A PENCIL OF ENRIQUES SURFACES OF INDEX ONE WITH NO SECTION

JASON MICHAEL STARR

Abstract. Monodromy arguments and deformation-and-specialization are used to prove existence of a pencil of Enriques surfaces with no section and index 1. The same technique completes the strategy from [GHMS05, §7.3] proving the family of witness curves for dimension d depends on the integer d.

1. Introduction

This paper uses monodromy and deformation-and-specialization to answer some questions related to [GHMS05]. Theorem 1.3 gives a new, elementary proof of existence of a pencil of Enriques surfaces over ℂ with no section, which moreover has index 1. Proposition 1.4 completes the strategy from [GHMS05, §7.3] proving the family H_{d} of witness curves depends on the relative dimension d.

The main theorem of [GHS03] proves a rationally connected variety defined over the function field of a curve over a characteristic 0 algebraically closed field has a rational point. A converse is proved in [GHMS05]; in particular [GHMS05, Cor. 1.4] proves there is an Enriques surface without a rational point that is defined over the function field of a curve (answering a question of Serre [CS01, p. 153]). Subsequently Lafon [Laf04] gave an explicit pencil of Enriques surfaces defined over ℤ[1/2] whose base-change to any field of characteristic ≠ 2 has no rational point. Hélène Esnault asked about the index of Enriques surfaces without a rational point.

Definition 1.1. Let X be a finite type scheme, algebraic space, algebraic stack, etc. over a field K. The index and the minimal degree are,

\begin{aligned}I(K, X) & = & gcd\{[L : K]|X(L) ≠ ∅\}, \\ M(K, X) & = & min\{[L : K]|X(L) ≠ ∅\}. \end{aligned}

Hélène Esnault asked, essentially, what is the possible index of an Enriques surface defined over a function field of a curve. In Lafon’s example, M(K, X_{K}) = I(K, X_{K}) = 2. In [GHMS05] the index is not computed, but likely there also I(K, X_{K}) > 1.

Question 1.2 (Esnault). If X is an Enriques surface defined over a function field of a curve K with no K-point, is I(K, X) > 1?

This has to do with whether there is an obstruction to K-points in Galois cohomology. If so and if the obstruction is compatible with restriction and corestriction, the order of the obstruction divides I(K, X). So if there is a cohomological obstruction “explaining” non-existence of K-points, then I(K, X_{K}) > 1. The main result proves there is an Enriques surface with no K-point whose index is 1.

Date: February 27, 2006.

Theorem 1.3. Let k be an algebraically closed field with char(k) ≠ 2, 3 that is “sufficiently big”, e.g. uncountable. There exists a flat, projective k-morphism π : X → ℙ_{k}^{1} with the following properties,

(i) the geometric generic fiber of π is a smooth Enriques surface, (ii) the invertible sheaf π_{∗}[ω_{π}^{⊗2}] has degree 6, (iii) for the function field K of ℙ_{k}^{1} and the generic fiber X_{K} of π, I(K, X_{K}) = 1

and M(K, X_{K}) = 3. Moreover every “very general” Enriques surface over k is a fiber of such a family.

The method is simple. Over ℙ^{1} a family of surfaces is given whose monodromy group acts as the full group of symmetries of the dual graph of the geometric generic fiber – which is the 2-skeleton of a cube. There is an action of ℤ/2ℤ acting fiberwise, and the quotient is a pencil X /ℙ^{1} of “Enriques surfaces”. The 8 vertices of the cube give a degree 4 multi-section of the pencil. The 6 faces of the cube give a degree 3 multi-section of the pencil. By monodromy considerations every multi-section of X has degree ≥ 3. The pencil X together with the degree 3 and degree 4 multisections deforms to a pencil whose geometric generic fiber is a smooth Enriques surface. For a general such deformation, M(K, X_{K}) = 3 and I(K, X_{K}) = 1.

The same method gives pencils of degree d hypersurfaces with minimal degree d, which is used to complete the argument from [GHMS05, Section 7.3].

Proposition 1.4. Let B be a normal, projective variety of dimension ≥ 2 and let M be an irreducible family of irreducible curves dominating B (i.e., the morphism from the total space of the family of curves to B is dominant). There is an integer d such that M is not a witness family for dimension d, i.e., there is a projective, dominant morphism of relative dimension d, π : X → B, whose restriction to each curve of M has a section, but whose restriction to some smooth curve in B has no section.

Acknowledgments: I am grateful to Igor Dolgachev, Hélène Esnault, Mike Stillman and Harry Tamvakis for useful conversations. This paper originated in meetings with Tom Graber, Joe Harris and Barry Mazur to whom I am very grateful. I was partially supported by NSF Grant DMS-0201423.

2. The construction for hypersurfaces

Let d, n > 0 be integers, let k be a field, and let V be a k-vector space of dimension n + 1. Degree d hypersurfaces in ℙ(V) are parametrized by the projective space,

ℙSym^{d}(V^{∨}) = Proj ⨁_{i} Sym^{i}(Syt^{d}(V)),

where Syt^{d}(V) is the vector space of symmetric tensors in ⊗^{d}V.

Let B, C be k-curves isomorphic to ℙ_{k}^{1}. There exists a degree d, separably-generated k-morphism f : C → B such that Gal(k(C)/k(B)) is the full symmetric group 𝔖_{d}. This is straightforward in every characteristic – in characteristic 0 any morphism with simple branching will do.

Let g : C → ℙ(V^{∨}) be a closed immersion whose image is a rational normal curve of degree n. Consider the pullback of the tautological surjection, V ⊗_{k} O_{C} → g^{∗}O(1). By adjointness, there is a map β : V ⊗_{k} O_{B} → f_{∗}(g^{∗}O(1)). For every locally free O_{C}-module E there is the norm sheaf on B,

Nm_{f}(E) = Hom_{O_{B}}(⋀^{d}(f_{∗}O_{C}), ⋀^{d}(f_{∗}E)),

together with the norm map of O_{B}-modules,

α′_{E} : ⨂^{d}(f_{∗}E) → Nm_{f}(E), e_{1}⊗⋯⊗e_{d} ↦ (c_{1} ∧ ⋯ ∧ c_{d} ↦ (c_{1} · e_{1}) ∧ ⋯ ∧ (c_{d} · e_{d})) ,

for e_{1} ⊗⋯⊗e_{d} ∈ ⨂^{d}(f_{∗}E) and c_{1} ∧⋯∧c_{d} ∈ ⋀^{d}(f_{∗}O_{B}). Only the restriction to the subsheaf of symmetric tensors is needed, α_{E} : Syt^{d}(f_{∗}E) → Nm_{f}(E). In particular, Nm_{f}(O_{C}) = O_{B} and α_{O_{C}} (b ⊗ ⋯ ⊗ b) ∈ O_{B} is the usual norm of b ∈ f_{∗}O_{C}.

Denote by γ the composition,

Syt^{d}(V) ⊗_{k} O_{B} ⟶^{Syt^{d}(β)} Syt^{d}(f_{∗}g^{∗}O(1)) ⟶^{α_{g^{∗}O(1)}} Nm_{f}(g^{∗}O(1)).

Because β is surjective, also γ is surjective. So there is an induced morphism h : B → ℙSym^{d}(V^{∨}). For every geometric point b ∈ B whose fiber f^{−1}(b) is a reduced set {c_{1}, …, c_{d}}, h(b) = [g(c_{1}) × ⋯ × g(c_{d})]. The degree of Nm_{f}(g^{∗}O(1)), and thus the degree of h, is n

Denote by X_{h} ⊂ B × ℙ(V) the preimage under (h, Id) of the universal hypersurface in ℙSym^{d}(V^{∨}) × ℙ(V), and by π : X_{h} → B the projection. Let m = min(d, n) and let S_{d,n} ⊂ ℤ_{≥0} denote the additive semigroup generated by \binom{d}{i} for i = 1, …, m. Denote K = k(B) and denote by X_{h,K} the generic fiber of π.

Proposition 2.1. Every irreducible multi-section of π has degree divisible by \binom{d}{i} for i = 1, …, m. The degree of every multi-section is in S_{d,n}. In particular, if d > n then M(K, X_{h,K}) = d and I(K, X_{h,K}) is divisible by gcd(d, \binom{d}{2}, …, \binom{d}{n}).

Proof. Denote by U ⊂ B the largest open subset over which f is étale and define W = f^{−1}(U). For each i = 1, …, m, denote by W_{i}/U the relative Hilbert scheme Hilb_{W/U}^{i}. Because W is étale over U, the fiber of f over a geometric point b of B is a set of d distinct points, f^{−1}(b) = {c_{1}, …, c_{d}}, and the fiber of Hilb_{W/U}^{i} is the set of subsets of f^{−1}(b) of size i. Every geometric fiber of X_{h} ×_{B} U → U is union of d hyperplanes. Denote by,

X_{h} ×_{B} U = X_{h}^{1} ⊔ X_{h}^{2} ⊔ ⋯ ⊔ X_{h}^{n},

the locally closed stratification where X_{h}^{i} is the set of points x in precisely i irreducible components of the geometric fiber X_{h} ⊗_{O_{B}} \overline{κ}(π(x)). Because every finite subset of distinct closed points on a rational normal curve over an algebraically closed field is in linearly general position, X_{h}^{i} = ∅ for i > m; in particular every geometric fiber of X_{h} ×_{B} U → U is a simple normal crossings variety. For each i = 1, …, m the morphism X_{h}^{i} → U factors as an 𝔸^{n−i}-bundle over W_{i} over U. The generic point of every irreducible multi-section is contained in X_{h}^{i} for some i = 1, …, m. Because Gal(k(C)/k(B)) is 𝔖_{d}, W_{i} is irreducible. Therefore the degree of the multi-section is divisible by deg(k(W_{i})/k(U)) = \binom{d}{i}. So the degree of every multi-section, irreducible or not, is in S_{d,n}. Moreover, the intersection of X_{h,K} with a general line in ℙ(V ⊗_{k}K) is a degree d multi-section, so M(K, X_{h,K}) = d. □

Let H_{n} ⊂ Hom(B, ℙSym^{d}(V^{∨})) denote the irreducible component of morphisms of degree n. Denote by X → H_{n} × B the pullback by the universal morphism of the universal hypersurface in ℙSym^{d}(V^{∨}) × ℙ(V). For every field k′ and every [h] ∈ H_{n}(k′), denote by X_{h} the restriction of X to Spec (k′)×B, by K′ the function field k′(B), and by X_{h,K′} the generic fiber of the projection to B.

Corollary 2.2. Assume d > n. In H_{n} there is a countable intersection of open dense subsets such that for every [h] in this set, M(K′, X_{h,K′}) = d and I(K′, X_{h,K′}) is divisible by gcd(d, …, \binom{d}{n}). In particular this holds for the geometric generic point of H_{n}.

Proof. The subset H_{n}^{good} ⊂ H_{n} where M(K′, X_{K′}) ≥ d and gcd(d, …, \binom{d}{n}) | I(K′, X_{h,K′}) is a countable intersection of open subsets by standard Hilbert scheme arguments: the complement of this set is the union over the countably many Hilbert polynomials P (t) of multi-sections of degree < d or not divisible by gcd(d, …, \binom{d}{n}) of the closed image in H_{n} of the relative Hilbert scheme Hilb_{X /H_{n}}^{P (t)}. By Proposition 2.1 H_{n}^{good} is nonempty, therefore it is a countable intersection of open dense subsets. Of course the intersection of X_{h,K′} with a general line in ℙ(V ⊗_{k} K′) gives a multi-section of degree d, therefore H_{n}^{good} is actually the set where M(K′, X_{K′}) = d and gcd(d, …, \binom{d}{n}) | I(K′, X_{h,K′}). □

2.1. Proof of Proposition 1.4. Let k be an uncountable, algebraically closed field. The main case of Proposition 1.4 is B = ℙ_{k}^{1} × ℙ_{k}^{1} and M is the complete linear system |O(a, b)|. Assume first that one of a, b = 0, say b = 0. Let f : Y → ℙ_{k}^{1} be a finite, separably-generated morphism of irreducible curves of degree > 1, and let X = Y × ℙ^{1} with projection π = (f, Id). Every divisor in |O(a, 0)| is a union of fibers of pr_{1}, so the restriction of π has a section. The restriction of π over every fiber of pr_{2} is just f, and so has no rational section. Thus assume a, b > 0.

Define n = 4ab and d = n − 1. Let V be a k-vector space of dimension n + 1. Let C ⊂ ℙ^{1} × ℙ^{1} be a smooth curve in the linear system |O(1, 2b)|. By Corollary 2.2, there exists a closed immersion of degree n, h : C → ℙSym^{d}(V^{∨}), such that M(k(C), X_{h,k(C)}) = d > 1. Of course h extends to a closed immersion j : ℙ^{1} × ℙ^{1} → ℙSym^{d}(V^{∨}) such that j^{∗}O(1) = O(2a − 1, 2b); after all, H^{0}(ℙ^{1} ×ℙ^{1}, O(2a−1, 2b)) → H^{0}(C, O_{C}(n)) is surjective. Define π : X → ℙ^{1} ×ℙ^{1} to be the base-change by j of the universal family of degree d hypersurfaces in ℙ(V). By construction, the restriction over C has no section.

Every divisor in |O(a, b)| is a curve in ℙSym^{d}(V^{∨}) of degree n − b whose span is a linear system of hypersurfaces in ℙ(V) of (projective) dimension ≤ n − b − (a − 1)(b − 1). Since n − b < n, this linear system has basepoints giving sections of the restriction of X to the divisor. This proves Proposition 1.4 for B = ℙ^{1} × ℙ^{1} and M = |O(a, b)|.

Let B be a normal, projective variety of dimension ≥ 2 and let M be an irreducible family of irreducible curves dominating B. There exists a smooth open subset U ⊂ B whose complement has codimension ≥ 2 and a dominant morphism g : U → ℙ^{1} × ℙ^{1}. Intersecting U with general hyperplanes, there exists an irreducible closed subset Z ⊂ U such that g|_{Z} : Z → ℙ^{1} ×ℙ^{1} is generically finite of some degree e > 0. For the geometric generic point of M, the intersection of the corresponding curve with U is nonempty, and the closure of the image under f is a divisor in the linear system |O(a′, b′)| for some integers a′, b′. Let a ≥ a′, and b ≥ b′ be integers such that 4ab > e + 1. There exists a projective, dominant morphism π : X → ℙ^{1} × ℙ^{1} whose restriction over every divisor in |O(a, b)| has a section, but whose restriction over a general divisor in |O(1, 2b)| has minimal degree 4ab − 1.

Define X_{B} ⊂ B × X to be the closure of U ×_{ℙ^{1}×ℙ^{1}} X. Then π_{B} : X_{B} → B is a projective dominant morphism. For the geometric generic point of M, the restriction of π_{B} to the curve has a section because the restriction of π to the image in ℙ^{1} × ℙ^{1} has a section. Let C_{B} ⊂ Z be the preimage of a general curve C in |O(1, 2b)|. The morphism C_{B} → C has degree e < 4ab − 1. Because every multi-section of π over C has degree ≥ 4ab − 1, π_{B} has no section over C_{B}.

3. The construction for Enriques surfaces

Let k be a field of characteristic ≠ 2, 3, and let V_{+} and V_{−} be 3-dimensional kvector spaces. Denote V = V_{+} ⊕ V_{−} and V′ = Sym^{2}(V_{+}^{∨}) ⊕ Sym^{2}(V_{−}^{∨}). Denote G = Grass(3, V′), parametrizing 3-dimensional subspaces of V′. This is a parameter space for Enriques surfaces. There are 2 descriptions of the universal family, each useful. First, let π_{Z} : Z → ℙ(V_{+}) × ℙ(V_{−}) be the projective bundle of the locally free sheaf pr_{+}^{∗}O_{ℙ(V_{+})}(−2) ⊕ pr_{−}^{∗}O_{ℙ(V_{−})}(−2). A general complete intersection of 3 divisors in |O_{Z}(1)| is an Enriques surface. Because H^{0}(Z, O_{Z}(1)) = V′, the parameter space for these complete intersections is G. Second, G parametrizes complete intersections in ℙ(V) of 3 quadric divisors that are invariant under the involution ι of ℙ(V) whose (−1)-eigenspace is V_{−} and whose (+1)-eigenspace is V_{+}. A general such complete intersection is a K3 surface on which ι acts as a fixedpoint-free involution; the quotient by ι is an Enriques surface. The two descriptions are equivalent: the involution extends to an involution ι̃ on the blowing up { \atop ℙ(V)} of ℙ(V) along ℙ(V_{+}) ∪ ℙ(V_{−}) and the quotient is Z. Denote by X → G the universal family of Enriques surfaces, and denote by Y → G the universal family of K3 covers.

Let B, C, D be k-curves isomorphic to ℙ_{k}^{1}. There exists a degree 2, separablygenerated morphism g : D → C and a degree 3, separably-generated morphism f : C → B such that Gal(k(D)/k(B)) is the full wreath product 𝔚_{3,2}, i.e., the semidirect product (𝔖_{2})^{3} ⋊ 𝔖_{3}. In characteristic 0, this holds whenever g and f have simple branching and the branch points of g are in distinct, reduced fibers of f. There is an involution ι_{D} of D commuting with g.

Let j : D → ℙ(V^{∨}) be a closed immersion equivariant for ι_{D} and ι whose image is a rational normal curve of degree 5. By the construction in Section 2, there is an associated morphism i : C → ℙSym^{2}(V^{∨}). Because j is equivariant, i factors through ℙ(V′). By a straightforward computation, i^{∗}O(1) = Nm_{g}(j^{∗}O(1)) ≅ O_{C}(5). The pushforward by f_{∗} of the pullback by i^{∗} of the tautological surjection is a surjection (V′)^{∨} ⊗ O_{B} → f_{∗}i^{∗}O(1). The sheaf f_{∗}i^{∗}O(1) is locally free, in fact f_{∗}i^{∗}O(1) ≅ f_{∗}O_{C}(5) ≅ O_{B}(1)^{3}, so there is an induced morphism h : B → G. Denote by π_{h} : X_{h} → B and ρ_{h} : Y_{h} → B the base-change by h of X and Y. Denote K = k(B) and denote by X_{h,K} the generic fiber of π_{h}.

Proposition 3.1. Every irreducible multi-section of π_{h} has degree divisible by 3 or 4. In particular M(K, X_{h,K}) = 3.

Proof. Denote by U ⊂ B the open set over which f ∘ g is étale, and denote by W ⊂ D the preimage of U. Denote by c : W̃ → U the Galois closure of W/U. Then c^{∗}f_{∗}O_{C}|_{U} ≅ OW̃{a_{1}, a_{2}, a_{3}} for idempotents a_{p}, p = 1, 2, 3. And c^{∗}g_{∗}f_{∗}O_{D}|_{U} ≅ OW̃{b_{1,1}, b_{1,2}, b_{2,1}, b_{2,2}, b_{3,1}, b_{3,2}} for idempotents b_{p,q}, p = 1, 2, 3, q = 1, 2. Of course a_{p} ↦ b_{p,1} + b_{p,2}, p = 1, 2, 3. The action of the Galois group 𝔚_{3,2} on a_{p} is by the symmetric group 𝔖_{3}, and the action on b_{p,q} is the standard representation of the wreath product.

For each p = 1, 2, 3 and q = 1, 2, denote by j_{p,q} : W̃ → ℙ(V^{∨}) the morphism obtained by composing the idempotent b_{p,q} : W̃ → W̃ ×_{U} W with the basechange of j. In particular, ι ∘ j_{p,1} = j_{p,2}. Denote by Λ_{p,q} ⊂ W̃ × ℙ(V) the pullback by (j_{p,q}, Id) of the universal hyperplane. Denote by YW̃ the base-change to W̃ of Y_{h}. Then,

YW̃ = ⋃_{(q_{1},q_{2},q_{3})∈{1,2}^{3}}(Λ_{1,q_{1}} ∩ Λ_{2,q_{2}} ∩ Λ_{3,q_{3}}).

There is a locally closed stratification,

YW̃ = Y3 ; W̃ ⊔ Y4 ; W̃ ⊔ Y5 ; W̃,

where Yl ; W̃ is the set of points lying in the intersection of precisely l of the Λ_{p,q}. The stratum Y3 ; W̃ is the union of 8 connected, open subsets,

Λ_{(q_{1},q_{2},q_{3})} ⊂ (Λ_{1,q_{1}} ∩ Λ_{2,q_{2}} ∩ Λ_{3,q_{3}}),

for q_{1}, q_{2}, q_{3} ∈ {1, 2}. Each connected component is a dense open subset of a ℙ^{2}- bundle over W̃. The stratum Y4 ; W̃ is the union of 12 connected, open subsets,

Λ_{(∗,q_{2},q_{3})} ⊂ (Λ_{1,1} ∩ Λ_{1,2}) ∩ Λ_{2,q_{2}} ∩ Λ_{3,q_{3}},
Λ_{(q_{1},∗,q_{3})} ⊂ Λ_{1,q_{1}} ∩ (Λ_{2,1} ∩ Λ_{2,2}) ∩ Λ_{3,q_{3}},
Λ_{(q_{1},q_{2},∗)} ⊂ Λ_{1,q_{1}} ∩ Λ_{2,q_{2}} ∩ (Λ_{3,1} ∩ Λ_{3,2})

for q_{1}, q_{2}, q_{3} ∈ {1, 2}. Each connected component is a dense open subset of a ℙ^{1}- bundle over W̃. Finally Y5 ; W̃ is the union of 6 connected sets,

Λ_{(∗,∗,q_{3})} = (Λ_{1,1} ∩ Λ_{1,2}) ∩ (Λ_{2,1} ∩ Λ_{2,2}) ∩ Λ_{3,q_{3}},
Λ_{(∗,q_{2},∗)} = (Λ_{1,1} ∩ Λ_{1,2}) ∩ Λ_{2,q_{2}} ∩ (Λ_{3,1} ∩ Λ_{3,2}),
Λ_{(q_{1},∗,∗)} = Λ_{1,q_{1}} ∩ (Λ_{2,1} ∩ Λ_{2,2}) ∩ (Λ_{3,1} ∩ Λ_{3,2})

for q_{1}, q_{2}, q_{3} ∈ {1, 2}. Each connected component projects isomorphically to W̃.

There is a bijection between multi-sections of Y_{h} over U and Galois invariant multisections of YW̃ over W̃. An irreducible multi-section of Y_{h} determines a multisection of YW̃ contained in a single stratum Yl ; W̃. The action of the Galois group 𝔚_{3,2} on the connected components of Yl ; W̃ is the obvious one; in particular, it acts transitively on the set of connected components. So every Galois invariant multisection in Y3 ; W̃ has degree divisible by 8, every Galois invariant multi-section in Y4 ; W̃ has degree divisible by 12, and every Galois invariant multi-section in Y5 ; W̃ has degree divisible by 6. Therefore every irreducible multi-section of Y_{h} has degree divisible by 8 or 6. Because Y_{h} is a double-cover of X_{h}, every irreducible multisection of X_{h} has degree divisible by 4 or 3. In particular, the minimal degree of a multi-section of X_{h} is 3. □

Because f_{∗}i^{∗}O(1) ≅ O_{B}(1)^{3}, the scheme X_{h} ⊂ B ×Z is a complete intersection of 3 divisors in the linear system |pr_{B}^{∗}O_{B}(1)⊗pr_{Z}^{∗}O_{Z}(1)|. A general deformation of this complete intersection is a pencil of Enriques surfaces satisfying Theorem 1.3 (i) and (ii) with M(K, X_{K}) ≥ 3, I(K, X_{K}) | 4 (this is valid so long as char(k) ≠ 2). For (iii), it is necessary to deform the pencil together with the degree 3 multi-section. This requires a bit more work, and the hypothesis char(k) ≠ 2, 3.

The stratum Y5 ; W̃ is Galois invariant and determines a degree 3 multi-section of

X_{h}. As a 𝔚_{3,2}-equivariant morphism to W̃, Y5 ; W̃ is just the base-change of D, and the morphism Y5 ; W̃ → ℙ(V) is Galois invariant. By étale descent it is the base-

change of a morphism j′ : D → ℙ(V). Now j′ induces a morphism to { \atop ℙ(V)}, the blowing up of ℙ(V) along ℙ(V_{+})∪ℙ(V_{−}). Because j′ is equivariant for ι and ι_{D}, the quotient morphism D → Z factors through C, i.e., there is an induced morphism i′ : C → Z. By a straightforward enumerative geometry computation, j′ has degree 5 with respect to O_{ℙ(V)}(1). Therefore i′ has degree 5 with respect to O_{Z}(1). The degree 3 multi-section of X_{h} is the image of (f, i′) : C → B × Z.

Lemma 3.2. If f, g and j are general, then (i′)^{∗} : H^{0}(Z, O_{Z}(1)) → H^{0}(C, O_{C}(5)) is surjective.

Proof. The condition that (i′)^{∗} is surjective is an open condition in families, hence it suffices to verify (i′)^{∗} is surjective for a single choice of f, g and j – even one for which Gal(k(D)/k(B)) is not 𝔚_{3,2}. Choose homogeneous coordinates [S_{0}, S_{1}] on D, [T_{0}, T_{1}] on C and [U_{0}, U_{1}] on B. Define g([S_{0}, S_{1}]) = [S_{0}^{2}, S_{1}^{2}] and f([T_{0}, T_{1}]) = [T_{0}^{3}, T_{1}^{3}]. Denote by μ_{6} the group scheme of 6^{th} roots of unity. There is an action of μ_{6} on D by ζ · [S_{0}, S_{1}] = [S_{0}, ζS_{1}]. This identifies μ_{6} with Gal(k(D)/k(B)).

Let e_{+,0}, e_{+,1}, e_{+,2} and e_{−,0}, e_{−,1}, e_{−,2} be ordered bases of V_{+} and V_{−} respectively, and let X_{+,0}, X_{+,1}, X_{+,2} and X_{−,0}, X_{−,1}, X_{−,2} be the dual ordered bases of V_{+}^{∨} and V_{−}^{∨} respectively. There is an action of μ_{6} on V by,

\begin{matrix}ζ · [X_{+,0}, X_{+,1}, X_{+,2}, X_{−,0}, X_{−,1}, X_{−,2}] = \\ [X_{+,0}, ζ^{2}X_{+,1}, ζ^{4}X_{+,2}, ζX_{−,0}, ζ^{3}X_{−,1}, ζ^{5}X_{−,2}] \end{matrix}

and a dual action on V^{∨}. Define j : D → ℙ(V) with respect to the ordered basis e_{+,0}, …, e_{−,2}, to be the μ_{6}-equivariant morphism,

j([S_{0}, S_{1}]) = [S_{0}^{5}, S_{0}^{3}S_{1}^{2}, S_{0}S_{1}^{3}, S_{0}^{4}S_{1}, S_{0}^{2}S_{1}^{3}, S_{1}^{5}].

In this case U = D_{+}(U_{0}U_{1}) ⊂ B and W̃ = W = D_{+}(S_{0}S_{1}) ⊂ C. It is straightforward to compute j′ with respect to the dual ordered basis X_{+,0}, …, X_{−,2},

j′([S_{0}, S_{1}]) = [S_{1}^{5}, S_{0}^{2}S_{1}^{3}, S_{0}^{4}S_{1}, S_{0}S_{1}^{4}, S_{0}S_{1}^{4}, S_{0}^{3}S_{1}^{2}, S_{0}^{5}].

As a double-check, observe this is μ_{6}-equivariant. The induced map (j′)^{∗} is,

\begin{matrix}X_{+,0}X_{+,0} & ↦ & T_{1}^{5}, & X_{+,0}X_{+,1} & ↦ & T_{0}T_{1}^{4}, & X_{+,0}X_{+,2} & ↦ & T_{0}^{2}T_{1}^{3}, \\ X_{+,1}X_{+,1} & ↦ & T_{0}^{2}T_{1}^{3}, & X_{+,1}X_{+,2} & ↦ & T_{0}^{3}T_{1}^{2}, & X_{+,2}X_{+,2} & ↦ & T_{0}^{4}T_{1}, \\ X_{−,0}X_{−,0} & ↦ & T_{0}T_{1}^{4}, & X_{−,0}X_{−,1} & ↦ & T_{0}^{2}T_{1}^{3}, & X_{−,0}X_{−,2} & ↦ & T_{0}^{3}T_{1}^{2}, \\ X_{−,1}X_{−,1} & ↦ & T_{0}^{3}T_{1}^{2}, & X_{−,1}X_{−,2} & ↦ & T_{0}^{4}T_{1}, & X_{−,2}X_{−,2} & ↦ & T_{0}^{5}. \end{matrix}

This is surjective by inspection. □

3.1. Proof of Theorem 1.3. The subvariety X_{h} ⊂ B×Z is a complete intersection of 3 divisors in the linear system |pr_{B}^{∗}O_{B}(1)⊗pr_{Z}^{∗}O_{Z}(1)|, each containing (f, i′)(C). Denote by I the ideal sheaf of (f, i′)(C) ⊂ B × Z, and denote I = H^{0}(B × Z, I ⊗ pr_{B}^{∗}O_{B}(1) ⊗ pr_{Z}^{∗}O_{Z}(1)). The projective space of I is the linear system of divisors on B × Z in the linear system |pr_{B}^{∗}O_{B}(1) ⊗ pr_{Z}^{∗}O_{Z}(1)| that contain (f, i′)(C). The Grassmannian G′ = Grass(3, I) is the parameter space for deformations of X_{h} that contain (f, i′)(C). For the same reason as in Corollary 2.2, in G′ there is a countable intersection of dense open subsets parametrizing subvarieties X′ ⊂ B ×Z with M(K, X′_{K}) ≥ 3 and I(K, X′_{K}) | 4. By construction, X′ contains the degree 3 multi-section (f, i′)(C). Therefore M(K, X′_{K}) = 3 and I(K, X′_{K}) = 1. It is straightforward to compute pr_{B} ∗ [ω_{X′/B}^{⊗2}] ≅ O_{B}(6). So to prove the theorem, it suffices to prove every “very general” Enriques surface occurs as a fiber of some X′, i.e., for a general [X] ∈ G, X occurs as pr_{Z}(X′ ∩ π_{B}^{−1}(b)) for some choice of f, g, i and b ∈ B.

A general 0-dimensional, length 3 subscheme of Z occurs as i′(f^{−1}(b)) for some choice of f, g, i and b ∈ B. So for a general Enriques surface [X] ∈ G and a general choice of 0-dimensional, length 3 subscheme of X, X is a complete intersection of 3 divisors in the linear system |O_{Z}(1)| containing i′(f^{−1}(b)) for some choice of f, g, i and b. To prove that a general [X] ∈ G is the fiber over b of X′ for some f, g, i and [X′] ∈ G′, it suffices to prove every divisor in the linear system |O_{Z}(1)| containing i′(f^{−1}(b)) is the fiber over b of a divisor in the linear system |I ⊗ O_{B}(1) ⊗ O_{Z}(1)|.

There is a short exact sequence,

0 → I ⊗ pr_{Z}^{∗}O_{Z}(1) → pr_{Z}^{∗}O_{Z}(1) → pr_{Z}^{∗}O_{Z}(1)|_{C} → 0,

giving a short exact sequence,

0 → pr_{B,∗}(I ⊗ pr_{Z}^{∗}O_{Z}(1)) → pr_{B,∗}pr_{Z}^{∗}O_{Z}(1) → pr_{B,∗}(pr_{Z}^{∗}O_{Z}(1)|_{C}) → 0.

Because (i′)^{∗} is surjective, pr_{B,∗}(I ⊗ pr_{Z}^{∗}O_{Z}(1)) is a locally free sheaf with h^{1} = 0. So it is ≅ O_{B}^{6} ⊕ O_{B}(−1)^{3}. Twisting by O_{B}(1), pr_{B,∗}(I ⊗ pr_{B}^{∗}O_{B}(1) ⊗ pr_{Z}^{∗}O_{Z}(1)) is generated by global sections. Therefore every divisor on Z in the linear system |O_{Z}(1)| containing the scheme i′(f^{−1}(b)) is the fiber over b of a divisor on B × Z in the linear system |I ⊗ pr_{B}^{∗}O_{B}(1) ⊗ pr_{Z}^{∗}O_{Z}(1)|.

References

[CS01] Pierre Colmez and Jean-Pierre Serre, editors. Correspondance Grothendieck-Serre. Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 2. Société Mathématique de France, Paris, 2001.

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

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

[Laf04] Guillaume Lafon. Une surface d’Enriques sans point sur ℂ((t)). C. R. Math. Acad. Sci. Paris, 338(1):51–54, 2004.

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

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