Asymptotic period relations for Jacobian elliptic surfaces

We find an asymptotic description of the period locus of simply connected Jacobian elliptic surfaces and of the period locus of hyperelliptic curves. The two descriptions are essentially the same, and are given by the alkanes of organic chemistry.


Introduction
The classical Schottky problem is that of describing the period locus J g of period matrices of complex algebraic curves of genus g as a subvariety of Siegel space H g .
This problem naturally extends to higher dimensions; for example, a simply connected algebraic surface of positive geometric genus g has a period matrix, and then the Schottky problem becomes that of describing the image of the moduli space under the multi-valued period map. This image we refer to as the period locus. As described below, we are only concerned here with local aspects of the geometry of the situation, for which the fact that the period map is multi-valued is irrelevant.
We consider this problem for simply connected Jacobian elliptic surfaces of geometric genus g. In the classification of surfaces these are the simplest beyond those which are covered by K3 or abelian surfaces (and for such surfaces the Schottky problem is well understood). Of course, beyond these lie the surfaces of general type, the well known complexity of whose moduli spaces suggests that it is reasonable to focus on some particular classes of surfaces such as the ones considered here.
The main result (summarized in Theorem 1.1 below) is that the picture for these elliptic surfaces is analogous to that for hyperelliptic curves and that the period loci P L g of elliptic surfaces and Hyp g of hyperelliptic curves are described by the alkanes of elementary organic chemistry. These are the acyclic saturated hydrocarbons and their molecular formula is C g H 2g+2 . They were first enumerated by Cayley, whose calculation is given (with some corrections) in [OEIS]. The connexion with algebraic geometry is that on a hyperelliptic curve of genus g the hyperelliptic involution has 2g + 2 fixed points and that when such curves degenerate to trees of elliptic curves then each elliptic curve E plays the rôle of a carbon atom C whose bonds correspond to the 2-torsion points of E.
In fact, there is a subdomain W 1 g of the period domain for elliptic surfaces which corresponds to trees of g special Kummer surfaces and which is isomorphic to H g+1 1 . We regard this as the analogue of the locus Diag g of diagonal matrices in H g , which corresponds to trees of elliptic curves.
Theorem 1.1 (1) There are one-to-one correspondences between the alkanes C g H 2g+2 , the branches through Diag g of Hyp g and the branches through W 1 g of P L g .
(2) Each of these branches has, to first order, a straightforward and explicit linear description in terms of matrices.
I am grateful to Paolo Cascini, Bob Friedman, Mark Gross and Dave Morrison for some valuable conversation and correspondence and to Hershel Farkas, Sam Grushevsky and Riccardo Salvati Manni for their encouragement.

Some further details
We begin by recalling some constructions that involve algebraic curves. Later we shall extend these constructions to include algebraic surfaces.
Fay [F1] constructed certain degenerating families of complex algebraic curves (that is, compact Riemann surfaces) via means of explicit plumbing constructions that are recalled below and then derived formulae for the derivative of the period matrix of each of these families. Akira Yamada [Y] then pointed out that Fay's formulae are wrong, and gave correct formulae for Fay's constructions.
In [F2] (bottom of page 123), Fay corrects his error by pointing out that his plumbing constructions should have been done differently, and that for these different plumbing constructions his formulae are correct.
In other words, Fay in [F1] in fact made two different plumbing constructions of 1-parameter degenerating families of curves without monodromy (so the curve degenerates but its Jacobian does not) for which there are explicit formulae for the derivative of the period matrix. For one construction the correct formula is that given in [F1] and for the other the correct formula is given in [Y]. (There are also two different plumbing constructions of families of curves with monodromy but we shall not use such constructions in this paper.) We shall refer to the plumbing constructions for which Yamada's formula [Y] is correct as Yamada plumbings, and those for which Fay's formula [F1] is correct as Fay plumbings.
We shall recall the detailed construction of Fay plumbings below. It is convenient to point out here that in our formulae there is a minus sign that does not appear in [F1]; this is because we have chosen a different normalization which slightly increases the flexibility of the construction.
In fact, and this is the crux of this paper, we do this in higher dimensions; one advantage of Fay plumbings is that they can be generalized to plumb not only curves but also, at least in certain circumstances, morphisms from curves to stacks.
We next recover Fay's version [F1] of Poincaré's "asymptotic period relations". These were discovered by Poincaré [P] when g = 4 and generalized by Fay to all values of g. According to Igusa's account (see p. 167 of [I]) Poincaré exploited the geometry of the theta divisor (specifically, that it is a hypersurface of translation type), so his argument cannot extend to the case of surfaces, while Fay uses his plumbing construction. These relations describe, both intrinsically and in terms of co-ordinates, the tangent cone to the closure J c g of the Jacobian locus along the locus Diag g of diagonal matrices in Siegel space H g in terms of the Grassmannian Grass(2, g).
These Poincaré-Fay asymptotic period relations have also been recovered by Farkas, Grushevsky and Salvati Manni [FGSM] in the course of proving their global weak solution to the Schottky problem. More precisely, they show that differentiating the identities obtained by substituting the Schottky-Jung proportionalities into Riemann's quartic theta identities leads to the Poincaré-Fay relations; they deduce their global weak solution as an immediate consequence.
Then we describe, to first order along Diag g , the closure Hyp c g of the hyperelliptic locus in H g . This description is given in terms of the alkanes of elementary organic chemistry, which were first enumerated by Cayley; see sequence A000602 in [OEIS] for corrections. These are the hydrocarbons whose molecular formula is of the type C g H 2g+2 (and so are exactly the saturated acyclic hydrocarbons) and have uses ranging from fuel to furniture polish, depending on their molecular weight. The g-alkane is the one whose carbon skeleton is a chain of length g. It is distinguished from the others of the same molecular formula by having a higher boiling point. We shall refer to g as the genus of the alkane. The number 2g + 2 is the number of fixed points of the hyperelliptic involution on a hyperelliptic curve of genus g. It turns out that Hyp c g has one branch along Diag g for each alkane, each branch is smooth and we give explicit first-order equations in terms of the entries τ ij of the period matrix. Here are more precise statements. Recall that a square matrix (a ij ) is tridiagonal if a ij = 0 whenever |i − j| ≥ 2.
Theorem 2.1 (a special case of Theorem 9.6) To first order (that is, modulo the square of the defining ideal) the branch of Hyp c g in H g that corresponds to the g-alkane is the locus of tridiagonal matrices.
There is a similar, although slightly more complicated, description of the branches corresponding to the other alkanes. Now turn to surfaces. We use Fay's plumbing to make similar constructions and calculations for degenerating families of simply connected Jacobian elliptic surfaces, which can be regarded as the simplest surfaces of strictly positive Kodaira dimension and also as the simplest surfaces for which, thanks mainly to our understanding of the period map for K3 surfaces, the Schottky problem is not vacuous.
Fix an integer h ≥ 2 and consider Jacobian elliptic surfaces X over P 1 of geometric genus h. These have 10h + 8 moduli, and the coarse moduli space is rational. The primitive (co)homology H = H prim of X is the orthogonal complement of the section and a fibre. We can describe a chart of the period domain Pick a totally isotropic sublattice L of H whose rank is h.
first to a basis of L ⊥ and then to a basis of H such that the induced basis of H/L ⊥ is dual to (A 1 , ..., A h ) and H is decomposed as Then the normalized period matrix of X is, when we ignore the h × h identity matrix that arises from integrating around the cycles A i , an h × (11h + 8) matrix whose final h × h block is skew-symmetric. So, if h = 1, this last block is zero and can be ignored to yield a vector of length 18. Different isotropic lattices L will give different charts.
Let P L h , the period locus, denote the image of the moduli stack in the period domain V h under the period map. We shall recall the definition of the domain of the period map later; the presence of RDPs creates a slight subtlety. Definition 2.3 A Jacobian elliptic surface is special if it is birational to a geometric quotient [C × E/ι] where C is a hyperelliptic curve, E is an elliptic curve and ι acts on C as its hyperelliptic involution and on E as (−1 E ).
Special surfaces X are characterized as the simply connected Jacobian elliptic surfaces whose global monodromy on the cohomology of the elliptic fibration is of order 2. We have p g (X) = g(C).
Here are some trivial remarks and some notation.
(1) If h = h i then the period domain V h contains a copy of V h 1 × · · · × V hr .
(2) Each V h i contains a copy P L h i ,special of the period locus of special elliptic surfaces of genus h i . P L h i ,special is isomorphic to Hyp h i × H 1 .
(3) If j ∈ [0, 4] then K j is the period domain for Jacobian elliptic K3 surfaces with j fibres of type D 4 . So K 4 = P L 1,special .
(4) Fix an alkane Γ and let γ j , for j ∈ [1, 4], denote the number of vertices (carbon atoms) in Γ that are joined to j other carbon atoms in Γ. Then there is a closed subvariety V Γ of the product K γ 1 1 × · · · × K γ 4 4 that is defined by the requirement that the D 4 -fibres on K3s that are adjacent in Γ should be isomorphic. Induction shows that dim V Γ = 9h + 9.
(5) In P L h 1 , special × · · · × P L hr, special , which is isomorphic to Hyp h i × H r 1 , there is a subvariety W h 1 ,...,hr defined by the property that the factors in each copy of H 1 are equal. This is the period locus for unions of special elliptic surfaces of genera h 1 , ..., h r where the relevant elliptic curves are isomorphic.
(6) For each Γ, W 1 h is the subvariety of V Γ defined by the condition that each K3 surface should be special.
We regard W 1 h , which is isomorphic to (H 1 ) h+1 , as the analogue in the period domain V h of the diagonal locus Diag g in H g .
The next result is stated in terms of a certain vector bundle E Γ → V Γ of rank h − 1. The fibre of E Γ over a point (Y 1 , ..., Y h ) of V Γ is the vector space spanned by certain matrices Π e of rank 1, where e = (i, j) runs over the edges (the carbon-carbon bonds) of Γ. Each Π e is a tensor product Π e = ω e ⊗ I e of two vectors, where each vector is computed from the surfaces Y i and Y j . The vector ω e is comprised of projective data while I e is a vector of integrals, so consists of transcendental data. These vectors are described explicitly in Proposition 6.20.
(1) There is a branch B Γ of P L h that contains V Γ .
(2) To first order, B Γ is, in a neighbourhood of W 1 h , the vector bundle E Γ .
(3) The zero section of E Γ is V Γ embedded in B Γ . This leads to the main result. It is an analogue of Theorem 9.6.
Theorem 2.5 (= Theorem 11.13) (1) The branch B Γ above is the unique branch that contains V Γ .
(2) In a neighbourhood of W 1 h , the period locus P L h is, to first order, the union of the vector bundles E Γ .

The domain of the period map for Jacobian elliptic surfaces
We shall refer to a given Jacobian elliptic surface f : X → C with section C 0 as smooth if it is smooth and also relatively minimal and as an RDP surface if it has only RDPs, C 0 lies in the smooth locus of f and C 0 is f -ample. That is, if every geometric fibre of f is a reduced and irreducible curve of arithmetic genus 1. They are objects of two of the stacks that we shall consider: (1) J E smooth (resp., J E smooth h ) is the stack of smooth Jacobian elliptic surfaces (resp., simply connected such surfaces of geometric genus h); (2) J E RDP (resp., J E RDP h ) is the stack of RDP Jacobian elliptic surfaces (resp., simply connected such surfaces of geometric genus h).
So J E RDP is separated and there is a natural morphism J E smooth → J E RDP given by passing to the relative canonical model. According to Artin's results [Ar74], which we now recall, this morphism is representable, 1-to-1 on field-valued points but not separated.
Suppose that (X → C → S, C 0 ) is an object of J E RDP over S. Then we have Artin's functor Res X/S , whose T -points, for an S-scheme T , are isomorphism classes of diagrams where F is smooth, projective and relatively minimal (this is equivalent to saying that X T → C T is an object of J E smooth ), and π is projective and birational, in the sense that π * O = O. Then Res X/S is represented by a locally quasi-separated algebraic space R over S, such that R → S is a bijection on all field-valued points. The morphism J E smooth → J E RDP described above can be localized to yield an isomorphism J E smooth × J E RDP S ∼ = →R. Now restrict attention to simply connected Jacobian elliptic surfaces of geometric genus p g = h and denote the corresponding stacks by J E smooth h and J E RDP h . Fix a unimodular lattice Λ = Λ h of rank 12h + 10 and signature (2h + 1, 10h + 9) and elements σ, φ ∈ Λ such that σ 2 = −(h + 1), σ.φ = 1 and φ 2 = 0. So Λ = Z{σ, φ} ⊥ H where H = H h is unimodular, its rank is 12h + 8 and its signature is (2h, 10h + 8). Assume also that H is even; these requirements specify Λ h and H h uniquely. Consider the subgroup G of the orthogonal group O Λ (Z) given by and an isometry Ψ : Λ T → R 2 F * Z such that Ψ maps σ to the class of the section (C 0 ) T of X T → C T and φ to the class of a fibre.
The discussion in the third paragraph on p. 228 of [C1] can be translated into the language of stacks to say that J E Λ is the domain of the period map. That is, the period map is a G-equivariant holomorphic morphism where V h is the period domain. Equivalently, the period map is the quotient morphism per : of quotient stacks. This fits into a commutative diagram If S is the henselization of J E RDP h at a closed point, so that S is the base of a miniversal deformation of an RDP Jacobian elliptic surface X s , then [Ar74] the henselization R hens of R = J E smooth h × J E RDP h S at its unique closed point is the base of a miniversal deformation of the minimal resolution X s of X s . (In fact Artin shows that R hens is the base of a versal deformation, but in characteristic zero his argument proves the miniversality of R hens .) So, for local purposes such as those of this paper, we can take the domain of the period map to be a miniversal deformation space of a smooth Jacobian elliptic surface.
In fact, slightly more is true. The finite Weyl group W associated to the configuration of singularities on X s acts on R hens in such a way that [R hens /W ] = S and there is a commutative diagram This is because there is a non-separated union R P = ∪ C R hens C of copies of R hens on which W acts freely, where there is one copy R hens C of R hens for each chamber C that defined in the usual way and two charts R hens C , R hens E in R P are glued in a way that depends upon the relative position of the chambers C, E with respect to W . (This glueing always induces an isomorphism over the complement of the discriminant locus in S.) Moreover W permutes the charts R hens C simply transitively, and R P /W = J E h × J E h S = R. The isomorphisms R hens C → R hens glue to a morphism α : R P → R hens that exhibits R hens as the maximal separated quotient of R P in the category of schemes. So W also acts on R hens , α is Wequivariant and S = [R hens /W ]. All this is documented in [SB17], to which we add one comment: since, in the commutative diagram the vertical maps are quotients by W , so that the square is 2-Cartesian, it follows that the scheme S is the maximal separated quotient of the algebraic space R in the category of algebraic spaces and that the stack R hens /W is the maximal separated quotient of R in the 2-category of algebraic stacks. Finally, consider the forgetful morphism α : S → x Def Xs,x from S to the product of the miniversal deformation spaces of the germs (X s , x) of X s at its singularities x and the morphisms Def Xs,x → Def Xs,x . These latter are geometric quotients by the relevant finite Weyl group and R hens is isomorphic to the fibre product S × so that if α is smooth then so is R hens .

Fay's plumbing for curves and stacky curves
In this section we give a detailed description of Fay plumbings. All of this is taken from [F1] and [F2] but we repeat it here to avoid confusion. Fix a real number δ > 0. Let ∆ = ∆ t be the open disc in the complex plane C with co-ordinate t defined by |t| < δ 2 and let F be the open submanifold of C 2 with co-ordinates q, v defined by |q| < δ 1/2 , |v| < δ and |v ± q| < δ (so that, in particular, |v 2 − q 2 | < δ 2 ). Set t = v 2 − q 2 . Note that the morphism t : F → ∆ is smooth outside the origin in ∆ and the fibre t −1 (0) consists of two discs U ′ a and U ′ b , where U ′ a is given by q − v = 0 and U ′ b by q + v = 0. These discs cross normally at the point 0 given by q = v = 0.
We shall use F , with these co-ordinates, as the basic plumbing fixture. Now suppose that C a , C b are curves of genus g a , g b , that a ∈ C a , b ∈ C b and that if i = a, b then we have chosen a local co-ordinate z i on some simply connected neighbourhood U i of i in C i such that U i is defined by the inequality |z i | < δ 1/2 . In particular, U i is an open disc that is embedded in C i .
The suffix i will stand for either a or b in what follows. Let U c i denote the closure of U i in C i . We shall assume that each U c i is a (closed) disc; this is slightly stronger than assuming U i is an open disc. We shall assume also either that C a and C b are distinct or that C a = C b and the closed discs U c a and U c b are disjoint. Of course, all of these assumptions can be fulfilled after decreasing δ if necessary.
In F we have open subsets V a and V b defined, respectively, by the inequalities |v − q| < |q|, |t| < δ|q| 2 and (4.1) |v + q| < |q|, |t| < δ|q| 2 . (4.2) Note that V a and V b are disjoint and that It is easy to check that π i is unramified (since the ramification locus is defined by v = 0) and is injective.
Proposition 4.3 We can glue the three charts W a , W b and F via the identifications V i ∼ = → Y i to get a separated 2-dimensional complex manifold C with a proper holomorphic map π : C → ∆ t such that π −1 (0) = C a ∪ a∼b C b , where C a and C b cross normally at a single point.
PROOF: This is routine.
We call this a Fay plumbing because it is, implicitly, constructed and considered by Fay in the final paragraph of p. 37 of [F1].
If C a and C b are distinct then for all t = 0 the fibre C t is a curve of genus g = g a + g b and there is no monodromy on its cohomology. This is what Fay calls "pinching a cycle homologous to zero" [F1], p. 37 et seq. We shall call it a homologically trivial Fay plumbing of C a to C b that identifies a with b or just a Fay plumbing of C a to C b without explicit mention of the points or co-ordinates that are chosen.
If C a = C b , of genus g, and also U c a ∩U c b = ∅ (so that the plumbing is possible), then for all t = 0 the fibre C t is of genus g + 1, C 0 is the nodal curve C/a ∼ b and there is non-trivial monodromy on the cohomology of C t . Fay calls this "pinching a non-zero homology cycle" [F1], p. 50.
Note that on F ∩ W i we have z i = q. In a homologically trivial Fay plumbing the derivative at t = 0 of the period matrix of C t is given as Corollary 3.2 on p. 41 of [F1]. We shall prove this, or, rather, a version of it in a higher-dimensional context, later on.
When a non-zero homology cycle is pinched in a Fay plumbing the period matrix is described by by Corollary 3.8 on p. 53 of [F1].
However, the families that are written down on pp. 37 and 50 of loc. cit., which we call the Yamada plumbing because they are considered explicitly by Yamada, are given by the glueing t = z a z b and (provided that g a + g b ≥ 1) are different families because, for example, their period matrices are different. Their expansions at t = 0 are given as Corollary 2 on p. 129 and Corollary 6 on pp. 137-138 of [Y].
In this paper we shan't have any need to pinch cycles that are not homologous to zero. For one application of such pinchings, however, see [CSB13]. Note that there Yamada plumbings are used, although an argument could also be based on the use of Fay plumbings.
It will also be useful to be able to plumb certain stacky curves.
Definition 4.4 A Z/2-curve is a reduced connected proper 1-dimensional Deligne-Mumford stack C such that at all geometric points C has at worst nodes, at every generic point the stabilizer (the isotropy group) is trivial, at every point the stabilizer is a subgroup of Z/2 and that if P is a node with non-trivial stabilizer then the stabilizer preserves each branch of the node.
there is a unique geometric point on C i lying over i. There are local co-ordinatesz i and z i on C i and on C i , respectively, such that the non-trivial element ι of Z/2 acts via ι * z i = −z i and z i =z 2 i . Take the plumbing fixture F , with co-ordinates q, v as before, and let ι act on F by ι * q = −q, ι * v = −v. So the quotient stack F = F/ ι is a smooth separated 2-dimensional Deligne-Mumford stack.
Suppose C i , for i = a, b, are Z/2-curves, that i is a smooth point of C i for each i and that each of a, b the stabilizer is ι ∼ = Z/2. Fix a local co-ordinatez i on C i at i such that ι * z i = −z i . Proposition 4.5 There are suitable open substacks W i of C i × ∆ such that the charts W a , W b and F can be glued via the formulaez i = q and t = q 2 − v 2 to get a smooth 2-dimensional Deligne-Mumford stack C with a proper separated morphism C → ∆ t .
PROOF: As before.
For future reference, put Taking geometric quotients gives the following result.
Proposition 4.6 The curves C i can be plumbed via the plumbing fixture G to PROOF: This is clear.
The minimal resolution of B gives a semi-stable family of curves over ∆ t .

Plumbing families of curves
The plumbing construction of the previous section can be extended so as to plumb families of curves.
So suppose that C a → S a , C b → S b are analytic families of semi-stable curves of genera g a and g b , respectively, with sections σ i : S i → C i each of which lies in the relevant smooth locus. Assume also that U i is a tubular neighbourhood of the section i = σ i (S i ) and that z i is a fibre co-ordinate on Such a neighbourhood and such a co-ordinate will exist if each S i is sufficiently small, for example a small polydisc.
Take the same 2-dimensional plumbing fixture F and disjoint open subsets V i as before and consider the morphisms As before, π i is an isomorphism to an open analytic subvariety In C i ×∆ t consider the closed subset that is the intersection of U c i ×∆ t and the complement of Y i , and define W i to be the open subvariety of C i × ∆ t obtained by deleting this closed subset.
The result of this glueing is a analytic space C with a proper morphism C → S a × S b × ∆ t that is a family of semi-stable curves of genus We can concatenate plumbings as follows: suppose that C i → S i are families of stable curves for i = a, b, c and that σ a , σ b 1 , σ b 2 , σ c are sections of C a , C b , C b , C c respectively and that σ b 1 and σ b 2 are disjoint.
Then there are two choices: we can first plumb C a to C b , obtaining C ′ → S a × S b × ∆, in a way that identifies the sections σ a and σ b 1 , and then plumb C ′ to C c to get a family over S a × S b × ∆ × S c × ∆, or we can first plumb C b to C c , obtaining C ′′ , by identifying σ b 2 with σ c , and then plumb C ′′ to C a to get another family over S a × S b × ∆ × S c × ∆. It will be important for us to notice that these two families are the same; that is, the final result is independent of the order of the plumbings.
Similarly we can construct stacks C i → S i by introducing isotropy groups Z/2 along each section σ(S i ) and then plumb together the stacks C i .

Homologically trivial plumbings of surfaces
Here we construct families of Jacobian elliptic surfaces that degenerate in a homologically trivial fashion, so that the period matrix specializes to a matrix lying in the interior of the period domain and there is an explicit formula for the derivative of the period matrix that involves no derivatives but is instead a matrix of rank one, just as for curves.
This leads to a result analogous to the asymptotic relations derived earlier for the locus of hyperelliptic curves.
We begin with a modification of Kodaira's degenerate fibres and his notation for them.
(1) I * n−4 = D n for n ≥ 4: contract the four (−2) curves at the extremities and call the result D n . This has four A 1 singularities.
(3) III = T a: blow up the tacnode twice, then contract the resulting (−2) curve and two (−4) curves and call the result III. This has one A 1 singularity and two singularities of type 1 4 (1, 1). (4) IV = T r: blow up the triple point, then contract the three resulting (−3) curves and call the result IV . This has three singularities of type 1 3 (1, 1).
(5) II * = E 8 , III * = E 7 and IV * = E 6 : contract all curves except the central one and call the result R * for R = II, III, IV . This has three singularities, all of type A.
Now suppose that Y i → C i are smooth simply connected Jacobian elliptic surfaces, each of which has a fibre φ i (with its structure as a scheme) of type D 4 over i. Let π i : Y i → Y i denote the contraction of the four (−2)-curves at the extremities, so that Y i → C i has a fibre φ i , with its structure as a scheme, of type D 4 . So φ i is a copy of P 1 but with multiplicity 2.
(2) C i → C i is an isomorphism outside i and the stabilizer group at the unique pointĩ of C i that lies over i is Z/2, (3) Y i → C i is smooth overĩ and the fibre φ˜i is E/(−1) for some elliptic curve E, (4) the quotient morphism ρ i : Y i → Y i is an isomorphism outside the four points that map to the four nodes on Y i , where the stabilizer group is Z/2, PROOF: Everything follows from the basic property of a D 4 fibre, which is that, PROOF: This is a well known local calculation depending on the fact that the non-trivial element ι of a stabilizer group acts trivially on a local generator where w i is a fibre co-ordinate.
Now assume that the two D 4 fibres are isomorphic. That is, two cross-ratios are equal. Corollary 6.3 Choose local co-ordinatesz i on C i atĩ such that ι * z i = −z i where ι is a generator of the stabilizer of the pointĩ.
(1) In terms of these local co-ordinates there is a Fay plumbing of the stacks (2) If Y → ∆ is any lifting of Y ′ → ∆ ′ then Y is a smooth 3-dimensional Deligne-Mumford stack, whose only non-trivial stabilizers are copies of Z/2 at each of the four 2-torsion points of Y a ∩ Y b (which is isomorphic to the quotient E/ι of an elliptic curve E by (−1)). (Y b ) and there is no monodromy on H 2 ( Y t , Z).
PROOF: (1): by Lemma 6.1 (6) the classifying morphisms C i → Eℓℓ Kod are isomorphic to first order at the points a, b. That is, there is a 2-commutative diagram (1) is proved. (2): in local co-ordinates the morphism Y ′ → ∆ ′ is given by t = xy, which proves smoothness. The rest is immediate. ( The absence of monodromy is a well known consequence. Now suppose that there is a lifting and fix one. Then there is a C 0 collapsing map γ : Y t → Y 0 , in the context of orbifolds, unique up to homotopy. There is a totally isotropic sublattice Lemma 6.5 Y t is in L-general position for t = 0.
PROOF: If not, then there is a holomorphic 2-form ω t on Y t such that A ω t = 0 for every A ∈ L. By the argument used in the proof of Lemma 6.3 3 ω t specializes Also, there are holomorphic 3-forms Ω (j) on the smooth 3-fold stack Y such that for every j and λ. We can expand Ω (j) on the inverse image in Y of the plumbing fixture F as Here and from now on all congruences are taken modulo t 2 . In a neighbourhood of Y a − ( Y a ∩ Y b ) we have, from the construction of the plumbing described in (4.1), (4.2) and the discussion immediately afterwards, On Y a we have q =z a and w = w a , so that is ι-invariant, we need only consider odd values of the index p in the expansions above. That is, and c (j) m,n = 0 if m + n ≡ 0 (mod 2). The next lemma is well known. It holds for any one-parameter degeneration of surfaces, not only for the families that we have constructed via Fay plumbings. However, the results such as Lemma 6.12 below that follow for the derivatives η (j) X i do not hold in such generality, because for more general glueing data a formula such as t ≡ q 2 − v 2 (mod t 2 ) will not hold. It follows that for a general oneparameter degeneration it is not possible to control the orders of the poles of the differentiated forms η (j) (4) These bases are normalized with respect to the given A-cycles on Y a and Y b .
Construct row vectors ] whose entries are differentials. Via the identifications of Lemma 6.2 we also regard these as vectors of differentials on Y i and on Y i and then write them as η Y i , etc. Abbreviate the notation by defining Proposition 6.13 (1) There is an equality of row vectors (2) In terms of the local co-ordinatesz i the forms η Y i are given by Examining the first few terms in the expansion provided by (6.7)-(6.10) gives 1,0 . The result now follows from (6.14)-(6.17).
Let Ψ Y i denote the period matrix of Y i with respect to the 2-cycles A 1 , ..., A h i above.
Proposition 6.18 (1) The period matrix Ψ t of Y t is given by where I i is a vector of integrals of the meromorphic differential η Y i over cycles on Y i .
(2) The derivative (dΨ t /dt)| t=0 of the period matrix of Y t at t = 0 is the rank PROOF: This is an immediate consequence of the calculations that we have made. Note that the vector [I a , I b ] can be seen to have the correct length (so that the matrix that describes (dΨ t /dt)| t=0 has the correct shape) by omitting the zeroes, four in each [I i ], that arise from integrating η Y i around the four exceptional (−2)-curves on Y i that are contracted in Y i .

Remark:
(1) Recall also that the final h×h block (with respect to an appropriate choice of basis) of the period matrix, and so of the derivative (dΨ/dt)| t=0 , is skew-symmetric. But a skew-symmetric matrix of rank 1 vanishes identically; for (dΨ/dt)| t=0 this is predicted by Griffiths transversality. In terms of the bases that we have used, this means that the final piece of length h i in each vector I i vanishes.
(2) (An analogue of Example 3.5 on p. 45 of [F1].) Suppose that Y b is rational. Then (dΨ t /dt)| t=0 is given by Note that Y b has 4 moduli if the constraint that its D 4 fibre φ b should be isomorphic to φ a is ignored. This constraint reduces the number of moduli to 3. There is a resolution Z → [ Y] of the geometric quotient such that the closed fibre Z 0 of Z → ∆ is semi-stable and has six components: two are the surfaces Y i for i = a, b and four are Veronese surfaces V 1 , ..., V 4 . So Y → ∆ has a birational model Z ′ → ∆ with good reduction, which is obtained from Z after flopping four times (each time in a curve in Y b that is a section of the elliptic fibration and meets one of the V i ), then contracting the strict transforms of the V i to curves and finally contracting the strict transform of Y b to a curve. The closed fibre Z ′ 0 is isomorphic to Y a . That is, a Fay plumbing gives a one-parameter deformation of a surface Y a where the derivative of the period matrix is given explicitly, provided that Y a contains a D 4 -fibre. Since Y b has three moduli the vector I b has three moduli, so (dΨ t /dt)| t=0 has three moduli.
Now suppose that f 1 : Y 1 → C 1 , ..., f r : Y r → C r are smooth simply connected Jacobian elliptic surfaces each of which is in L i -general position for an appropriate L i . Assume also that Y i → C i has D 4 -fibres over a finite non-empty set {P ij } of points in C i . Let Y i → Y i denote the contraction of each of these designated D 4 -fibres to a D 4 -fibre.
Suppose also that Γ is a connected tree with r vertices and that we can associate the surfaces Y i to the vertices of Γ such that two vertices i, j are joined in Γ if and only if the D 4 -fibre on Y i that lies over P ij is isomorphic to the D 4 -fibre on Y j that lies over P ji .
Then the plumbings modulo t 2 of the stacks Y i that has just been described can be iterated to give where e runs over the edges of Γ and the closed fibre is Y i arranged according to the tree Γ. Take the normalized basis and the leading terms of the forms η ij and η ji have opposite signs; recall that these leading terms are ± 1 4z −2 i . Proposition 6.20 Assume that Y ′ → ∆ ′ r−1 can be lifted to a family Y → S te over an (r − 1)-dimensional polydisc.
Then Y is smooth and the general fibre Y t is in L-general position for an appropriate L, there is no monodromy on where I ij is a vector of integrals of η ij around 2-cycles on Y i and the sum is over the edges of Γ.
PROOF: This is a straightforward consequence of Proposition 6.18. We shall see, in Lemma 11.8, that the liftings Y → S te exist when each surface Y i is a special elliptic surface. Then, in Theorem 11.12, which will lead to the main result, Theorem 11.13, we shall take all the surfaces Y i to be K3 surfaces and Γ will be an alkane. Note that K3 surfaces are in L-general position for all appropriate L, so that the general fibre Y t will also be in L-general position.

Fay's formulae for homologically trivial plumbings of curves
For a homologically trivial Fay plumbing of curves, the arguments of the previous section go through to recover Fay's Corollary 3.2, as follows. No change is required in the argument except that the fibre co-ordinates w and w i should be suppressed.
Suppose that C → ∆ t is a homologically trivial Fay plumbing of C a to C b that identifies a with b and (ω (1) (t), ..., ω (g) (t)) is a normalized basis of H 0 (C t , ω Ct ).
there is a unique elementη i ∈ H 0 (C i , ω C i (2i)), normalized by the requirements that, firstly, A kη i = 0 for every A-cycle A k on C i and that, secondly, of row vectors of length g.
It follows that and v i is the vector of integrals of the formη i around the Bcycles on C i . Since, by the bilinear relations for integrals of the first kind, τ (C t ) is symmetric, it follows that v = λ ω Ca (a), −ω C b (b) for some scalar λ and We can calculate λ from the bilinear relations for integrals of the second kind on C a with only poles at a: where γ is a loop in C a around a and φ = df . Taking ψ = ω (j) Ca and φ =η a gives, via the expansions above in terms of power series of ω j Ca and η j Ca , . This differs from the formula given in Fay's Corollary 3.2, in which λ = 1 4 , because our 1-forms are normalized by the requirement that A l ω (j) = δ jl and not 2π √ −1δ jl .
Remark: As already mentioned, on p. 41 of [F1] the minus sign in front of ω C b (b) is missing. The reason is a different choice of normalization in the plumbing construction, which replaces z b by −z b . Logically, however, there is no difference.

Poincaré's asymptotic period relations
Suppose that E 1 , ..., E g are disjoint elliptic curves and that D is a copy of P 1 . Fix points a i ∈ E i and a local co-ordinate z i on E i at a i . On D fix a point ∞ and a global co-ordinate u on D − {∞}. Fix distinct points b 1 , ..., b g ∈ D − {∞} given by u − b j = 0. Then successively making Fay plumbings of the E i to D using these data in a way that identifies a j ∈ E j to b j ∈ D − {∞} leads to a family C → S of genus g curves over a g-dimensional polydisc S = S g = ∆ t 1 ,...,tg with co-ordinates t 1 , ..., t g . We fix a symplectic basis (A j , B j ) of each H 1 (E j , Z) and let v j be the corresponding normalized 1-form on E j . We write v i (a i ) = v i dz i (a i ); this notation will be used many times.
The next result is due to Poincaré when g = 4 and to Fay in general. According to Igusa ([I], p. 167) Poincaré's proof exploited the fact that the theta divisor on a Jacobian is of translation type, and so can not be obviously extended to an analysis of period matrices in higher dimensions. However, Fay's approach, of which we include some details that he omitted, goes via his plumbing construction and so can be extended. It turns out that hyperelliptic curves are particularly interesting from this point of view.
As already mentioned, this result is derived in [FGSM] from their global results.
Theorem 8.1 ("Poincaré's asymptotic period relations", [F1], p.45) For i = j the entries τ ij of the period matrix τ of C t can be written as τ ij = τ ij u ij where u ij ≡ 1 (mod (t)) and PROOF: Induction on g.
Construct a family C ′ → S 1 of curves of genus 1 by plumbing a 1 ∈ E 1 to b 1 ∈ D. This has a normalized 1-form ω 1 (C ′ t 1 ). Then near a point of D − {b 1 } we have Now construct a genus 2 family C → ∆ t 1 ,t 2 by plumbing C ′ to E 2 in a way that identifies b 2 ∈ D with a 2 ∈ E 2 . There are normalized 1-forms Ω j (C t 1 ,t 2 ) on C t 1 ,t 2 , where j = 1, 2. Then, near a point in E 2 , we have , where η 2 is a meromorphic 1-form on E 2 with a double pole at a 2 and η 2 = 1 4 v 2 (a 2 )(z −2 2 + h.o.t.)dz 2 . Moreover, A 2 η 2 = 0. So Now assume that g ≥ 3 and that the result is true for plumbings of genus ≤ g −1. Write Note that c ij ∈ C * since v i (a i ) = 0.
Suppose that k ∈ [3, g] and that j ∈ [1, 2]. Then, by induction, where u 12k ≡ 1 (mod (t)) and X 12k is some function. Moreover, τ 12 ≡ 0 (mod t j ) since setting t j = 0 gives a plumbing where the curve C t remains singular and one of its irreducible components is the non-varying elliptic curve E j . Therefore X 12k is divisible by t j and then we can write Since c 12 = 0 the induction is complete. It follows that the off-diagonal quantities y ij = τ −1/2 ij satisfy the Plücker relations y ij y kl − y ik y jl + y il y jk = 0.
These translate into octic relations amongst the τ ij , which are "asymptotic relations" amongst the τ ij . Let J c g denote the closure of the Jacobian locus J g inside the stack A g along the closed substack Diag g of A g that parametrizes products of elliptic curves. Note that Diag g is isomorphic to the quotient (A 1 ) g /S g of A g 1 by the symmetric group. In transcendental terms, A g = H g /Sp 2g (Z) and Diag g = Diag g /(SL 2 (Z) ≀ S g ). Let M c g denote the open substack of the stack M g of stable curves of genus g that parametrizes curves of compact type. Then the Jacobian morphism M g → A g extends to a proper morphism M c g → A g whose image is J c g . We let J g and J c g denote the inverse images in H g of J g and J c g . We show next, in Proposition 8.2, that these "asymptotic relations" are exactly the defining equations, in terms of the entries τ ij of the period matrix, of the associated graded ring belonging to the closed subvariety Diag g of J c g . We shall use the term multi-elliptic to refer to a stable curve of genus g that contains g elliptic components (and maybe some smooth rational components). Such a curve is necessarily of compact type. Then Diag g is the image of the stack of multi-elliptic stable curves.
Let Gr J c g denote the sheaf of graded O Diagg -algebras that is associated to the closed embedding Diag g ֒→ J c g and Gr J c g its pullback to O Diag g .
The Grassmannian Grass(2, g) is embedded in Proj C[{y ij }] = P ( g 2 )−1 y via the Plücker co-ordinates y ij . Let X denote the closure of the image of Grass(2, g) under the generically finite rational map given by y ij → y −2 ij = τ ij and let X ⊆ A ( g 2 ) be the affine cone over X with affine co-ordinate ring C[ X]. The quadratic Plücker identities relating the y ij give rise to octic relations between the τ ij that are the defining equations of X (or of X).
Proposition 8.2 Gr J c g is isomorphic, as a sheaf of graded O Diag g -algebras, to the constant sheaf O Diag g ⊗ C C[ X].
PROOF: Let E → Diag g be the g-fold universal elliptic curve over Diag g , so that dim E = 2g. Set U = ((P 1 ) g − ∆)/P GL 2 where ∆ is the union of all the diagonals. Let S be a g-dimensional polydisc with co-ordinates t 1 , ..., t g and put L = E × U × S. Then a suitable Fay plumbing is a family of curves over L which then gives a period map h : L → H g such that τ ij = τ ij u ij as above. Moreover, along the sublocus of L defined by t 1 = · · · = t g = 0 the space L is everywhere locally the base of a miniversal deformation of a singular stable curve C 0 which is of the form C 0 = P 1 + g 1 E ′ i for varying elliptic curves E ′ i . Therefore, by the local Torelli theorem for smooth curves, there is a dense open subspace of L along which the derivative of h is injective. So the image of h is open inside some branch of J c g along Diag g . In particular, its dimension is 3g − 3.
Lemma 8.3 J c g is unibranched along Diag g .

PROOF:
Suppose that E 1 , ..., E g are elliptic curves and that M E i ⊂ M g is the locus of multi-elliptic curves C that contain each E i . That is, C is a tree whose components are the curves E i together with some copies of P 1 . To prove the lemma it is enough to show that M E i is connected. We do this by induction on g. This is clearly true when g = 1. Set R = O Hg . Let T denote the ideal in R generated by {τ ij |i = j}. This is the defining ideal of Diag g . Set S = Gr T R, the associated sheaf of graded O Diag g -algebras. Since the spaces concerned are Stein we shall not emphasize the distinction between a ring and a sheaf of rings. Given a graded O Diag g -algebra such as S, Proj S will denote the projectivization relative to Diag g in the analytic category.
Let I be the ideal of R that defines J c g and K the ideal of R that defines (Diag g × X) ∩ H g . Let I, K be the associated graded ideals of S.
Note that K is the defining ideal of Diag g × X inside Proj S and so is prime. The existence of Fay's octic identities of Plücker type given above mean that there exist f 1 , ...f r ∈ I such that, if g i = f i (mod I), then g 1 , ..., g r generate K. Therefore K ⊆ I.
Suppose that K = I. Now Proj(S/K) = Diag g × X, which is reduced and irreducible of dimension 3g − 4. So dim Proj(S/I) ≤ 3g − 5. On the other hand, Proj(S/I) is the exceptional divisor in the blow-up Bl Diag g J g and so has dimension 3g − 4. Therefore K = I and the proposition follows.
Corollary 8.4 Up to graded equivalence the singularity of J c g at a given point x of Diag g is isomorphic to the cone X and is independent of x.
When g = 4 (the case considered in detail by Poincaré) X is then an octic hypersurface in A 6 so that J c 4 does not have rational singularities along Diag 4 .

Hyperelliptic curves and alkanes
We continue with the notation of the previous section. Suppose instead that we choose points a 1 on E 1 , b i−1 and a i on E i for i = 2, ..., g − 1 and b g−1 on E g and a local co-ordinate z x at each point x, and then plumb E i to E i+1 by identifying a i to b i in a chain. This leads to a family C → S of genus g curves where S = ∆ t 1 ,...,t g−1 is a (g − 1)-dimensional polydisc with co-ordinates t 1 , ..., t g−1 . Let τ i be the period of E i , defined, as before, after the choice of a symplectic basis (A j , B j ) of H 1 (E j , Z) and normalized 1-form ω j on E j . Construct a symplectic basis of H 1 (C t , Z) as a union of the symplectic bases (A j , B j ).
Lemma 9.1 If each difference b i−1 −a i is 2-torsion on E i and at each point b i−1 , a i on E i a local co-ordinate is chosen that is anti-invariant under the involution [−1 E i ] then the family C → S is hyperelliptic.
PROOF: The constraint on the points a 1 , ..., a g−1 implies that the chain C 0 = E 1 ∪ · · · ∪ E g is a double cover of a chain B 0 = Γ 1 ∪ · · · ∪ Γ g of copies of P 1 and that the map C 0 → B 0 is not étale at the nodes. So, via Proposition 4.6, Fay plumbings can be constructed simultaneously to give a double cover C → B over S where B → S is a family of genus zero curves.
We next give a description of the closure HYP c g of the hyperelliptic locus in M c g near Diag g in combinatorial terms.
Lemma 9.2 Suppose that C is multi-elliptic. Choose a node on each component E i and take it to be the origin of E i .
(1) C is hyperelliptic if and only if C contains no rational curves, each component E i of C contains at most 4 nodes and each node of C that lies on E i is a 2-torsion point of E i .
(2) If C is hyperelliptic then the hyperelliptic involution ι C preserves each component of C and the fixed point locus of ι C consists of the nodes of C and the 2-torsion points of each component E i .
PROOF: This is an elementary exercise.
Recall from elementary chemistry that carbon has valence 4, hydrogen has valence 1 and that an alkane is a hydrocarbon whose molecular formula is of the form C g H 2g+2 . Note that the hyperelliptic involution of a hyperelliptic curve of genus g has 2g + 2 fixed points. Alkanes were first enumerated by Cayley and again, more recently, by Rains and Sloane (who corrected some errors by Cayley); see [OEIS], sequence A000602. We shall refer to g as the genus of the alkane. We can describe stable multi-elliptic hyperelliptic curves C in terms of alkanes of the same genus.
Corollary 9.3 Each fibre in HYP c g over a point in Diag g is a finite set that corresponds naturally to the set of alkanes of genus g.
PROOF: This is a translation of the preceding lemma. The carbon atoms correspond to the elliptic curves and the hydrogen atoms to the 2-torsion points on each elliptic curve that are not nodes. That is, the the hydrogen atoms correspond to the fixed points of the hyperelliptic involution that are not nodes.
Corollary 9.4 The branches of HYP c g through Diag g in A g correspond to the alkanes of genus g.
We shall also use the term alkane to refer to a hyperelliptic multi-elliptic stable curve.
Suppose that C = g 1 E i is an alkane. Suppose that K is the set of edges, so that {i, j} ∈ K if and only if i = j and E i meets E j . Choose a local co-ordinate on each E i at each node of C (not necessarily anti-invariant under the involution [−1 E i ]) and let C → S g−1 = ∆ {t k |k∈K} be the result of plumbing together the curves E i according to these data.
Theorem 9.5 Modulo ({t l |l ∈ K}) 2 , the off-diagonal entry τ (C t ) k of the period matrix of C t is a multiple of t k if k ∈ K. The other off-diagonal entries vanish. PROOF: We can suppose the curves E i ordered so that E g meets only E g−1 and then argue by induction on g.
Suppose that a ∈ E g−1 is identified with b ∈ E g . Suppose that C ′ → S ′ = ∆ t 1 ,...,t g−2 is the result of plumbing E 1 , ..., E g−1 according to the data, so that C → S is the plumbing of C to E g × S ′ that identifies {a} × S ′ with {b} × S ′ .
Write (t ′ ) = (t 1 , ..., t g−2 ) and (t) = (t ′ , t g−1 ) and suppose that (Ω ′ 1 , ..., Ω ′ g−1 ) is a basis of normalized 1-forms on C ′ . By Fay's formula, τ (C t ) is congruent to Moreover, according to the induction hypothesis, τ (C ′ t ′ ) has the required form, and the result follows. Let Hyp c g denote the closure of the hyperelliptic locus in H g . Theorem 9.6 (Asymptotic period relations for the hyperelliptic locus) To first order in a neighbourhood of D g each branch of Hyp c g is described as a subvariety of H g by the vanishing of the entries τ k in the period matrix τ where k runs over the set of pairs that are not edges of the corresponding alkane.
In particular, the branch corresponding to the linear alkane equals, to first order, the locus T g of tridiagonal matrices.
PROOF: We prove this only for the linear alkane. The proof in general is the same but with more complicated notation.
Let E τ = C/(Zτ ⊕Z) be the elliptic curve corresponding to τ ∈ H 1 . Pick a coordinate z on C such that the involution [−1 Eτ ] acts via z → −z. Let F → D g be the family whose fibre over (τ 1 , ..., τ g ) is E τ 2 × · · ·× E τ g−1 . Then the Fay plumbing just described, using the co-ordinates provided, gives a family C → F × S g−1 of stable curves which is minimally versal at each point where t 1 = · · · = t g−1 = 0 and so is minimally versal everywhere.
Fixing a non-zero 2-torsion point on each of E 2 , ..., E g−1 defines a section of F → D g . Then the restriction of C to the corresponding subvariety D g × S g−1 of F ×S g−1 is then, over the complement of the discriminant, a family of hyperelliptic curves.
Since dim(D g × S g−1 ) = 2g − 1 this is everywhere a minimally versal family of hyperelliptic curves. Then, by the local Torelli theorem for hyperelliptic curves, the image T of D g × S g−1 in H g is of dimension 2g − 1. To first order T lies in T g , and the theorem is proved.

Stable reduction
Chakiris proved [C1] [C2] a generic Torelli theorem for Jacobian elliptic surfaces over P 1 by reducing the problem to special elliptic surfaces.
To achieve this reduction he extended the domain of the period map to make the map proper. In turn, he did this by first showing that a one-parameter degeneration of Jacobian elliptic surfaces over P 1 without monodromy can be put into a certain standard form. See, e.g., the statement ( * * ) in [C2], top of p. 174 or the "stable reduction" theorem on p. 231 of [C1]. Note, however, that this version of a stable reduction theorem gives a closed fibre that contains curves of cusps, and so is not semi log canonical (slc) in the sense of the Minimal Model Program (MMP).
In this section we shall refine his result, so that the period map becomes proper over each locus W h 1 ,...,hr . For this, we use his ideas strengthened by the MMP, which was not available to him.
Theorem 10.1 Suppose that X → ∆ is a 1-parameter degeneration of simply connected Jacobian elliptic surfaces of geometric genus h ≥ 1 which is semi-stable (in the usual sense that the closed fibre X 0 is reduced with normal crossings) and that there is no monodromy on the cohomology of the geometric generic fibre Xη. Assume also that, under the period map, the image of 0 ∈ ∆ is the direct sum of the period matrices of special elliptic surfaces V 1 , ..., V r .
Then there is a birationally equivalent model Y → ∆ with the following properties: (1) Y has Q-factorial canonical singularities; (2) the closed fibre Y 0 has slc singularities; (3) the irreducible components of Y 0 are the singular models V i of the V i with D 4 -fibres; (4) if V i ∩ V j is not empty then it is a copy of P 1 and contains 4 points at all of which V i and V j each has a node; (5) each triple intersection V i ∩ V j ∩ V k is empty; (6) Y 0 is formed by arranging the surfaces V i in a tree.
PROOF: First assume only that the generic fibre Xη is a smooth minimal elliptic surface and is not ruled. To begin, run the MMP in two steps, as follows.
Step 1: Run a K X /∆ MMP on X → ∆ and let X 1 → ∆ be the result. Then X 1 has Q-factorial terminal singularities and K X 1 /∆ is semi-ample So some relative pluricanonical system |mK X 1 /∆ | defines an algebraic fibre space f : X 1 → S where S → ∆ is a semi-stable family of curves (so that S has singularities of type A) and K X 1 /∆ pulls back from an ample Q-line bundle on S. Moreover, the closed fibre X 1,0 has slc singularities. Replace X by X 1 .
Step 2: If there are surfaces E i in X such that f (E i ) is a point then there are only finitely many such. For suitable α i ∈ Q with 0 < α i ≪ 1 run a (K X /S , α i E i ) MMP on X → S. The result is a birational map X − → X 1 under which the strict transform of each E i is of dimension at most 1. Since K X is trivial in a neighbourhood of E i , the rational map X 1 − → S is a morphism, all its fibres are 1-dimensional, X 1 has canonical singularities and X 1,0 has slc singularities. Moreover, X 1 is Q-factorial, by general properties of the MMP. Replace X by X 1 .
At this point X has Q-factorial canonical singularities and X 0 has slc singularities. Moreover, there is a surface S, a semi-stable morphism S → ∆ (so that S has singularities of type A) and a morphism f : X → S with only onedimensional fibres such that K X /∆ is the pullback of an ample Q-line bundle on S.
Now we assume also that Xη is Jacobian and simply connected. Then X η is also Jacobian (maybe after a finite base change ∆ → ∆, which we can and do ignore), so that X → S has a generic section (that is, a closed subscheme S such that the induced morphism S → S is proper and birational), the closed fibre S 0 is a tree of copies C i of P 1 and the inverse image f −1 (C i ) → C i has a section. Moreover, the generic fibre of f −1 (C i ) → C i is either elliptic or a cycle of rational curves.
Lemma 10.2 The irreducible components of X 0 are either models of the special elliptic surfaces V i or are ruled.
PROOF: Recall that a Jacobian elliptic surface over P 1 whose geometric genus is zero is either rational or (elliptic) ruled. So we only have to consider components Z of X 0 that are Jacobian elliptic surfaces of positive genus. Now the Hodge structure of Z embeds into the product of the Hodge structures of the special elliptic surfaces V i ; this is enough [C1] to ensure that the global monodromy of the elliptic fibration Z → P 1 that is provided by the fact that Z is a component of X 0 is of order at most 2, so that Z is special, and then the Torelli theorem for hyperelliptic curves shows that Z is birational to one of the V i .
From now on let C i denote a typical irreducible component of S 0 , put X i = f −1 (C i ) and let f i : X i → C i denote the restriction of f : X → S. Let ν i : X ν i → X i be the normalization and X i → X ν i the minimal resolution; a priori, these objects might be disconnected. We assume that there is no monodromy on H 2 (Xη, Z); this is equivalent to the statement that p g (Xη) = p g ( X i ) (recall that every surface that has been contracted is ruled).

Lemma 10.3 The maps H
Since the formation of the groups H i (X t , O) commutes with specialization the lemma follows from the assumption that p g (Xη) = p g ( X i ).
Fix P ∈ S 0 and denote localizations at P and along fibres over P by a superscript h.
Lemma 10.4 If P is a node of S 0 then H 1 (f −1 (P ), O) = 0 and f −1 (P ) red is a simply connected configuration (a tree) of P 1 's. PROOF: Say Γ ij = X i ∩ X j if this is not empty. Then the cohomology of the short exact sequence is non-zero for some Γ jk , then there is a birational morphism π : then (since a disjoint union of trees has more vertices than edges) #{X i |H 1 (O X i ) = 0} ≥ r + 1, which is impossible.
Now suppose that C, D are irreducible components of S 0 that meet at a point P .
Recall that S has an RDP of type A N −1 at P for some N ≥ 1. Then the henselization S hens can be written as a geometric quotient S hens = [S ′ /(Z/N)] where S ′ is smooth and local and S ′ → ∆ is semi-stable. Let X ′ denote the normalization of X hens × S hens S ′ with induced morphism f ′ : X ′ → S ′ and let P ′ denote the inverse image of P in S ′ .
Since Z/N acts freely in codimension 1 on S ′ , it also does so on X ′ . So the quotient map π : X ′ → [X ′ /(Z/N)] = X hens is étale outside the 1-dimensional sublocus f −1 (P ) of X hens and so, in particular, is étale in codimension 1. Therefore X ′ has canonical singularities, so is Cohen-Macaulay. Since f ′ : X ′ → S ′ is dominant with equi-dimensional fibres and S ′ is smooth, f ′ is flat.
Suppose that ∆ → ∆ is a finite base change, say of order L. Then the same argument shows that X ′ × ∆ ∆ also has canonical singularities, and so X ′ 0 has slc singularities.
Note that X hens . Of course, the induced morphisms p : Y ′ → C ′ and q : Z ′ → D ′ are proper; they are partial normalizations of Y × C C ′ and Z × D D ′ . Since Z/N acts freely outside a finite set, and since Y, Z are Cohen-Macaulay, we can identify Y = [Y ′ /(Z/N)] and Z = [Z ′ /(Z/N)].
Since C ′ , D ′ are principal divisors on S ′ , the divisors Y ′ and Z ′ are principal on X ′ . We can write Y ′ = (x = 0) and Z ′ = (y = 0), where x, y are eigenfunctions of Z/N and xy is Z/N-invariant.
Suppose that ξ is a closed point of f ′−1 (P ′ ). Localize X ′ , Y ′ , Z ′ at ξ to get a 3-fold germ X ′′ and principal divisors Y ′′ , Z ′′ on it. The stabilizer of ξ is a subgroup H ∼ = Z/M of Z/N that acts freely on X ′′ − {ξ}. Put X loc = [X ′′ /H] etc.; these are localizations of X , Y and Z.
Lemma 10.5 There are just two possibilities.
(2) X ′′ 0 is a degenerate cusp of multiplicity at most 4, Y ′′ , Z ′′ are of type A, Y ′′ ∩ Z ′′ is a nodal curve, M ∈ {1, 2} and X loc 0 is either a degenerate cusp or the geometric quotient of a degenerate cusp by Z/2. PROOF: We know that X ′′ is canonical, X ′′ 0 = Y ′′ ∪ Z ′′ is slc and that Y ′′ , Z ′′ are principal divisors on X ′′ .
Next, the classification of slc singularities with at least two branches shows that the curve Y ′′ ∩ Z ′′ is either smooth or a plane node. Then in the first case X ′′ is smooth, X ′′ 0 = A ∞ and X loc 0 = A ∞ / 1 M (a, −a, 1) while in the second case X ′′ 0 is a degenerate cusp and Y ′′ , Z ′′ are of type A.
Because p : Y ′ → C ′ is a Jacobian semi-stable family of elliptic curves and H acts effectively on Y ′ , the classification of automorphism groups of elliptic curves shows that M ∈ {1, 2, 3, 4, 6}. If the fibre Lemma 10.9 Suppose that p : Y → C is generically smooth and that p −1 (P ) is of type either D 4 or R or R * . Then Z → D is also generically smooth.
PROOF: If q : Z → D were generically a cycle of rational curves then q −1 (P ) would be of type D n for some n ≥ 5.
For any irreducible component C i of S 0 let C 0 i denote the complement in C i of the singular points of S 0 .
Lemma 10.10 Suppose that X i = f −1 (C i ) → C i is generically smooth. Then X i is normal, and over C 0 i it is Gorenstein.
is a Jacobian elliptic fibration, so that K X hens −X hens i pulls back from a line bundle on S hens −S hens 0 and so is trivial. Since X hens i is an irreducible and principal Weil divisor in X hens , the homomorphism Cl(X hens ) → Cl(X hens − X hens i ) is an isomorphism. Therefore K X hens is trivial. So X is Gorenstein along f −1 (P ) and K X hens pulls back from a line bundle on S hens . Therefore X hens i is also Gorenstein and ω X hens i also pulls back from a line bundle on C hens i .
Suppose that X i → X min i is the minimal model. Then K X min i pulls back from a line bundle on C i ; comparison of this with the description of ω X hens i just given . So X i is normal and Gorenstein over C 0 i . Normality follows from Lemma 10.8.
Lemma 10.11 X i has rational singularities.
PROOF: From our knowledge of the structure of X 0 along the curves X i ∩ X j we only need to prove that X i has RDPs over C 0 i . Since X 0 has slc singularities, we need only exclude cusps and simply elliptic singularities.
So suppose that there is at least one such irrational singularity. Let X i → X i be the minimal resolution. Since X i is rational or elliptic ruled, it follows that H 1 (X i , O X i ) = 0. Consider the natural morphism X i → X 0 . From the assumption on monodromy, the induced homomorphism φ : ; this latter map is surjective, and so both φ and the natural map Lemma 10.12 Suppose S 0 is regarded as a tree and that C i is an end of S 0 . Suppose that C i meets C j with P = C i ∩C j , that X i = f −1 (C i ) → C i is generically smooth and that f −1 (P ) is irreducible. Then X i is a special elliptic surface. PROOF: Assume not; then X i is rational. Pick a (−1)-curve E on the minimal resolution X i of X i , and let E denote its image on X i . Now f −1 (P ) is in fact of type D, R or R * . Examination of each case leads to an evaluation of K X i .E and (from Corollary 10.1 3) X j .E; since X i .E = −X j .E the adjunction formula then shows that K X .E < 0, which is absurd.
For example, if f −1 (P ) is of type D, then K X i .E = −1 and X j .E = 1/2, so that K X .E = −1/2. The other six cases are handled similarly.
In particular, if X i is an end (that is, if C i is an end of the tree S 0 ), is generically smooth over C i and is rational, then X i must meet its neighbour X j in a fibre of type D ≥5 .
Suppose that X 0 is special elliptic and that X 0 , X 1 , ..., X n is a maximal chain such that each of X 1 , ..., X n is generically smooth over its base curve and is rational.
For T = D n , R, R * , define T * = D n , R * , R, respectively. So each φ i is of type T on X i and of type T * on X i+1 , for some T . Say that φ is of type (T , T * ).
Let X i → C i be the minimal smooth model of X i → C i . Say that the total inverse image of φ i on X i has Euler characteristic a i and that the total inverse image of φ i on X i+1 has Euler characteristic b i . Inspection shows that a i +b i ≥ 12 and that a i + b i = 12 if and only if φ is of type (T , T * ) for T = D 4 , II, III, IV .
Since φ 0 is of type D 4 , D 4 , a 0 = b 0 = 6. Moreover, c 2 ( X i ) ≥ b i−1 +a i , and equality holds of and only if X i has no other singular fibres. Then the preceding discussion shows that there are, a priori, three possibilities.
(2) Besides X n−1 , X n only meets special elliptic surfaces and meets at least one such surface. Then, besides φ n−1 , X n contains a D 4 fibre ψ. So which shows that none of X 1 , ..., X n meets any other X j and that each of X 1 , ..., X n has only two singular fibres.
(3) X n meets a further surface X n+1 which is not generically smooth over its base and X n ∩ X n+1 = φ n , say, is of type (D ≥5 , D ≥5 ). Then 12n = n 1 c 2 ( X i ) ≥ b 0 + 12(n − 1) + a n , which is impossible.
Lemma 10.13 If n ≥ 2 then X n meets exactly one special elliptic surface. If n = 1 then X n meets exactly two special elliptic surfaces.
PROOF: Assume that n ≥ 2. If X n meets more one special elliptic surface then X n contains at least three singular fibres φ, at least two of which are of type (D ≥4 , D ≥4 ) This contradicts c 2 ( X n ) = 12. It follows that X 0 is a tree of singular models of special elliptic surfaces with D 4 fibres and some rational surfaces Y 1 , ..., Y m . The surfaces intersect in D 4 fibres and the Y 1 , ..., Y m contain exactly two such fibres. The Y i are arranged in disjoint chains that join pairs of special surfaces, and no Y i is an end of the tree. Each Y i is of the form Y i = [(E × P 1 )/ι] and so possesses a projection r i : Y i → [E/ι] ∼ = P 1 that is, generically, a ruling whose fibres φ i are sections of f j : Y j → C j . Then φ i .K X /∆ = 0. However, K X /∆ pulls back from an ample Q-line bundle on S, so this is absurd and there are no rational surfaces in X 0 .
Theorem 10.1 is now proved. Note that Y → ∆ factors as Y → T where S → T is obtained by contracting those curves C j ⊂ S 0 that are images of the surfaces Y j . Let /ι] denote the quotient map. Note that we now know that the elliptic factor E is independent of the index i. Let V i → V i be the minimal resolution; this factorizes as V i → V i → V i is the resolution of those nodes that do not lie on any double curve in Y 0 .
Say that V i meets s i other surfaces V j in Y 0 . That is, V i contains s i double curves δ ij , each of which is of type D 4 .
Since each V i has only nodes, we can assume that, after some finite base change ∆ → ∆ if necessary, each V i is smooth outside the double curves δ ij .
Lemma 10.14 If B i ⊂ V i is a section of f i : V i → C i then B i meets each δ ij in a node and B 2 i = −(h i + 1) + s i /2. PROOF: The strict transform B i of B i on V i is a section, so meets each D 4 fibre in an end curve of the D 4 configuration. So B i meets δ ij is a node. Since B 2 i = −(h i + 1) the lemma follows. Proposition 10.15 Y → T has a section.
PROOF: There is an irreducible Weil divisor D ⊂ Y that restricts to a section of Y → T over the generic point of ∆. We can write Lemma 10.16 The singularities of Y are of index at most 2.

The main results
Our goal is to give a first order description of the period locus P L h in a neighbourhood of W 1 h .
Definition 11.1 Denote by J E RDP,n.m. (resp., J E smooth,n.m. ) the stack of generalized RDP (resp., smooth) semi-stable Jacobian elliptic surfaces with no monodromy. Its objects over a scheme B are pairs (X f →S → B, S ′ ) with the following properties: (1) f : X → S and S → B are projective; (2) X → B and S → B are flat, Cohen-Macaulay and of relative dimensions 2 and 1, respectively; (3) S ′ is a section of X → S; (4) S → B is a pre-semi-stable family of curves of genus zero; (5) for every geometric point b of B the fibre X b of X → B is a reduced sum X b = V i of irreducible components V i each of which is of the form (6) over the complement of the nodes of S b each V i has only RDPs (resp., is smooth) and V i → C i is relatively minimal; (7) 2S ′ is a Cartier divisor on X ; (8) 2S ′ is f -ample (resp., is f -nef); (9) the restriction f i = f | V i : V i → C i gives the minimal resolution V ′ i of V i the structure of an elliptic surface of genus h i over C i on which the strict transform of S ′ ∩ V i is the identity section; (10) X b has slc singularities; (11) if P = C i ∩ C j is a node of S b then f −1 (P ) is a fibre of type D 4 on each of V i and V j and the section S ′ meets f −1 (P ) in one of the four points on it that is an A 1 singularity on both V i and V j ; (12) the rank one sheaf ω X /B is the pullback of an invertible sheaf on S.
Lemma 11.4 Given an object (X f → S → B, S ′ ) of J E * ,n.m. , there are Deligne-Mumford stacks X and S such that (1) there is a flat and projective (so representable) morphismf : X → S such that f : X → S is the geometric quotient off , (2) there is a section S ′ off whose quotient is S ′ , (3) S → B is a family of Z/2-curves, (4) each geometric fibre X b is a reduced tree X b = V i where each V i is a 2-dimensional stack that is an elliptic surface over a component of S b , (5) X b has normal crossings except (when * = RDP ) for RDPs, PROOF: This is merely an observation.
PROOF: According to Lemma 11.4 it is enough to show that the deformation theory of one of the 2-dimensional stacks X b is unobstructed. Write X b = X ; then there is a morphismf : X → S with a section S 0 that describes X as an RDP Jacobian elliptic surface over a Z/2-curve S. The section S 0 is then a relatively ample Cartier divisor on X . Define L =f * O X ( S 0 ) and then define . This is a P(1, 2, 3)-bundle over S and X is a sextic divisor in P L . Therefore to give the stack X lying over S is to give a line bundle L on S and a sextic divisor in P L . It follows that the deformation theory of X is unobstructed.
Denote by subscript h the value of the geometric genus.  (1) X = r 1 X i and Y = r 1 Y i where, by assumption, each Y i is birational to a geometric quotient [(E i × C i )/ι].
(2) The elliptic curves E i are all isomorphic since Y is connected.
(3) The configurations X = X i , Y = Y i and S = C i are isomorphic trees.
(4) S ′′ = r(S ′ ) is a section of Y → S. 2S ′′ is Cartier and is ample relative to S.
(5) Say σ ij = X i ∩ X j if this is non-empty and φ ij = r(σ ij ). Then σ ij and φ ij are fibres of type D 4 on each of X i , X j , Y i , Y j , as appropriate.
(6) Y i has 4 singularities of type A 1 on each φ ij and has D 4 -singularities disjoint from S ′′ and from the φ ij .
The divisor 2S ′′ defines a finite morphism ρ : Y → Z = r 1 Z i of degree 2. Say ψ ij = ρ(φ ij ). Then, because of the nature of special Jacobian elliptic surfaces, (1) Z → S is a P 1 -bundle, (2) the branch locus B ⊂ Z is B = B 0 + B 1 + ij ψ ij where (3) B 0 = ρ(S ′′ ) is a section of Z → S and a Cartier divisor on Z, (4) B 1 is disjoint from B 0 , (5) B 1 is a sum B 1 = 3 1 D i of three sections D i of Z → S, (6) each D i is a Cartier divisor on Z, (7) the D i are linearly equivalent and (8) B 1 has ordinary triple points over the complement of the nodes in S and, as does B 0 , meets each ψ ij transversely.
Conversely, given (Z, B), we recover Y as the double cover of Z branched along B + ij ψ ij .
We next prove that we can deform the triple points of B 1 independently, while fixing Z. For this, let Σ denote the set of triple points of B 1 and I Σ its sheaf of ideals, and consider the short exact sequence of coherent sheaves on Z. It is straightforward to verify that H 0 (Z, I 3 Σ (B 1 )) = Sym 3 H 0 (Z, I Σ (D 1 )) and that then a count of dimensions shows that the map H 0 (Z, O Z (B 1 )) → H 0 (Z, O Z /I 3 Σ ) is surjective. It follows that the morphism Def Y → Def Y,P of deformation spaces, where P runs over the D 4 singularities of Y , is formally smooth.
(2) The conclusions of Proposition 6.20 hold for Y → S te . Now fix an alkane Γ of genus h. Recall the 9h + 9-dimensional closed subvariety V Γ of the period space V h . Via the Torelli theorem and the surjectivity of the period map for K3 surfaces, we regard the points of V Γ as h-tuples (Y 1 , ..., Y h ) of Jacobian elliptic K3 surfaces, one surface for each vertex of Γ, where adjacent surfaces (that is, surfaces that correspond to adjacent vertices of Γ) have isomorphic D 4 -fibres.
Let J E Γ denote the reduced closed substack of J E smooth,n.m. h whose geometric points are configurations x = (V i |i ∈ Γ). So each V i is K3 and is smooth outside the double locus D i = V i ∩ (∩ j =i V j ). Each component of D i is a fibre of type D 4 . So J E Γ equals (per + ) −1 (V Γ ). Let J E 1 h denote the substack defined by the condition that each V i is a special Kummer surface. So J E Γ equals (per + ) −1 (V Γ ) and J E 1 h equals (per + ) −1 (W 1 h ).
Lemma 11.10 J E Γ is smooth along J E 1 h . PROOF: Recall first the easy fact that dim J E Γ = 9h + 9.
Let x = (V i |i ∈ Γ) be a point in the subvariety J E 1 h of J E Γ . Let V i → V i be the minimal resolution and D i ⊂ V i denote the total transform of D i . So D i is a sum of D 4 fibres. Let σ i ⊂ V i be the given section. Then the Zariski tangent space T x J E Γ is given by this is because each H 1 classifies first order deformations of V i that preserve the combinatorial structure of the configuration D i + σ i , and then we must impose the condition that when the cross-ratio of the four marked points on each D 4 fibre varies, it does so in a way that is compatible with the fact that it lies on V i and V j .
Then, if V i is a vertex of Γ whose valency is r, dim H 1 ( V i , T V i (− log( D i + σ i ))) = 20 − (1 + 5r) + r − 1 = 18 − 4r; this is because the first Chern classes of the curves in the configuration D i + σ i on V i are not linearly independent in H 1 ( V i , Ω 1 V i ), but rather satisfy r − 1 linear conditions. So, if Γ has γ j vertices of valency j, then dim T x J E Γ = 4 1 γ j (18 − 4j) − (h − 1) = 9h + 9, since jγ j = 2e, where e = h − 1 is the number of edges in Γ. Therefore dim T x J E Γ = dim J E Γ and the smoothness is established.
Lemma 11.11 (1) The period map per + restricts to a morphism per + Γ : J E Γ → V Γ that is an isomorphism over a neighbourhood of W 1 h .
PROOF: We show first that the derivative per + * of per + is injective at all points x of J E Γ .
It follows that per + Γ is étale over a neighbourhood of W 1 h and then that V Γ is smooth along W 1 h . Finally, the surjectivity of the period map for K3 surfaces completes the proof of the lemma.
Define the vector bundle E Γ → V Γ by the property that its fibre over the point (Y 1 , ..., Y h ) of V Γ is the vector space spanned by the h − 1 matrices Π e , each of rank one, where e = (i, j) runs over the edges of Γ and, in the notation of Proposition 6.20, This is a vector bundle of rank h − 1.
Theorem 11.12 (1) There is a branch B Γ of P L h that contains V Γ .
(2) To first order B Γ is, in a neighbourhood of W 1 h , the vector bundle E Γ .
PROOF: Choose a smooth neighbourhood V 0 Γ of W 1 h in V Γ and let E 0 Γ denote the restriction of E Γ to V 0 Γ . Then Proposition 6.20 gives a family of surfaces of genus h parametrized by V 0 Γ × S h−1 , where S h−1 is an h − 1-dimensional polydisc, and the image of V 0 Γ × S h−1 under the period map equals, to first order, precisely E 0 Γ . Since dim E Γ = 10h + 8, which is the number of moduli, and, by the results of [C1] and [C2] (see also the remark below) the period map is generically injective, E 0 Γ is, to first order, the image of some open piece of the moduli space.
Theorem 11.13 (1) The branch B Γ of P L h is the unique branch of P L h that contains V Γ .
(2) To first order, the period locus P L h equals the union ∪ Γ E Γ of the vector bundles E Γ → V Γ in a neighbourhood of W 1 h . PROOF: Proposition 11.7 shows that, in particular, per + is proper over some neighbourhood N h of W 1 h in P L h . Set J E 0 h = (per + ) −1 (N h ). Note that V Γ ⊂ P L h and put V 0 Γ = N h ∩ V Γ . (We could have used this choice of V 0 Γ in the proof of Theorem 11.12.) Let E 0 Γ → V 0 Γ be the restriction of E Γ to V 0 Γ . The plumbing construction of Section 6 and the formula of Proposition 6.18 for the derivative of the period matrix show that E 0 Γ is, to first order, a closed subvariety of N h . That is, there is, for each Γ, a closed substack F Γ of J E 0 h such that per + induces an isomorphism F Γ → E 0 Γ to first order. Now on one hand J E smooth,n.m. h , and so J E 0 h , is smooth and, on the other hand, for each x = (V 1 , ..., V h ) ∈ W 1 h , Theorem 10.1 shows that (per + ) −1 (x) is a finite set which consists of exactly one point for each alkane Γ of genus h. Therefore (per + ) −1 (W 1 h ) ∼ = Γ W 1 h and, in a neighbourhood of W 1 h , J E 0 h = Γ F Γ . So P L h = ∪ Γ E 0 Γ in a neighbourhood of W 1 h .
Remark: Note also that the properness of per + over a neighbourhood of W 1 h is enough to provide a slight variant of Chakiris' proof [C1], [C2] of generic Torelli for simply connected Jacobian elliptic surfaces.