X_{1}Ñ_{o} X_{0}^{r_{o}+1}M̃ ] ,

resp.

S_{ϵ} = \frac{−1}{2(r_{ϵ} − 2)} [ 2X_{1}^{r_{ϵ}+1}M̃ X_{0}X_{1}^{r_{ϵ}−1}Ñ_{ϵ} X_{0}^{r_{ϵ}−1}X_{1}Ñ_{ϵ} 2X_{0}^{r_{ϵ}+1}M̃ ] .

The normal bundle N_{C/X} is the kernel of the sheaf morphism S. To describe this map, we write out

\{ \begin{matrix}M_{o} = c_{0,o}Y_{0} + c_{1,o}Y_{1} + c_{2,o}Y_{2} + c_{3,o}Y_{3} + c_{4,o}Y_{4} + c_{5,o}Y_{5}, \\ N_{o} = d_{0,o}Y_{0} + d_{1,o}Y_{1} + d_{2,o}Y_{2} + d_{3,o}Y_{3} + d_{4,o}Y_{4} + d_{5,o}Y_{5} \end{matrix} ,

resp.

\{ \begin{matrix}M_{ϵ} = c_{0,ϵ}Y_{0} + c_{1,ϵ}Y_{1} + c_{2,ϵ}Y_{2} + c_{3,ϵ}Y_{3} + c_{4,ϵ}Y_{4} + c_{5,ϵ}Y_{5}, \\ N_{ϵ} = d_{0,ϵ}Y_{0} + d_{1,ϵ}Y_{1} + d_{2,ϵ}Y_{2} + d_{3,ϵ}Y_{3} + d_{4,ϵ}Y_{4} + d_{5,ϵ}Y_{5} \end{matrix}

Then we have

\{ \begin{matrix}M̃_{o} = c_{0,o}X_{0}^{2r_{o}+1} + c_{1,o}X_{0}^{2r_{o}}X_{1} + c_{2,o}X_{0}^{r_{o}+1}X_{1}^{r_{o}} + c_{3,o}X_{0}^{r_{o}}X_{1}^{r_{o}+1} + c_{4,o}X_{0}X_{1}^{2r_{o}} + c_{5,o}X_{1}^{2r_{o}+1}, \\ Ñ_{o} = d_{0,o}X_{0}^{2r_{o}+1} + d_{1,o}X_{0}^{2r_{o}}X_{1} + d_{2,o}X_{0}^{r_{o}+1}X_{1}^{r_{o}} + d_{3,o}X_{0}^{r_{o}}X_{1}^{r_{o}+1} + d_{4,o}X_{0}X_{1}^{2r_{o}} + d_{5,o}X_{1}^{2r_{o}+1} \end{matrix} ,

resp.

\{ \begin{matrix}M̃_{ϵ} = c_{0,ϵ}X_{0}^{2r_{ϵ}} + c_{1,ϵ}X_{0}^{2r_{ϵ}−1}X_{1} + c_{2,ϵ}X_{0}^{r_{ϵ}+1}X_{1}^{r_{ϵ}−1} + c_{3,ϵ}X_{0}^{r_{ϵ}−1}X_{1}^{r_{ϵ}+1} + c_{4,ϵ}X_{0}X_{1}^{2r_{ϵ}−1} + c_{5,ϵ}X_{1}^{2r_{ϵ}}, \\ Ñ_{ϵ} = d_{0,ϵ}X_{0}^{2r_{ϵ}} + d_{1,ϵ}X_{0}^{2r_{ϵ}−1}X_{1} + d_{2,ϵ}X_{0}^{r_{ϵ}+1}X_{1}^{r_{ϵ}−2} + d_{3,ϵ}X_{0}^{r_{ϵ}−1}X_{1}^{r_{ϵ}+1} + d_{4,ϵ}X_{0}X_{1}^{2r_{ϵ}−1} + d_{5,ϵ}X_{1}^{2r_{ϵ}} \end{matrix}

Denote by n_{o}, n′_{o}, m_{o}, m′_{o} the following expressions,

\{ \begin{matrix}n_{o} = d_{4,o}X_{0}^{2}X_{1}^{r_{o}} + d_{5,o}X_{0}X_{1}^{r_{o}+1}, \\ n′_{o} = d_{0,o}X_{0}^{r_{o}+1}X_{1} + d_{1,o}X_{0}^{r_{o}}X_{1}^{2} + d_{2,o}X_{0}X_{1}^{r_{o}+1} + d_{3,o}X_{1}^{r_{o}+2}, \\ m_{o} = c_{4,o}X_{0}X_{1}^{r_{o}+1} + c_{5,o}X_{1}^{r_{o}+2}, \\ m′_{o} = c_{0,o}X_{0}^{r_{o}+2} + c_{1,o}X_{0}^{r_{o}+1}X_{1} + c_{2,o}X_{0}^{2}X_{1}^{r_{o}} + c_{3,o}X_{0}X_{1}^{r_{o}+1} \end{matrix}

Denote by n_{ϵ}, n′_{ϵ}, m_{ϵ}, m′_{ϵ} the following expressions,

\{ \begin{matrix}n_{ϵ} = d_{3,ϵ}X_{0}X_{1}^{r_{ϵ}} + d_{4,ϵ}X_{0}^{2}X_{1}^{r_{ϵ}−1} + d_{5,ϵ}X_{0}X_{1}^{r_{ϵ}}, \\ n′_{ϵ} = d_{0,ϵ}X_{0}^{r_{ϵ}}X_{1} + d_{1,ϵ}X_{0}^{r_{ϵ}−1}X_{1}^{2} + d_{2,ϵ}X_{0}X_{1}^{r_{ϵ}}, \\ m_{ϵ} = 2c_{4,ϵ}X_{0}X_{1}^{r_{ϵ}+1} + 2c_{5,ϵ}X_{1}^{r_{ϵ}+2}, \\ m′_{ϵ} = 2c_{0,ϵ}X_{0}^{r_{ϵ}+2} + 2c_{1,ϵ}X_{0}^{r_{ϵ}+1}X_{1} + 2c_{2,ϵ}X_{0}^{3}X_{1}^{r_{ϵ}−1} + 2c_{3,ϵ}X_{0}X_{1}^{r_{ϵ}+1} \end{matrix}

Then X_{0}X_{1}Ñ_{o} = X_{1}^{r_{o}+1}n_{o} + X_{0}^{r_{o}+1}n′_{o} and M̃_{o} = X_{1}^{r_{o}−1}m_{o} + X_{0}^{r_{o}−1}m′_{o}, resp. X_{0}X_{1}Ñ_{ϵ} = X_{1}^{r_{ϵ}+1}n_{ϵ} + X_{0}^{r_{ϵ}+1}n′_{ϵ} and 2M̃_{ϵ} = X_{1}^{r_{ϵ}−2}m_{ϵ} + X_{0}^{r_{ϵ}−2}m′_{ϵ}.

Consider the sheaf morphism R_{o} : O_{ℙ^{1}}(2r_{o})⊕O_{ℙ^{1}}(2r_{o} +2)⊕O_{ℙ^{1}}(2r_{o} −1) → N_{C/ℙ^{5}}, resp. R_{ϵ} : O_{ℙ^{1}}(2r_{ϵ} −2)⊕O_{ℙ^{1}}(2r_{ϵ} +2)⊕O_{ℙ^{1}}(2r_{ϵ} −2) → N_{C/ℙ^{5}} given by the matrices,

R_{o} = [ \begin{matrix}X_{0}^{r_{o}+1} & 0 & n_{o} \\ 0 & X_{0}^{r_{o}−1} & −m_{o} \\ 0 & −X_{1}^{r_{o}−1} & −m′_{o} \\ −X_{1}^{r_{o}+1} & 0 & n′_{o} \end{matrix} ] ,

resp.

R_{ϵ} = [ \begin{matrix}X_{0}^{r_{ϵ}+1} & 0 & n_{ϵ} \\ 0 & X_{0}^{r_{ϵ}−2} & −m_{ϵ} \\ 0 & −X_{1}^{r_{ϵ}−2} & −m′_{ϵ} \\ −X_{1}^{r_{ϵ}+1} & 0 & n′_{ϵ} \end{matrix} ]

The composition S ∘R is zero. The matrix R has rank 3 generically (in particular it has rank 3 at [0 : 1] and [1 : 0] by the hypothesis that M̃ is nonzero at those points). By degree considerations, R has rank 3 everywhere and gives an isomorphism of O_{ℙ^{1}}(2r_{o})⊕O_{ℙ^{1}}(2r_{o} +2)⊕O_{ℙ^{1}}(2r_{o} −1), resp. O_{ℙ^{1}}(2r_{ϵ} −2)⊕O_{ℙ^{1}}(2r_{ϵ})⊕O_{ℙ^{1}}(2r_{ϵ} −2) with the kernel of S, i.e. with N_{C/X}. In particular, h^{1}(ℙ^{1}, N_{C/X}) = 0, so ([X], [C]) is a point of U_{e}.

7.3. Initial description of the pairing. In this subsection we begin the description of the skew-symmetric bilinear pairing on H^{0}(C, N_{C/X}) induced by ω_{e}. We complete the description in the next subsection. Elements in H^{0}(ℙ^{1}, N_{C/X}) are denoted by (g_{1}, g_{2}, g_{3}) or g_{1}e_{1} + g_{2}e_{2} + g_{3}e_{3} where e_{i} is the ith column of the matrix R and where g_{1} ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{o})), g_{2} ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{o} + 2)) and g_{3} ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{o} − 1)), resp. g_{1} ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{ϵ} − 2)), g_{2} ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{ϵ} + 2)) and g_{3} ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{ϵ} − 2)).

By Theorem 5.1, to compute the bilinear pairing ω_{e} on H^{0}(ℙ^{1}, N_{C/X}) it is equivalent (up to a nonzero scalar) to compute the boundary map

δ : H^{0}(ℙ^{1}, ⋀^{2} N_{C/X}) → H^{1}(ℙ^{1}, O_{ℙ^{1}}(−2)).

The next term in the long exact sequence of cohomology is H^{1}(ℙ^{1}, O_{ℙ^{1}}(3r_{o}))^{⊕4}, resp. H^{1}(ℙ^{1}, O_{ℙ^{1}}(3r_{ϵ} − 1))^{⊕2} ⊕ H^{1}(ℙ^{1}, O_{ℙ^{1}}(3r_{ϵ} − 2))^{⊕2}, both of which are zero. Therefore the connecting homomorphism is the cokernel of the map on global sections

R_{o}^{†} : H^{0}(ℙ^{1}, O_{ℙ^{1}}(3r_{o}))^{⊕4} → H^{0}(ℙ^{1}, ⋀^{2} N_{C/X})

resp.

R_{ϵ}^{†} : H^{0}(ℙ^{1}, O_{ℙ^{1}}(3r_{ϵ} − 1) ⊕ O_{ℙ^{1}}(3r_{ϵ} − 2) ⊕ O_{ℙ^{1}}(3r_{ϵ} − 2)) ⊕ O_{ℙ^{1}}(3r_{ϵ} − 1))

⟶ H^{0}(ℙ^{1}, ⋀^{2} N_{C/X}),

determined by the sheaf morphism R^{†} : N_{C/ℙ^{5}}^{∨} ⊗ ⋀^{3} N_{C/X} → ⋀^{2} N_{C/X} that is adjoint to R. (The adjoint R^{†} = diag(1, −1, 1) ∘ R^{t}, where R^{t} is the transpose of R.) If we use as “ordered basis” for ⋀^{2} N_{C/X} the elements e_{2} ∧ e_{3}, e_{1} ∧ e_{3} and e_{1} ∧ e_{2}, then the matrix of R^{†} is

R_{o}^{†} = [ \begin{matrix}X_{0}^{r_{o}+1} & 0 & 0 & −X_{1}^{r_{o}+1} \\ 0 & −X_{0}^{r_{o}−1} & X_{1}^{r_{o}−1} & 0 \\ n_{o} & −m_{o} & −m′_{o} & n′_{o} \end{matrix} ] ,

resp.

R_{ϵ}^{†} = [ \begin{matrix}X_{0}^{r_{ϵ}+1} & 0 & 0 & −X_{1}^{r_{ϵ}+1} \\ 0 & −X_{0}^{r_{ϵ}−2} & X_{1}^{r_{ϵ}−2} & 0 \\ n_{ϵ} & −m_{ϵ} & −m′_{ϵ} & n′_{ϵ} \end{matrix} ] .

In other words, the pairing ω_{e} is given by

[(g_{1}e_{1} + g_{2}e_{2} + g_{3}e_{3}), (h_{1}e_{1} + h_{2}e_{2} + h_{3}e_{3})] = (g_{1}h_{2} − g_{2}h_{1})e_{1} ∧ e_{2}
+(g_{1}h_{3} − g_{3}h_{1})e_{1} ∧ e_{3} + (g_{2}h_{3} − g_{3}h_{2})e_{2} ∧ e_{3} mod Im(R^{†}).

7.4. The image of the map R^{†}. To compute an explicit formula for the pairing [·, ·], we need to find the image of R^{†}. First consider the intersection of H^{0}(ℙ^{1}, O_{ℙ^{1}}(4r_{o}+2))e_{1}∧e_{2} with the image of R_{o}^{†}, resp. H^{0}(ℙ^{1}, O_{ℙ^{1}}(4r_{ϵ}))e_{1}∧e_{2} with the image of R_{ϵ}^{†}. A global section of O_{ℙ^{1}}(3r_{o})^{⊕4}, resp. O_{ℙ^{1}}(3r_{ϵ} −1)⊕O_{ℙ^{1}}(3r_{ϵ} −2)⊕ O_{ℙ^{1}}(3r_{ϵ} − 2) ⊕ O_{ℙ^{1}}(3r_{ϵ} − 1) is mapped under R^{†} into H^{0}(ℙ^{1}, O_{ℙ^{1}}(4r_{o} + 2))e_{1} ∧ e_{2}, resp. H^{0}(ℙ^{1}, O_{ℙ^{1}}(4r_{ϵ}))e_{1} ∧ e_{2} iff it is of the form

v_{o} = [ \begin{matrix}X_{1}^{r_{o}+1}p_{o} \\ −X_{1}^{r_{o}−1}q_{o} \\ −X_{0}^{r_{o}−1}q_{o} \\ X_{0}^{r_{o}+1}p_{o} \end{matrix} ]

for some p_{o} ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{o} − 1)) and q_{o} ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{o} + 1)), resp. iff it is of the form

v_{ϵ} = [ \begin{matrix}X_{1}^{r_{ϵ}+1}p_{ϵ} \\ −X_{1}^{r_{ϵ}−2}q_{ϵ} \\ −X_{0}^{r_{ϵ}−2}q_{ϵ} \\ X_{0}^{r_{ϵ}+1}p_{ϵ} \end{matrix} ]

for some p_{ϵ} ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{ϵ} − 2)) and q_{ϵ} ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{ϵ})). The image of such an element is,

R_{o}^{†}(v_{o}) = (X_{0}X_{1}Ñ_{o}p_{o} + M̃_{o}q_{o})e_{1} ∧ e_{2},

resp.

R_{ϵ}^{†}(v_{ϵ}) = (X_{0}X_{1}Ñ_{ϵ}p_{ϵ} + 2M̃_{ϵ}q_{ϵ})e_{1} ∧ e_{2}.

There is one last simplification. Assume that c_{1} = c_{2} = c_{3} = c_{4} = 0 and d_{1} = d_{2} = d_{3} = d_{4} = 0, in other words L_{a} and L_{b} are 2 linearly independent, linear combinations of Y_{0} and Y_{5} and c_{0}, c_{5}, d_{0} and d_{5} are all nonzero. Consider those q such that q = X_{0}X_{1}q′ for some q′ ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{o} − 1)), resp. q′ ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{ϵ} − 2)). Then R_{o}^{†}(v_{o}) equals X_{0}X_{1}(Ñ_{o}p_{o} + M̃_{o}q′_{o}), resp. R_{ϵ}^{†}(v_{ϵ}) equals X_{0}X_{1}(Ñ_{ϵ}p_{ϵ} + 2M̃_{ϵ}q_{ϵ}). Since M̃ and Ñ are linearly independent elements in the span of X_{0}^{2r_{o}+1} and X_{1}^{2r_{o}+1}, resp. in the span of X_{0}^{2r_{ϵ}} and X_{1}^{2r_{ϵ}}, as p and q′ vary the expression R^{†}(v) varies over the whole linear span of

X_{0}^{4r_{o}+1}X_{1}, …, X_{0}^{2r_{o}+2}X_{1}^{2r_{o}}, X_{0}^{2r_{o}}X_{1}^{2r_{o}+2}, …, X_{0}X_{1}^{4r_{o}+1},

resp.

X_{0}^{4r_{ϵ}−1}X_{1}, …, X_{0}^{2r_{ϵ}+1}X_{1}^{2r_{ϵ}−1}, X_{0}^{2r_{ϵ}−1}X_{1}^{2r_{ϵ}+1}, …, X_{0}X_{1}^{4r_{ϵ}−1}.

Notice that X_{0}^{4r_{o}+2}, X_{0}^{2r_{o}+1}X_{1}^{2r_{o}+1} and X_{1}^{4r_{o}+2} are missing, resp. X_{0}^{4r_{ϵ}}, X_{0}^{2r_{ϵ}}X_{1}^{2r_{ϵ}} and X_{1}^{4r_{ϵ}} are missing. Taking q_{o} = X_{0}^{2r_{o}+1} and q_{o} = X_{1}^{2r_{o}+1} gives c_{0,o}X_{0}^{4r_{o}+2} + c_{5,o}X_{0}^{2r_{o}+1}X_{1}^{2r_{o}+1} and c_{0,o}X_{0}^{2r_{o}+1}X_{1}^{2r_{o}+1} + c_{5,o}X_{1}^{4r_{o}+2}. And taking q_{ϵ} = X_{0}^{2r_{ϵ}} and q_{ϵ} = X_{1}^{2r_{ϵ}} gives c_{0,ϵ}X_{0}^{4r_{ϵ}} + c_{5,ϵ}X_{0}^{2r_{ϵ}}X_{1}^{2r_{ϵ}} and c_{0,ϵ}X_{0}^{2r_{ϵ}}X_{1}^{2r_{ϵ}} + c_{5,ϵ}X_{1}^{4r_{ϵ}}. Thus the intersection of H^{0}(ℙ^{1}, O_{ℙ^{1}}(4r + 2))e_{1} ∧ e_{2} with the image of R^{†} is the subspace with basis

c_{0,o}X_{0}^{4r_{o}+2} + c_{5,o}X_{0}^{2r_{o}+1}X_{1}^{2r_{o}+1}, X_{0}^{4r_{o}+1}X_{1}, X_{0}^{4r_{o}}X_{1}^{2}, …, X_{0}^{2r_{o}+2}X_{1}^{2r_{o}}, X_{0}^{2r_{o}}X_{1}^{2r_{o}+2}, …

X_{0}X_{1}^{4r_{o}+1}, c_{0,o}X_{0}^{2r_{o}+1}X_{1}^{2r_{o}+1} + c_{5,o}X_{1}^{4r_{o}+2},

resp.

c_{0,ϵ}X_{0}^{4r_{ϵ}} + c_{5,ϵ}X_{0}^{2r_{ϵ}}X_{1}^{2r_{ϵ}}, X_{0}^{4r_{ϵ}−1}X_{1}, X_{0}^{4r_{ϵ}−2}X_{1}^{2}, …, X_{0}^{2r_{ϵ}+1}X_{1}^{2r_{ϵ}−1}, X_{0}^{2r_{ϵ}−1}X_{1}^{2r_{ϵ}+1}, …

X_{0}X_{1}^{4r_{ϵ}−1}, c_{0,ϵ}X_{0}^{2r_{ϵ}}X_{1}^{2r_{ϵ}} + c_{5,ϵ}X_{1}^{4r_{ϵ}}.

For each pair of nonnegative integers (i, j), denote by α_{i,j} : H^{0}(ℙ^{1}, O_{ℙ^{1}}(i + j)) → ℂ the linear functional such that for every homogeneous polynomial g of degree d,

g(X_{0}, X_{1}) = ∑_{i+j=d} α_{i,j}(g)X_{0}^{i}X_{1}^{j},

i.e. α_{i,j}(g) is the coefficient of X_{0}^{i}X_{1}^{j} in g. Then the linear functional c_{5,o}^{2}α_{4r_{o}+2,0} − c_{0,o}c_{5,o}α_{2r_{o}+1,2r_{o}+1} + c_{0,o}^{2}α_{0,4r_{o}+2}, resp. c_{5,ϵ}^{2}α_{4r_{ϵ},0} − c_{0,ϵ}c_{5,ϵ}α_{2r_{ϵ},2r_{ϵ}} + c_{0,ϵ}^{2}α_{0,4r_{ϵ}}, is a nonzero linear functional on H^{0}(ℙ^{1}, O_{ℙ^{1}}(4r + 2)) whose kernel is precisely the intersection with the image of R^{†}.

Using the first two rows of R^{†}, every element in H^{0}(ℙ^{1}, ⋀^{2} N_{C/X}) is congruent to some element in H^{0}(ℙ^{1}, O_{ℙ^{1}}(4r_{o} +2))e_{1} ∧e_{2}, resp. H^{0}(ℙ^{1}, O_{ℙ^{1}}(4r_{ϵ}))e_{1} ∧e_{2} modulo the image of R^{†}. Carrying this out, up to a nonzero scalar, the pairing [·, ·] is,

[(g_{1}e_{1} + g_{2}e_{2} + g_{3}e_{3}), (h_{1}e_{1} + h_{2}e_{2} + h_{3}e_{3})]_{o} =
(c_{5,o}^{2}α_{4r_{o}+2,0} − c_{0,o}c_{5,o}α_{2r_{o}+1,2r_{o}+2} + c_{0,o}^{2}α_{0,4r_{o}+2})(g_{1}h_{2} − g_{2}h_{1}) +
c_{0,o}c_{5,o}(c_{5,o}α_{3r_{o},r_{o}−1} − c_{0,o}α_{r_{o}−1,3r_{o}})(g_{1}h_{3} − g_{3}h_{1}) +
c_{0,o}c_{5,o}(d_{5,o}α_{3r_{o}+1,r_{o}} − d_{0,o}α_{r_{o},3r_{o}+1})(g_{2}h_{3} − g_{3}h_{2}),

resp.

[(g_{1}e_{1} + g_{2}e_{2} + g_{3}e_{3}), (h_{1}e_{1} + h_{2}e_{2} + h_{3}e_{3})]_{ϵ} =
(c_{5,ϵ}^{2}α_{4r_{ϵ},0} − c_{0,ϵ}c_{5,ϵ}α_{2r_{ϵ},2r_{ϵ}} + c_{0,ϵ}^{2}α_{0,4r_{ϵ}})(g_{1}h_{2} − g_{2}h_{1}) +
2c_{0,ϵ}c_{5,ϵ}(c_{5,ϵ}α_{3r_{ϵ}−2,r_{ϵ}−2} − c_{0,ϵ}α_{r_{ϵ}−2,3r_{ϵ}−2})(g_{1}h_{3} − g_{3}h_{1}) +
c_{0,ϵ}c_{5,ϵ}(d_{5,ϵ}α_{3r_{ϵ},r_{ϵ}} − d_{0,ϵ}α_{r_{ϵ},3r_{ϵ}})(g_{2}h_{3} − g_{3}h_{2}).

7.5. Diagonalizing the pairing. The antisymmetric bilinear map [·, ·] gives a linear transformation ω̃_{e} : H^{0}(ℙ^{1}, N_{C/X}) → H^{0}(ℙ^{1}, N_{C/X})^{∨} and we want to find the kernel of this linear transformation. This is done by “diagonalizing” the pair (H^{0}(ℙ^{1}, N_{C/X}), [·, ·]), i.e. by finding a direct sum decomposition

H^{0}(ℙ^{1}, N_{C/X})_{o} = ⨁_{i=0}^{r_{o}−2} E_{i,o} ⊕ E_{r_{o}−1,o} ⊕ E_{r_{o},o},

resp.

H^{0}(ℙ^{1}, N_{C/X})_{ϵ} = ⨁_{i=0}^{r_{ϵ}−3} E_{i,ϵ} ⊕ E_{r_{ϵ}−2,ϵ} ⊕ E_{r_{ϵ}−1,ϵ} ⊕ E_{r,ϵ}

into pairwise orthogonal subspaces with respect to [·, ·]. In the odd case, to show [·, ·]_{o} has trivial kernel, it suffices to show the restriction to each space E_{i,o} has trivial kernel.

In the even case, there is a vector w in E_{r_{ϵ},ϵ} lying in the kernel. On the quotient vector space H^{0}(ℙ^{1}, N_{C/X})/ℂ {w}, there is an induced alternating bilinear form [·, ·]′_{ϵ} and an induced direct sum decomposition ⨁_{i=0}^{r_{ϵ}} E′_{i,ϵ} by pairwise orthogonal subspaces. To show [·, ·]′_{ϵ} has trivial kernel, it suffices to show the restriction to each space E′_{i,ϵ} has trivial kernel. In both case, this is done by computing the determinant of the matrix of [·, ·]_{o}, resp. [·, ·]′_{ϵ} with respect to a suitable basis.

For i = 0, …, r_{o} − 2, denote by E_{i,o} ⊂ H^{0}(ℙ^{1}, N_{C/X})_{o} the subspace generated by

v_{i,1,o} = X_{0}^{r_{o}+1+i}X_{1}^{r_{o}−1−i}e_{1}
v_{i,2,o} =  X_{0}^{r_{o}−i}X_{1}^{r_{o}+2+i}e_{2}
v_{i,3,o} =  X_{0}^{2r_{o}−1−i}X_{1}^{i}e_{3}
v_{i,4,o} =  X_{0}^{i}X_{1}^{2r_{o}−1−i}e_{3}
v_{i,5,o} =  X_{0}^{r_{o}+2+i}X_{1}^{r_{o}−i}e_{2}
v_{i,6,o} = X_{0}^{r_{o}−1−i}X_{1}^{r_{o}+1+i}e_{1}

For i = 0, …, r_{ϵ} − 3, denote by E_{i,ϵ} ⊂ H^{0}(ℙ^{1}, N_{C/X})_{ϵ} the subspace generated by

v_{i,1,ϵ} = X_{0}^{r_{ϵ}+i}X_{1}^{r_{ϵ}−2−i}e_{1}
v_{i,2,ϵ} = X_{0}^{r_{ϵ}−i}X_{1}^{r_{ϵ}+2+i}e_{2}
v_{i,3,ϵ} = X_{0}^{2r_{ϵ}−2−i}X_{1}^{i}e_{3}
v_{i,4,ϵ} = X_{0}^{i}X_{1}^{2r_{ϵ}−2−i}e_{3}
v_{i,5,ϵ} = X_{0}^{r_{ϵ}+2+i}X_{1}^{r_{ϵ}−i}e_{2}
v_{i,6,ϵ} = X_{0}^{r_{ϵ}−2−i}X_{1}^{r_{ϵ}+i}e_{1}

For i = r_{o} − 1 denote by E_{r_{o}−1,o} ⊂ H^{0}(ℙ^{1}, N_{C/X})_{o} the subspace generated by

v_{r−1,1,o} =  X_{0}^{2r_{o}}e_{1}
v_{r−1,2,o} =  X_{0}^{2r_{o}+2}e_{2}
v_{r−1,3,o} = X_{0}^{2r_{o}+1}X_{1}e_{2}
v_{r−1,4,o} = X_{0}^{r_{o}}X_{1}^{r_{o}+1}e_{3}
v_{r−1,5,o} = X_{0}^{r_{o}+1}X_{1}^{r_{o}}e_{3}
v_{r−1,6,o} = X_{0}X_{1}^{2r_{o}+1}e_{2}
v_{r−1,7,o} =  X_{1}^{2r_{o}+2}e_{2}
v_{r−1,8,o} =  X_{1}^{2r_{o}}e_{1}

For i = r_{ϵ} − 2, denote by E_{r_{ϵ}−2,ϵ} ⊂ H^{0}(ℙ^{1}, N_{C/X})_{ϵ} the subspace generated by

v_{r−2,1,ϵ} =  X_{0}^{2r_{ϵ}−2}e_{1}
v_{r−2,2,ϵ} =  X_{0}^{2r_{ϵ}+2}e_{2}
v_{r−2,3,ϵ} = X_{0}^{r_{ϵ}}X_{1}^{r_{ϵ}−2}e_{3}
v_{r−2,4,ϵ} = X_{0}^{2}X_{1}^{2r_{ϵ}}e_{2}
v_{r−2,5,ϵ} = X_{0}^{2r_{ϵ}}X_{1}^{2}e_{2}
v_{r−2,6,ϵ} = X_{0}^{r_{ϵ}−2}X_{1}^{r_{ϵ}}e_{3}
v_{r−2,7,ϵ} =  X_{1}^{2r_{ϵ}+2}e_{2}
v_{r−2,8,ϵ} =  X_{1}^{2r_{ϵ}−2}e_{1}

For i = r_{o} denote by E_{r_{o},o} ⊂ H^{0}(ℙ^{1}, N_{C/X})_{o} the subspace generated by

\{ \begin{matrix}v_{r,1,o} & = & X_{0}^{r,o}X_{1}^{r,o}e_{1} \\ v_{r,2,o} & = & X_{0}^{r_{o}+1}X_{1}^{r_{o}+1}e_{2} \end{matrix}

For i = r_{ϵ} − 1 denote by E_{r_{ϵ}−1,ϵ} ⊂ H^{0}(ℙ^{1}, N_{C/X})_{ϵ} the subspace generated by

\{ \begin{matrix}v_{r−1,1,ϵ} & = & X_{0}^{r_{ϵ}−1}X_{1}^{r_{ϵ}−1}e_{1} \\ v_{r−1,2,ϵ} & = & X_{0}^{r_{ϵ}+1}X_{1}^{r_{ϵ}+1}e_{2} \end{matrix}

Finally, for i = r_{ϵ}, denote by E_{r_{ϵ},ϵ} ⊂ H^{0}(ℙ^{1}, N_{C/X})_{ϵ} the subspace generated by

\{ \begin{matrix}v_{r,1,ϵ} & = & X_{0}^{2r_{ϵ}+1}X_{1}e_{2} \\ v_{r,2,ϵ} & = & X_{0}^{r_{ϵ}−1}X_{1}^{r_{ϵ}−1}e_{3} \\ v_{r,3,ϵ} & = & X_{0}X_{1}^{2r_{ϵ}+1}e_{2} \end{matrix}

Each of these generating sets is a subbasis of the standard monomial basis of H^{0}(ℙ^{1}, N_{C/X}). Visibly, every monomial basis vector is in precisely one of the subspaces E_{i}, and thus these spaces give a direct sum decomposition of H^{0}(ℙ^{1}, N_{C/X}). As a consistency check, observe that for i = 0, …, r_{o} − 2, resp. i = 0, …, r_{ϵ} − 3, dim(E_{i,o}) = 6, resp. dim(E_{i,ϵ}) = 6, dim(E_{r_{o}−1,o}) = 8, resp. dim(E_{r_{ϵ}−2,ϵ}) = 8, dim(E_{r_{o},o}) = 2, resp. dim(E_{r_{ϵ}−1,ϵ}) = 2, and dim(E_{r_{ϵ},ϵ}) = 3. So the sum of the dimensions of the spaces E_{i,o} is

6(r_{o} − 1) + 8 + 2 = 6r_{o} + 4 = (2r_{o} + 1) + (2r_{o} + 3) + 2r_{o},

i.e.,

dimH^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{o}))e_{1} + dimH^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{o} + 2))e_{2} + dimH^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{o} − 1))e_{3}.

Similarly, the sum of the dimensions of the spaces E_{i,ϵ} is

6(r_{ϵ} − 2) + 8 + 2 + 3 = 6r_{ϵ} + 1 = (2r_{ϵ} − 1) + (2r_{ϵ} + 3) + (2r_{ϵ} − 1),

i.e.,

dimH^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{ϵ}−2))e_{1}+dimH^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{ϵ}+2))e_{2}+dimH^{0}(ℙ^{1}, O_{ℙ^{1}}(2r_{ϵ}−2))e_{3}.

Checking the spaces E_{i} are pairwise orthogonal with respect to [·, ·] is straightforward, but tedious. One way to think of it is to consider the graph whose vertices are the standard monomial basis vectors of H^{0}(ℙ^{1}, N_{C/X}), and where there is an edge between two such basis vectors iff the pairing is nonzero for this pair. Thus there is never an edge between g_{1}e_{1} and h_{1}e_{1}, nor between g_{2}e_{2} and h_{2}e_{2}, nor between g_{3}e_{3} and h_{3}e_{3}. There is an edge between g_{1}e_{1} and h_{2}e_{2} iff g_{1}h_{2} = X_{0}^{4r_{o}+2}, X_{0}^{2r_{o}+1}X_{1}^{2r_{o}+1} or X_{1}^{4r_{o}+2}, resp. iff g_{1}h_{2} = X_{0}^{4r_{ϵ}}, X_{0}^{2r_{ϵ}}X_{1}^{2r_{ϵ}} or X_{1}^{4r_{ϵ}}. There is an edge between g_{1}e_{1} and h_{3}e_{3} iff g_{1}h_{3} = X_{0}^{3r_{o}}X_{1}^{r_{o}−1} or X_{0}^{r_{o}−1}X_{1}^{3r_{o}}, resp. iff g_{1}h_{3} = X_{0}^{3r_{ϵ}−2}X_{1}^{r_{ϵ}−2} or X_{0}^{r_{ϵ}−2}X_{1}^{3r_{ϵ}−2}. And there is an edge between g_{2}e_{2} and h_{3}e_{3} iff g_{2}h_{3} = X_{0}^{3r_{o}+1}X_{1}^{r_{o}} or X_{0}^{r_{o}}X_{1}^{3r_{o}+1}, resp. iff g_{2}h_{3} = X_{0}^{3r_{ϵ}}X_{1}^{r_{ϵ}} or X_{0}^{r_{ϵ}}X_{1}^{3r_{ϵ}}. Thus, the valences of X_{0}^{2r_{o}}e_{1} , X_{1}^{2r_{o}}e_{1}, X_{0}^{r_{o}}X_{1}^{r_{o}−1}e_{3} and X_{0}^{r_{o}−1}X_{1}^{r_{o}}e_{3} are each 3, resp. the valences of X_{0}^{2r_{ϵ}−2}e_{1}, X_{1}^{2r_{ϵ}−2}e_{1}, X_{0}^{r_{ϵ}}X_{1}^{r_{ϵ}−2}e_{3} and X_{0}^{r_{ϵ}−2}X_{1}^{r_{ϵ}}e_{3} are each 3. Also the valences of X_{0}^{r_{o}}X_{1}^{r_{o}}e_{1} and X_{0}^{r_{o}+1}X_{1}^{r_{o}+1}e_{2} are each 1, resp. the valences of X_{0}^{r_{ϵ}−1}X_{1}^{r_{ϵ}−1}e_{1}, X_{0}^{r_{ϵ}+1}X_{1}^{r_{ϵ}+1}e_{2}, X_{0}^{2r_{ϵ}+1}X_{1}e_{2} and X_{0}X_{1}^{2r_{ϵ}+2}e_{2} are each 1. Every other vertex has valence two. Moreover, there is an symmetry of the graph by permuting the variables X_{0} and X_{1}. Using this, it is straightforward to compute the maximal connected subgraph containing the vector v_{i,1} for each i. The vertices of this subgraph are the generators of E_{i}. Therefore the E_{i} are pairwise orthogonal.

7.6. Computing the determinants. Finally, we will compute the matrix and determinant of the restriction of ω̃_{e} to each of the subspace E_{i}. In the odd case, each determinant is nonzero, proving that ω̃ has trivial kernel. In the even case, all but one of the determinants is nonzero, and for E_{r_{ϵ}}, the restriction of ω̃ has a 1-dimensional kernel.

For i = 0, …, r_{o} − 2, resp. for i = 0, …, r_{ϵ} − 3 denote by A_{i} the matrix of ω̃_{e} : E_{i} → E_{i}^{∨} with respect to the ordered basis v_{i,1}, …, v_{i,6} and the dual ordered basis of E_{i}^{∨}. The computation gives,

A_{i,o} = [ \begin{matrix}0 & c_{0}c_{5} & −c_{0}c_{5}^{2} & 0 & 0 & 0 \\ −c_{0}c_{5} & 0 & 0 & c_{0}c_{5}d_{0} & 0 & 0 \\ c_{0}c_{5}^{2} & 0 & 0 & 0 & c_{0}c_{5}d_{5} & 0 \\ 0 & −c_{0}c_{5}d_{0} & 0 & 0 & 0 & −c_{0}^{2}c_{5} \\ 0 & 0 & −c_{0}c_{5}d_{5} & 0 & 0 & −c_{0}c_{5} \\ 0 & 0 & 0 & c_{0}^{2}c_{5} & c_{0}c_{5} & 0 \end{matrix} ] ,

resp.

A_{i,ϵ} = [ \begin{matrix}0 & c_{0}c_{5} & −2c_{0}c_{5}^{2} & 0 & 0 & 0 \\ −c_{0}c_{5} & 0 & 0 & c_{0}c_{5}d_{0} & 0 & 0 \\ 2c_{0}c_{5}^{2} & 0 & 0 & 0 & c_{0}c_{5}d_{5} & 0 \\ 0 & −c_{0}c_{5}d_{0} & 0 & 0 & 0 & −2c_{0}^{2}c_{5} \\ 0 & 0 & −c_{0}c_{5}d_{5} & 0 & 0 & −c_{0}c_{5} \\ 0 & 0 & 0 & 2c_{0}^{2}c_{5} & c_{0}c_{5} & 0 \end{matrix} ]

The Pfaffian of each matrix is Pfaff(A_{i,o}) = c_{0}^{3}c_{5}^{3}(c_{0}d_{5} −c_{5}d_{0}), resp. Pfaff(A_{i,ϵ}) = 2c_{0}^{3}c_{5}^{3}(c_{0}d_{5} − c_{5}d_{0}). Thus the determinant is Det(A_{i,o}) = c_{0}^{6}c_{5}^{6}(c_{0}d_{5} − c_{5}d_{0})^{2}, resp. Det(A_{i,ϵ}) = 4c_{0}^{6}c_{5}^{6}(c_{0}d_{5} − c_{5}d_{0})^{2}. By hypothesis, c_{0}, c_{5} are nonzero and (c_{0}, c_{5}) is linearly independent from (d_{0}, d_{5}). Thus each determinant is nonzero.

For i = r_{o} − 1, resp. i = r_{ϵ} − 2, denote by A_{i} the matrix of ω̃_{e} : E_{i} → E_{i}^{∨} with respect to the ordered basis v_{i,1}, …, v_{i,8} and the dual ordered basis of E_{i}^{∨}. The computation gives,

A_{r_{o}−1,o} = [ \begin{matrix}0 & −c_{5}^{2} & 0 & −c_{0}c_{5}^{2} & 0 & c_{0}c_{5} & 0 & 0 \\ c_{5}^{2} & 0 & 0 & 0 & −c_{0}c_{5}d_{5} & 0 & 0 & 0 \\ 0 & 0 & 0 & −c_{0}c_{5}d_{5} & 0 & 0 & 0 & −c_{0}c_{5} \\ c_{0}c_{5}^{2} & 0 & c_{0}c_{5}d_{5} & 0 & 0 & 0 & −c_{0}c_{5}d_{0} & 0 \\ 0 & c_{0}c_{5}d_{5} & 0 & 0 & 0 & −c_{0}c_{5}d_{0} & 0 & −c_{0}^{2}c_{5} \\ −c_{0}c_{5} & 0 & 0 & 0 & c_{0}c_{5}d_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{0}c_{5}d_{0} & 0 & 0 & 0 & c_{0}^{2} \\ 0 & 0 & c_{0}c_{5} & 0 & c_{0}^{2}c_{5} & 0 & −c_{0}^{2} & 0 \end{matrix} ] ,

resp.

A_{r_{ϵ}−2,ϵ} = [ \begin{matrix}0 & −c_{5}^{2} & −2c_{0}c_{5}^{2} & c_{0}c_{5} & 0 & 0 & 0 & 0 \\ c_{5}^{2} & 0 & 0 & 0 & 0 & −c_{0}c_{5}d_{5} & 0 & 0 \\ 2c_{0}c_{5}^{2} & 0 & 0 & 0 & c_{0}c_{5}d_{5} & 0 & −c_{0}c_{5}d_{0} & 0 \\ −c_{0}c_{5} & 0 & 0 & 0 & 0 & c_{0}c_{5}d_{0} & 0 & 0 \\ 0 & 0 & −c_{0}c_{5}d_{5} & 0 & 0 & 0 & 0 & −c_{0}c_{5} \\ 0 & c_{0}c_{5}d_{5} & 0 & −c_{0}c_{5}d_{0} & 0 & 0 & 0 & −2c_{0}^{2}c_{5} \\ 0 & 0 & c_{0}c_{5}d_{0} & 0 & 0 & 0 & 0 & c_{0}^{2} \\ 0 & 0 & 0 & 0 & c_{0}c_{5} & 2c_{0}^{2}c_{5} & −c_{0}^{2} & 0 \end{matrix} ]

The Pfaffian of this matrix is Pfaff(A_{r_{o}−1,o}) = c_{0}^{3}c_{5}^{3}(c_{0}d_{5}−c_{5}d_{0})^{2}, resp. Pfaff(A_{r_{ϵ}−2,ϵ}) = c_{0}^{3}c_{5}^{3}(c_{0}d_{5}−c_{5}d_{0})^{2}. Thus the determinant is Det(A_{r_{o}−1,o}) = c_{0}^{6}c_{5}^{6}(c_{0}d_{5}−c_{5}d_{0})^{4}, resp. the determinant is Det(A_{r_{ϵ}−2,ϵ}) = c_{0}^{6}c_{5}^{6}(c_{0}d_{5} − c_{5}d_{0})^{4}. By hypothesis, c_{0}, c_{5} are nonzero and (c_{0}, c_{5}) is linearly independent from (d_{0}, d_{5}). Thus each determinant is nonzero.

For i = r_{o}, resp. i = r_{ϵ} − 1, denote by A_{i} the matrix of ω̃_{e} : E_{i} → E_{i}^{∨} with respect to the ordered basis v_{i,1}, v_{i,2} and the dual ordered basis of E_{i}^{∨}. The computation gives,

A_{r_{o},o} = [ \begin{matrix}0 & c_{0}c_{5} \\ −c_{0}c_{5} & 0 \end{matrix} ] ,

resp.

A_{r_{ϵ}−1,ϵ} = [ \begin{matrix}0 & c_{0}c_{5} \\ −c_{0}c_{5} & 0 \end{matrix} ] .

Visibly the Pfaffian of this matrix is c_{0}c_{5} and the determinant is c_{0}^{2}c_{5}^{2}. By hypothesis, c_{0}, c_{5} are nonzero, thus each determinant is nonzero.

In the odd case, since the determinant of each matrix A_{i,o} is nonzero, the kernel of ω̃_{e} is trivial. This proves Theorem 1.2 in case e ≥ 5 is an odd integer. From this point on, suppose that e ≥ 6 is even.

Denote by A_{r_{ϵ},ϵ} the matrix of ω̃_{e} : E_{r_{ϵ},ϵ} → E_{r_{ϵ},ϵ}^{∨} with respect to the ordered basis v_{r_{ϵ},1,ϵ}, v_{r_{ϵ},2,ϵ}, v_{r_{ϵ},3,ϵ} and the dual ordered basis of E_{r_{ϵ},ϵ}^{∨}. The computation gives,

A_{r_{ϵ},ϵ} = [ \begin{matrix}0 & −c_{0}c_{5}d_{5} & 0 \\ c_{0}c_{5}d_{5} & 0 & −c_{0}c_{5}d_{0} \\ 0 & c_{0}c_{5}d_{0} & 0 \end{matrix} ]

This matrix is singular: the kernel contains the vector w = d_{0,ϵ}v_{r_{ϵ},1} + d_{5,ϵ}v_{r_{ϵ},3}, i.e. (d_{0,ϵ}X_{0}^{2r_{ϵ}} + d_{5,ϵ}X_{1}^{2r_{ϵ}})X_{0}X_{1}e_{2}. So this vector is in the kernel of ω̃_{e}. Consider the quotient vector space V′ = H^{0}(ℙ^{1}, N_{C/X})/ℂ{w}. There is an induced alternating bilinear pairing ω′_{e} on V′. Since w′ ∈ E_{r_{ϵ},ϵ}, there is an induced direct sum decomposition V′ = ⨁_{i=0}^{r_{ϵ}} E′_{i,ϵ} by pairwise orthogonal subspaces where for i = 0, …, r −1 the quotient map E_{i,ϵ} → E′_{i,ϵ} is an isomorphism. And E′_{r_{ϵ},ϵ} has as basis the images of the vectors v_{r_{ϵ},1,ϵ}, v_{r_{ϵ},2,ϵ} provided d_{5,ϵ} ≠ 0, and has as basis the images of the vectors v_{r_{ϵ},2,ϵ}v_{r_{ϵ},3,ϵ} provided d_{0,ϵ} ≠ 0.

First consider the case, d_{5,ϵ} ≠ 0. Denote by A′_{r_{ϵ},ϵ} the matrix of ω′_{e} : E′_{r_{ϵ},ϵ} → (E′_{r_{ϵ},ϵ})^{∨} with respect to the ordered basis v′_{r_{ϵ},1,ϵ}, v′_{r_{ϵ},2,ϵ} and the dual ordered basis of (E′_{r_{ϵ},ϵ})^{∨}. The computation gives,

A′_{r_{ϵ},ϵ} = [ \begin{matrix}0 & −c_{0}c_{5}d_{5} \\ c_{0}c_{5}d_{5} & 0 \end{matrix} ]

The Pfaffian of this matrix is c_{0}c_{5}d_{5} and the determinant is c_{0}^{2}c_{5}^{2}d_{5}^{2}. By hypothesis c_{0}, c_{5}, d_{5} are nonzero, thus the determinant is nonzero.

The remaining case is that d_{0} ≠ 0. Again denote by A′_{r_{ϵ},ϵ} the matrix of ω′_{e} : E′_{r_{ϵ},ϵ} → (E′_{r,ϵ})^{∨} with respect to the ordered basis v′_{r_{ϵ},2,ϵ}, v′_{r_{ϵ},3,ϵ} and the dual ordered basis of (E′_{r_{ϵ},ϵ})^{∨}. The computation gives,

A′_{r_{ϵ},ϵ} = [ \begin{matrix}0 & −c_{0}c_{5}d_{0} \\ c_{0}c_{5}d_{0} & 0 \end{matrix} ]

The Pfaffian of this matrix is c_{0}c_{5}d_{0} and the determinant is c_{0}^{2}c_{5}^{2}d_{0}^{2}. By hypothesis c_{0}, c_{5}, d_{0} are nonzero, thus the determinant is nonzero.

In both cases, the determinant of the restriction of ω′_{e} to each subspace E′_{i,ϵ} is nonzero. Thus the kernel of ω̃_{e} is spanned by w = (d_{0,ϵ}X_{0}^{2r_{ϵ}} + d_{5,ϵ}X_{1}^{2r_{ϵ}})X_{0}X_{1}e_{2}. In particular, the kernel of ω̃_{e} is 1-dimensional. This proves Theorem 1.2 in case e ≥ 6 is an even integer.

8. Comments and Questions

There are some generalizations of Theorem 1.2 to stable maps with marked points. The cubic hypersurface X ⊂ ℙ^{5} and the stable map f : ℙ^{1} → ℙ^{5} are the same as in Section 7. The marked points are [0 : 1], [1 : 0] ∈ ℙ^{1}. The same method as Section 7 proves the following.

Theorem 8.1. Let X ⊂ ℙ^{5} a smooth cubic hypersurface, let \overline{M}_{0,n}(X, e) denote the Kontsevich moduli space of pointed stable maps to X of arithmetic genus 0 and degree e, and let \overline{M}_{e,n} be a nonsingular projective model of the coarse moduli space. Denote by ev : \overline{M}_{e,n} → X^{n} evaluation at each marked point. And denote by T_{ev}, the kernel of the derivative, dev : T_{\overline{M}_{e,n}} → ev^{∗}T_{X^{n}}.

There is a canonical section ω_{e} ∈ H^{0}(\overline{M}_{e,n}, Ω_{\overline{M}_{e,n}}^{2}). Suppose that X is general.

(1) (i) If n = 1, if e ≥ 5 is an odd integer, and if ζ = (C, p, f : C → X) is a general point of \overline{M}_{e,1}, the restriction of ω_{e} to T_{ev}|_{ζ} has a 1-dimensional kernel.

(2) (ii) If n = 1, if e ≥ 6 is even, e ≥ 6, and if ζ = (C, p, f : C → X) is a general point of \overline{M}_{e,1}, the restriction of ω_{e} to T_{ev}|_{ζ} is nondegenerate. Therefore a general fiber of ev has Kodaira dimension ≥ 0 and, in particular, it is not uniruled.

(3) (iii) If n = 2, if e ≥ 5 is odd, and if ζ = (C, p_{1}, p_{2}, f : C → X) is a general point of \overline{M}_{e,2}, the restriction of ω_{e} to T_{ev}|_{ζ} is nondegenerate. Therefore a general fiber of (ev_{1}, ev_{2}) has Kodaira dimension ≥ 0 and, in particular, it is not uniruled.

(4) (iv) If n = 2, if e ≥ 6 is even, and if ζ = (C, p_{1}, p_{2}, f : C → X) is a general point of \overline{M}_{e,2}, the restriction of ω_{e} to T_{ev}|_{ζ} has a 1-dimensional kernel.

Proof. Most of the details are left to the reader. The technique is almost identical to the proof of Theorem 1.2 and is roughly as follows: For (i) and (ii), consider the special pairs ([X], [C]) used in Section 7. In addition, assume that d_{0,o}, d_{5,o} are both nonzero. For the marked point on C, use either f([0 : 1]) or f([1 : 0]). The tangent space to T_{ev}|_{ζ} is the subspace of sections of H^{0}(ℙ^{1}, N_{C/X}) vanishing at [0 : 1], resp. [1 : 0]. The form ω_{e} on this subspace is the form computed in Section 7. In particular, since the space of sections vanishing at [0 : 1] is generated by standard monomial basis vectors of H^{0}(ℙ^{1}, N_{C/X}), the direct sum decomposition into pairwise orthogonal subspaces yields a direct sum decomposition of the space of sections vanishing at [0 : 1].

In the odd case, the kernel is generated by c_{5,o}v_{0,2,o} + v_{0,3,o} + d_{5,o}v_{0,6,o}. And the induced pairing on the quotient space is nondegenerate. In the even case, the kernel is nontrivial: it is generated by d_{0,ϵ}v_{r_{ϵ},1,ϵ} + d_{5,ϵ}v_{r_{ϵ},3,ϵ} and v_{r_{ϵ}−2,1} + 2c_{5,ϵ}v_{r_{ϵ}−2,3} − d_{5,ϵ}v_{r_{ϵ}−2,6,ϵ}. However, under a nontrivial first-order deformation of the pointed curve not changing the map f : ℙ^{1} → X, only moving the point [0 : 1] on ℙ^{1}, the kernel becomes trivial (this is a simple deformation theory exercise).

Parts (iii) and (iv) are the same. In the odd case, the kernel is trivial. In the even case, the kernel is generated by d_{0,ϵ}v_{r_{ϵ},1,ϵ} + d_{5,ϵ}v_{r_{ϵ},3,ϵ} (no deformation theory is needed). □

Question 8.2. What is the Kodaira dimension of \overline{M}_{e}, resp. what is the dimension of a fiber of the MRC quotient of \overline{M}_{e}, when the form ω_{e} does have a kernel? If e ≥ 6 is even, is \overline{M}_{e} uniruled?

We are convinced that \overline{M}_{e} is not uniruled, but we do not have a proof for e ≥ 8. In case e is 6, we can prove that \overline{M}_{6} is not uniruled by an ad hoc argument. It is possible this could be used as the base case of an induction by considering how the kernel of ω_{e+2} specializes on the boundary divisor Δ_{e,2} ⊂ \overline{M}_{e}.

Proposition 8.3. If the cubic hypersurface X ⊂ ℙ^{5} is general, then M_{6} is not uniruled. More precisely, there exists a rational transformation f : M_{6} −−→ Hilb_{X}^{6t} whose general fiber is a genus 1 curve which is a leaf of the distribution Ker(ω_{e}).

Here is a rough sketch of the proof. The method of proof is similar to that in [13], but instead of using residual curves in an intersection of X with a cubic scroll, we use residual curves in an intersection of X with a quartic scroll. For a general nondegenerate, rational, degree 6 curve C ⊂ ℙ^{5}, there is a unique quartic scroll Σ ⊂ ℙ^{5} containing C. If X is general, then X contains no quartic scrolls (although special smooth cubic fourfolds can contain a quartic scroll, [14, Section 4.1.3]). The intersection Σ ∩ X is a degree 12 curve in Σ that is a local complete intersection (in particular it is Gorenstein) and contains C as a subcurve of degree 6. By Gorenstein liaison, the residual curve C′ to C in Σ is a degree 6 curve of arithmetic genus 1, and is a smooth, connected curve if C general. This gives a rational transformation from M_{6} to the open subset U of the Chow variety/Hilbert scheme parametrizing degree 6 curves in X of arithmetic genus 1; [C] ↦ [C′]. The fiber of this rational transformation containing [C] is isomorphic to Pic^{2}(C′), i.e., it is a connected, smooth curve of genus 1 (actually it will only be a dense open subset since we are working on the non-complete variety M_{6}).

On M_{6} there is the 2-form ω_{6} constructed in Section 4. On U there is a 2-form by the same process as in Section 4 corresponding to the family of degree 6 curves of arithmetic genus 1. On the domain of definition of the rational transformation M_{6} → U, form the pullback of the 2-form on U; denote this pullback 2-form by ω′. Over a dense open set of M_{6}, the curve Σ∩X is a connected, reduced at-worst-nodal curve and the process from Section 4 produces a 2-form ω′′ corresponding to this family of curves. The relation between these forms is

ω_{6} + ω′ = ω′′

on the open, dense locus where all three are defined.

On the other hand, there is a unirational space W ⊂ Hilb^{(2t+1)(t+1)}(ℙ^{5}) parametrizing all smooth, nondegenerate quartic scrolls in Σ ⊂ ℙ^{5} (in fact this is a homogeneous space for PGL_{6} since any two such scrolls are projectively equivalent). Over a dense open subset of W the process from Section 4 produces a 2-form corresponding to the family of curves whose fiber over [Σ] is Σ ∩ X. And ω′′ is the pullback of this 2-form by the obvious rational map M_{6} −−→ W. Since W is unirational, it does not support any nonzero 2-form, i.e., ω′′ = 0. So ω_{6} = −ω′. In particular, the kernel of ω_{6} coincides with the kernel of ω′. Since ω′ is a pullback by the rational transformation M_{6} → U, in particular the tangent space of the fiber of this rational transformation is contained in ω_{6}. We know the fiber is one-dimensional. By Theorem 1.2, also the kernel of ω_{6} is one-dimensional. Thus the kernel of ω_{6} at a general point of M_{6} is precisely the tangent space to the fiber of M_{6} → U. In other words, the foliation determined by the kernel of ω_{6} is algebraically integrable on a dense (Zariski) open subset of M_{6}, the leaf space is (birationally) an open subset U of the Hilbert scheme of smooth, degree 6 curves in X of genus 1, and the projection to the leaf space is (birationally) the rational transformation M_{6} −−→ U.

From this it follows that U has Kodaira dimension ≥ 0, in particular it is not uniruled. By the special case of the Iitaka conjecture proved in [17], the Kodaira dimension of M_{6} is ≥ 0. In particular, M_{6} is not uniruled.

There are lots of missing details in this argument. They each follow by straightforward arguments of projective geometry, and are left to the reader.

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

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

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

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