The moduli space of cubic surface pairs via the intermediate Jacobians of Eckardt cubic threefolds
aa r X i v : . [ m a t h . AG ] N ov THE MODULI SPACE OF CUBIC SURFACE PAIRS VIA THEINTERMEDIATE JACOBIANS OF ECKARDT CUBIC THREEFOLDS
SEBASTIAN CASALAINA-MARTIN AND ZHENG ZHANG
Abstract.
We study the moduli space of pairs consisting of a smooth cubic surface and a smoothhyperplane section, via a Hodge theoretic period map due to Laza, Pearlstein, and the second namedauthor. The construction associates to such a pair a so-called Eckardt cubic threefold, admittingan involution, and the period map sends the pair to the anti-invariant part of the intermediateJacobian of this cubic threefold, with respect to this involution. Our main result is that the globalTorelli theorem holds for this period map; i.e., the period map is injective. To prove the result,we describe the anti-invariant part of the intermediate Jacobian as a Prym variety of a branchedcover. Our proof uses results of Naranjo–Ortega, Bardelli–Ciliberto–Verra, and Nagaraj–Ramanan,on related Prym maps. In fact, we are able to recover the degree of one of these Prym maps bydescribing positive dimensional fibers, in the same spirit as a result of Donagi–Smith on the degreeof the Prym map for connected ´etale double covers of genus 6 curves.
Introduction
Moduli spaces of pairs consisting of a variety, together with a boundary divisor, have become acentral focus in moduli theory. In light of the success of Hodge theory in studying the geometry ofmoduli spaces, it is natural to construct Hodge theoretic period maps associated with such modulispaces of pairs. In this paper, we focus on a special case, namely we consider the moduli space M of cubic surface pairs ( S, E ), where S is a smooth cubic surface, and E ⊂ S is a smooth hyperplanesection. Moduli spaces of cubic surfaces with boundary divisors have been studied before in anumber of contexts, e.g., [HKT09] and [Fri16]. More recently, in the context of log K-stability,some compactifications of the moduli space M were described in [GMGS18], by reducing to theGIT compactifications of cubic surface pairs analyzed in [GMG19]. Cubic surface pairs ( S, E ) alsoprovide examples of log Calabi–Yau surfaces, which arise in the study of mirror symmetry [GHK15];the mirror family to a particular cubic surface pair was recently given in [GHKS19].In another direction, the Zilber–Pink conjecture [Pin05, Conj. 1.3] provides a framework thatcalls attention to unlikely intersections in Shimura varieties; i.e., subvarieties of Shimura varietiesthat meet special subvarieties in higher than expected dimension. The moduli space M is anunlikely intersection of this type. More precisely, let C ⊂ A denote the moduli space of interme-diate Jacobians of cubic threefolds sitting inside the moduli space of principally polarized abelianvarieties of dimension 5. There is a generically injective finite map M → C , with image M ′ param-eterizing intermediate Jacobians of Eckardt cubic threefolds without a choice of Eckardt point (thisis explained below, see also § M ′ is an unlikely intersection of C with a special subvarietyof A (denoted by A τ in the proof of Proposition 5.13). Unlikely intersections in the Torelli locuswere investigated recently in [MO13].In this paper, we study the moduli space M of cubic surface pairs ( S, E ) via a Hodge theoreticperiod map constructed in [LPZ18]. More precisely, for our purposes, for the pair (
S, E ), it ismore convenient to consider the associated pair ( S, Π), where Π ⊂ P is the hyperplane such that Date : November 26, 2020.2010
Mathematics Subject Classification. = S ∩ Π. Then, associated with the pair ( S, Π) is a so-called Eckardt cubic threefold X ⊂ P ,which in coordinates can be described as the zero set of f ( x , . . . , x ) + l ( x , . . . , x ) x , where f (respectively, l ) is the polynomial of degree 3 (respectively, 1) on P defining S (respectively, Π).Such a cubic threefold X has a natural involution τ , given by x
7→ − x . The point p = [0 , , , , X . We then define J X − τ to be the anti-invariant part of the interme-diate Jacobian J X with respect to the induced involution τ on J X . The principal polarization on
J X induces a polarization on
J X − τ of type (1 , , ,
2) so that in the end we obtain a period map: P : M −→ A (1 , , , ( S, Π) J X − τ . As the moduli space of cubic surface pairs has dimension 7, while the moduli space of polarizedabelian fourfolds has dimension 10, the period map cannot be dominant. Our main result is thefollowing global Torelli theorem for P . Theorem 0.1 (Theorem 6.1, Corollary 5.8, Torelli theorem for P ) . The period map P : M →A (1 , , , (which sends ( S, Π) to the anti-invariant part J X − τ ) is injective and has injective differ-ential. We note that several other period maps for cubic hypersurfaces, and cubic hypersurface pairshave been studied recently. First, a related period map, for cubic threefold pairs, was constructed in[LPZ18] and [YZ18] in essentially the same way we constructed the period map P above, motivatingthe construction we use here. In another direction, Allcock, Carlson and Toledo [ACT02, ACT11]studied the moduli space of cubic surfaces (respectively, cubic threefolds) via the period map forcubic threefolds (respectively, cubic fourfolds) by considering the pairs ( P , S ) (respectively, ( P , X ))where S (respectively, X ) is a cubic surface (respectively, cubic threefold).Turning now to the proof of Theorem 0.1, it is well-known that two cubic threefolds are iso-morphic if and only if their intermediate Jacobians are isomorphic as principally polarized abelianvarieties. This can be proved either by studying the Gauss map on the theta divisor [CG72] orby analyzing the singularities of the theta divisor [Mum74, Bea82]. In our situation, the abelianvariety is not principally polarized, so that it is not immediately clear how to formulate a proof ofTheorem 0.1 in these terms. However, a third proof of the Torelli theorem for cubic threefolds canbe given by studying the fibers of the Prym map R → A over intermediate Jacobians of cubicthreefolds: by [DS81] (see also [Don92]) the fiber of the Prym map over an intermediate Jacobianof a cubic threefold is (a Zariski open subset of) the Fano surface of lines, which determines thecubic threefold up to isomorphism. We prove Theorem 0.1 in a similar way, via fibers of a Prymmap. We explain this in more detail below. The proof that the period map has injective differentialis proven via a more direct Hodge theoretic approach (Corollary 5.8).Further motivation for our approach to Theorem 0.1 via Prym varieties comes from the factthat realizing J X − τ as a Prym variety turns out to be important for an ongoing project of Laza,Pearlstein and the second named author, on the LSV construction [LSV17] applied to an Eckardtcubic fourfold; recall that the Prym construction of the intermediate Jacobian of a cubic threefoldis central to the work in [LSV17].In this direction, we prove that J X − τ is isomorphic to the dual abelian variety of the Prymvariety of a double cover of a smooth genus 3 curve branched at 4 points. For the associated Prymmap P , : R , −→ A (1 , , , , which is known to be dominant, and generically finite of degree 3 ([BCV95, NR95]; see also [NO19,Thm. 0.3]), we show that the fiber over the dual abelian variety ( J X − τ ) ∨ is isomorphic to (a Zariski pen subset of) the elliptic curve E = S ∩ Π, whose Jacobian is in turn isomorphic to the invariantpart
J X τ : Theorem 0.2 (Theorem 5.1, fiber of the Prym map) . Let X be an Eckardt cubic threefold comingfrom a cubic surface pair ( S, Π) , with involution τ . Consider the proper Prym map P , : R a , −→ A (1 , , , where R a , denotes the moduli space of branched allowable double covers (see §
5) of genus curvesbranched at points. The fiber of P , over the dual abelian variety of the anti-invariant part, ( J X − τ ) ∨ ∈ A (1 , , , , is isomorphic to the elliptic curve E = S ∩ Π . This is an analogue of the result of Donagi and Smith, mentioned above, that the fiber of thePrym map R a → A over the intermediate Jacobian of a smooth cubic threefold is isomorphic tothe Fano surface of lines on the cubic threefold [DS81, Don92]. We prove Theorem 0.2 using asimilar technique.In fact, Donagi and Smith develop a technique to study the degree of a generically finite map bystudying the local degree along positive dimensional fibers. Applying this to the case of the Prymmap over the intermediate Jacobian locus, they show that the Prym map R → A has degree 27.Geometrically, this comes down to the fact that a general hyperplane section of a cubic threefoldis a smooth cubic surface, which contains 27 lines. In a similar way, as a corollary of Theorem 0.2,we can recover the degree of the Prym map P , . Corollary 0.3 ([BCV95, NR95], [NO19, Thm. 0.3], degree of the Prym map) . The degree of thePrym map P , : R , → A (1 , , , is . More precisely, we show that the local degree of R , → A (1 , , , along a fiber E as in Theorem 0.2is 3, which implies the degree of the map is 3 (see § E with the hyperplane section).We now give an outline of the paper. We work throughout over the complex numbers C . InSection 1, we review the construction of Eckardt cubic threefolds X from cubic surface pairs ( S, Π)following [LPZ18]. We consider lines on Eckardt cubic threefolds, for future use with the Prymconstruction. We also explain the period map P : M → A (1 , , , . In Section 2 we review variousrealizations of cubic surfaces and cubic threefolds as fibrations in quadrics, as well as the associatedPrym construction due to Mumford showing that the intermediate Jacobian of a cubic threefold X is isomorphic to the Prym variety associated to the double cover of the discriminant curve obtainedby projection from a line ℓ ⊂ X (see [CG72, App. C]) and also [Bea77a, Bea77b]). Since our cubicthreefold X has an involution τ , in order to realize the involution on the intermediate Jacobian viathe Prym construction, we must choose a line ℓ ⊂ X that is preserved by the involution.It turns out there are two types of τ -invariant lines in X : the 27 lines ℓ on the cubic surface S ⊂ X , which are point-wise fixed by τ , and the lines ℓ ′ passing through the Eckardt point p , andare not point-wise fixed by τ (Lemma 1.16). This latter class of lines ℓ ′ form the ruling of the cone X ∩ T p X , and are parametrized by the elliptic curve E . In other words, the τ -fixed locus of theFano surface F ( X ) of lines in X consists of 27 isolated points and an elliptic curve isomorphic to E .The Prym construction then falls into two cases, depending on the type of τ -invariant linechosen. When projecting X from one of the 27 point-wise fixed lines ℓ (Section 7), we obtain an´etale double cover e D → D of a smooth plane quintic D (Proposition 7.9). Moreover, the involution τ acts naturally on e D , and we obtain a ( Z / Z ) automorphism group acting on e D . Using the ideas f Mumford [Mum74] (see also [Rec74, Don92, RR03, RR06]), one obtains a tower of double coverscorresponding to the various subgroups of ( Z / Z ) (Lemma 7.10), and one can identify J X − τ withthe quotient by a nontrivial 2-torsion point of the Prym of a double cover of a smooth genus 2curve branched at six points (Theorem 7.15). This naturally leads us to the Prym map: P , : R , → A (1 , , , for connected double covers of smooth curves of genus 2 branched at 6 points, which was studiedby Naranjo and Ortega [NO19, NO20], who show that the map is injective. In fact, a carefulanalysis using the results in [LO11] on the differential allows us to conclude that our period map P : M → A (1 , , , is generically finite onto its image. As a consequence, one obtains that P hasgenerically injective differential; we give a direct proof that P has injective differential at all pointsin Corollary 5.8.In contrast, when projecting X from a τ -invariant line ℓ ′ through the Eckardt point p (Section3), we get an allowable double cover of a reducible plane quintic obtained as the union a planequartic C and a residual line (Proposition 3.6). Restricting the cover to C one obtains a doublecover e C → C of a genus 3 curve branched at the 4 points of intersection of the residual line with C .We use Beauville’s theory [Bea77a, Bea77b] to study this cover, and find that J X − τ is isomorphicto the dual abelian variety of the Prym variety P ( e C, C ) (Theorem 3.11), leading us to the Prymmap: P , : R , → A (1 , , , for connected double covers of smooth curves of genus 3 branched at four points. As describedafter Theorem 0.2 above, we show that the fiber of P , over ( J X − τ ) ∨ is equal to the elliptic curve E parameterizing the τ -invariant lines through p . For this, we review in Section 4 Donagi–Smith’sargument for the fiber of the Prym map over the intermediate Jacobian of a cubic threefold. Wethen build on this in Section 5, to complete the proof of Theorem 0.2.In Section 6, we complete the proof of Theorem 0.1. The basic idea is that given two Eckardtcubics X and X , with J X − τ ∼ = J X − τ then Theorem 0.2 implies that the two polarized abelianvarieties arise from choosing different lines on the same Eckardt cubic threefold. The only ambiguityis the Eckardt point. For this, we have Theorem 3.8, showing that one can recover the Eckardtpoint.We make one final remark on automorphisms. Automorphisms of prime order of smooth cubicthreefolds have been classified in [GAL11]. Let us use the notation in [GAL11, Thm. 3.5]. Allcock,Carlson and Toledo study cubic threefolds admitting an automorphism of type T (which is oforder 3) in [ACT02]. An isogenous decomposition of the intermediate Jacobian of a cubic threefoldadmitting an automorphism of type T (which has order 5) is given by van Geemen and Yamauchiin [vGY16]. We study the intermediate Jacobian of an Eckardt cubic threefold (which admits aninvolution of type T ) in this paper. The study of intermediate Jacobians of cubic threefolds withother types of automorphisms will appear elsewhere. Acknowledgements.
We thank heartily Radu Laza and Gregory Pearlstein for many useful dis-cussions related to the subject. We thank Brendan Hassett for mentioning the connection withunlikely intersections in Shimura varieties. The first named author thanks Claire Voisin for a con-versation on isogenies of intermediate Jacobians, and Angela Ortega for a conversation on Prymvarieties of branched covers, both of which played an important role in our approach to this paper.The first named author also thanks Jeff Achter for some conversations about Shimura varieties.The second named author is grateful to Giulia Sacc`a for several valuable suggestions at the earlystage of the project. . Cubic threefolds with an Eckardt point
Definition of an Eckardt cubic threefold.
We follow [LPZ18, §
1] and briefly recall theconstruction and some geometric properties of a cubic threefold admitting an Eckardt point. Wealso fix the notation that will be used in the remainder of the paper.1.1.1.
Definition and characterizations of Eckardt cubic threefolds.
Definition 1.1 ([LPZ18] Definition 1.5) . Let X be a smooth cubic threefold. A point p of X is an Eckardt point if X ∩ T p X (where T p X denotes the projectivized tangent space of X at p ) is a conewith vertex p over an elliptic curve E .We call a pair ( X, p ) consisting of a smooth cubic threefold X with an Eckardt point p an Eckardtcubic threefold . Eckardt cubic threefolds can be characterized in the following way.
Proposition 1.2.
Let X be a smooth cubic threefold. The following statements are equivalent. • X admits an Eckardt point p . • X admits an involution τ which fixes point-wise a hyperplane section S ⊂ X and a point p ∈ ( X − S ) (which is an Eckardt point). • One can choose coordinates on P such that X has equation (1.3) f ( x , . . . , x ) + l ( x , . . . , x ) x = 0 , (in which case p = [0 , , , , is an Eckardt point).Proof. See [LPZ18, Lem. 1.6, Lem. 1.8]. (cid:3)
We now discuss Proposition 1.2 in coordinates. Let X be a cubic threefold in the coordinates ofEquation (1.3). Let p = [0 , , , , T p X = ( l = 0)is the projectivized tangent space to X at p and the intersection X ∩ T p X is a cone with vertex p over the elliptic curve(1.5) E = ( f = l = x = 0) . As a result, p is an Eckardt point of X . The involution τ of Proposition 1.2 is given by(1.6) τ : [ x , . . . , x , x ] [ x , . . . , x , − x ] , and the point-wise fixed hyperplane section S ⊂ X is given by(1.7) S = ( f = x = 0) . Equivalence between Eckardt cubic threefolds and cubic surface pairs.
We now review theone-to-one correspondence between Eckardt cubic threefolds (
X, p ) and cubic surface pairs ( S, Π)where S is a smooth cubic surface and Π is a transverse plane in P .First, an Eckardt cubic threefold ( X, p ) determines (up to projective linear transformation) acubic surface pair ( S, Π). Namely, the cubic surface S ⊂ X is the point-wise fixed hyperplane sectionof Proposition 1.2. The plane Π is easiest to define in coordinates. Starting with an Eckardt cubicthreefold X cut out by Equation (1.3), we have(1.8) S = ( f = x = 0) , Π = ( l = x = 0) , and view S (respectively, Π) as a cubic surface (respectively, a plane) in ( x = 0) ∼ = P . The baseelliptic curve E of the cone X ∩ T p X is the intersection of S and Π:(1.9) E = ( f = l = x = 0) = S ∩ Π . ecause X is smooth, by [LPZ18, Lem. 1.4] the cubic surface S is smooth, and S intersects theplane Π transversely (which together imply that the elliptic curve E is smooth).Conversely, a cubic surface pair ( S, Π) determines an Eckardt cubic threefold (
X, p ) (up to pro-jective linear transformation). Indeed, we assume the equation of the cubic surface S (respectively,the plane Π) in P is f ( x , . . . , x ) = 0 (respectively, l ( x , . . . , x ) = 0), and consider the cubicthreefold X with equation f ( x , . . . , x ) + l ( x , . . . , x ) x = 0. By rescaling x , we see that this isindependent of the choice of equations of S and Π. Since S and Π meet transversally, we have by[LPZ18, Lem. 1.3] that the cubic threefold X is smooth. Remark . A coordinate free description of X is to take the double cover Z → P branched alongthe singular quartic S ∪ Π, and then perform some birational modifications to obtain X (explicitly,one blows up Z along the reduced inverse image of S ∩ Π and then blows down the strict transformof the inverse image of Π in Z ; see [LPZ18, Prop. 1.9]). A similar correspondence between Eckardtcubic fourfolds and cubic threefold pairs has been used in [LPZ18] to construct a period map forthe moduli space of cubic threefold pairs.1.2. The period map for cubic surface pairs via Eckardt cubics.
In this subsection wedefine a period map P for cubic surface pairs ( S, Π) using Eckardt cubic threefolds. We shall provethe global Torelli theorem for P in Section 6.Let X be an Eckardt cubic threefold constructed from a cubic surface pair ( S, Π), as discussedabove. By abuse of notation, we use τ to denote the involution on the principally polarized inter-mediate Jacobian J X induced by the involution on X in (1.6). Define the invariant part J X τ andthe anti-invariant part J X − τ respectively by(1.11) J X τ = Im(1 + τ ) , and J X − τ = Im(1 − τ ) . By [BL04, Prop. 13.6.1],
J X τ and J X − τ are τ -stable complementary abelian subvarieties of J X .The dimensions of
J X τ and J X − τ are computed in the following lemma. Lemma 1.12.
The abelian subvarieties
J X τ and J X − τ have dimensions and respectively. Theprincipal polarization of J X induces polarizations of type (2) and (1 , , , on J X τ and J X − τ ,respectively.Proof. One may compute the dimensions of
J X τ and J X − τ by computing the dimensions of thepositive and negative eigenspaces for the action of τ on H , ( X ), or H , ( X ), for any particularEckardt cubic threefold X , such as the one given by F = x + x + x + x + x x = 0. Identifying theeigenspaces is a standard computation using Griffiths residues (see e.g., [CMSP17, Thm. 3.2.12]).The statement on the polarization type follows from Lemma 1.13, below. (cid:3) Lemma 1.13 ([Rod14, Thm. 5.3]) . Let ( A, Θ) be a principally polarized abelian variety, and assumethat τ : A → A is an involution such that τ ∗ Θ ≡ alg Θ . Defining the invariant and anti-invariantsub-abelian varieties A τ := Im(1 + τ ) , and A − τ := Im(1 − τ ) , respectively, we have that A τ and A − τ are complementary sub-abelian varieties of ( A, Θ) in thesense of [BL04, p.125, p.365] , and there is some non-negative integer r such that the polarizationtypes of the restrictions of Θ to A τ and A − τ are both of the form (1 , · · · , , , · · · , | {z } r ); (i.e., the number of s in the polarization type may differ for A τ and A − τ , but the number of s isthe same). Moreover, the exponent of A τ and A − τ is , unless r = 0 , in which case the exponentis , and then A = A τ × A − τ as principally polarized abelian varieties. roof. The assertion on polarizations follows from [Rod14, Thm. 5.3]. The entire lemma in factfollows from standard arguments via norm maps for abelian subvarieties of principally polarizedabelian varieties (see e.g., [BL04, Ch. 12]). (cid:3)
Remark . Given a polarized abelian variety ( A, Θ) of type ( d , . . . , d g ), the isogeny φ Θ : A → A ∨ induces the isogeny φ − : A ∨ → A . One can check that this is induced by a polarization Θ ∨ on A ∨ of type ( d g d g , d g d g − , . . . , d g d ), which is characterized by the fact that up to translation it is the onlyample line bundle on A ∨ such that φ Θ ∨ = φ − and φ ∗ Θ Θ ∨ ≡ alg d g Θ (e.g., the proof of [BL04,Prop. 14.4.1], where they take d φ − in place of φ − ; see also [BL03]). In particular, in the languageof Lemma 1.13, we have A/A τ ∼ = ( A − τ ) ∨ and A/A − τ ∼ = ( A τ ) ∨ have induced polarization of type(1 , · · · , | {z } r , , . . . , M be the moduli space of cubic surface pairs ( S, Π), where S is a smooth cubic surfaceand Π is a plane in P meeting S transversely, constructed for instance using GIT [GMG19]. Let A (1 , , , be the moduli space of abelian fourfolds with a polarization of type (1 , , , M = 4 + 3 = 7 and dim A (1 , , , = (cid:0) (cid:1) = 10. We define via Lemma 1.12 a period map:(1.15) P : M −→ A (1 , , , ( S, Π) J X − τ which sends a cubic pair ( S, Π) to the anti-invariant part
J X − τ of the intermediate Jacobian ofthe Eckardt cubic threefold X associated with ( S, Π).1.3.
Lines on an Eckardt cubic.
To study the period map P , we need to analyze how theintermediate Jacobian J X of the Eckardt cubic threefold (
X, p ) coming from a cubic surface pair( S, Π) decomposes with respect to the involution τ . Our strategy will be to project X from a τ -invariant line to exhibit J X as the Prym variety of the associated discriminant double cover. Inthe following lemma, we classify lines in X that are invariant under the involution τ . Lemma 1.16.
Let X be a cubic threefold with an Eckardt point p cut out by Equation (1.3). Let ℓ ⊂ X be a τ -invariant line (i.e., preserved by τ ). Then either ℓ is a line on the cubic surface S = ( f = x = 0) , or ℓ passes through the Eckardt point p = [0 , , , , and is contained in thecone X ∩ T p X over the elliptic curve E = ( f = l = x = 0) .In other words, there are two types of τ -invariant lines in X : the lines ℓ on the cubic surface S ⊂ X (which are point-wise fixed by τ ), and the lines ℓ ′ that pass through the Eckardt point p (which have only the points p , and p ′ := ℓ ′ ∩ E , fixed by τ ), and form the ruling of the cone X ∩ T p X over the elliptic curve E .Proof. The fixed locus of the involution τ : x
7→ − x of P consists of the Eckardt point p andthe hyperplane ( x = 0). If τ fixes every point of ℓ ⊂ X , then ℓ ⊂ ( x = 0) which implies ℓ ⊂ S .Otherwise, τ fixes two points of ℓ ⊂ X . One of the points needs to be off the hyperplane ( x = 0)and hence must be the Eckardt point p . Now, since ℓ ⊂ X , we have that ℓ ⊂ T p X , so that bydefinition, the other fixed point of ℓ is on the elliptic curve E . (cid:3) Cubic surfaces and cubic threefolds as fibrations in quadrics
In this section, we recall some facts about cubic hypersurfaces and fibrations in quadrics. Thekey point is to connect constructions that arise in the projection of a cubic threefold from a line,to the case of the projection of a cubic surface from a point or line. This connection between thetwo cases is central to our analysis of the intermediate Jacobian of an Eckardt cubic threefold. The definition of ψ L in the first line of the proof of [BL04, Prop. 14.4.1] should be ψ L = d φ − L , not ψ L = d d g φ − L . .1. Cubic surfaces as fibrations in quadrics.
Projecting a cubic surface from a point.
Let (
S, p ′ ) be a smooth cubic surface S ⊂ P togetherwith a point p ′ ∈ S . Projecting S from the point p ′ determines a rational map S P , which isresolved by blowing up the point p ′ , yielding a morphism π p ′ : Bl p ′ S −→ P . The discriminant curve (the locus in P where the fibers of the projection are singular) is a planequartic C , which is smooth if and only if p ′ does not lie on a line of S , in which case π p ′ is a fibrationin 0-dimensional quadrics (equivalently, a double cover of P ).Focusing on the case where the discriminant curve (equivalently, the branch locus of π p ′ ) issmooth, it is well-known that the 27 lines on the cubic surface, together with the exceptional divisoron the blow-up Bl p ′ S map isomorphically under π q to the 28 bitangents (odd theta characteristics)of C . In particular, pairs ( S, p ′ ) consisting of smooth cubic surfaces with a point not contained ina line on the surface, up to projective linear transformations, are in bijection with pairs ( C, κ C ),where C is a smooth plane quartic, and κ C is an odd theta characteristic, up to projective lineartransformations. One recovers ( S, q ) from (
C, κ C ) by taking the double cover of b S → P branchedalong C , and then, viewing κ C as a bitangent to C , one blows down the unique ( − b S that is in the pre-image of the bitangent line, to obtain S (see e.g., [Huy19, § S be a smooth cubic surface, and let p ′ ∈ S be a point (which for now may becontained in a line on S ). Without loss of generality, we assume that p ′ = [0 , , , S from p ′ to P x ,x ,x , we write the equation of S as(2.1) k ( x , x , x ) x + 2 q ( x , x , x ) x + c ( x , x , x ) = 0where k , q and c are polynomials of degree 1, 2 and 3 respectively. Note that p ′ is contained in a lineof S if and only if there is a fiber of π p ′ of dimension 1; i.e., if and only if the locus ( k = q = c = 0)is nonempty.Let N be the matrix(2.2) N = (cid:18) k qq c (cid:19) . The discriminant curve C ⊂ P is then defined by det N = kc − q = 0:(2.3) C = (det N = 0) = ( kc − q = 0) . We can now see that p ′ is not contained in a line of S if and only if the discriminant curve C issmooth. Indeed, suppose first that p ′ is not contained in a line of S . Then π p ′ is a branched doublecover, and if C had a node, then the branched double cover, Bl p ′ S , would be singular, giving acontradiction. Conversely, if p ′ is contained in a line of S , then we saw above that ( k = q = c = 0) isnonempty. But then at a point in that locus, one can easily see that the discriminant ( kc − q = 0)vanishes to order at least 2.We note also that the theta characteristic κ C on C is defined by the bitangent line ( k = 0):(2.4) κ C = p ( k = 0)where by the square root, we mean the unique effective divisor with twice the divisor giving thedivisor ( k = 0). Remark . Note that [Bea00, Prop. 4.2] gives another approach to recovering the smooth cubicsurface S and point p ′ from the smooth plane quartic C and the odd theta characteristic κ C . On he one hand, given ( S, p ′ ) in the coordinates of (2.1), there is a short exact sequence(2.6) 0 / / O P ( − ⊕ O P ( − N / / O P ( − ⊕ O P / / κ C / / . Conversely, given a smooth plane quartic C and an odd theta characteristic κ C , there is a presen-tation of κ C as in (2.6), with N as in (2.2). The cubic surface defined by (2.1) using the entries of N is smooth, and together with point p ′ = [0 , , , C and the associated theta characteristic is κ C .2.1.2. Projecting a cubic surface from a line.
Let (
S, ℓ ) be a smooth cubic surface S ⊂ P togetherwith a line ℓ ⊂ S . Projecting S from the line ℓ determines a rational map π ℓ : S P ; as theinduced projection P → P is resolved by blowing up the line ℓ , which is a divisor on S , we havethat π ℓ is a morphism, yielding a fibration in conics π ℓ : S −→ P . The discriminant is a degree 5 divisor D ⊂ P . A second degeneracy locus Q ⊂ P can be definedas the degree 2 divisor in P over which the associated conics in S meet ℓ with multiplicity 2 at apoint (we can put a well-defined scheme structure on Q ; see the description below in coordinates).We now describe this in coordinates. We may assume ℓ ⊂ P is cut out by x = x = 0, so thatthe equation of S is of the form(2.7) l ( x , x ) x + 2 l ( x , x ) x x + l ( x , x ) x +2 q ( x , x ) x + 2 q ( x , x ) x + c ( x , x ) = 0where l i , q j and c are homogeneous polynomials in x , x of degree 1, 2 and 3 respectively. Let M be the matrix(2.8) M = l l q l l q q q c . The degree 5 discriminant D ⊂ P x ,x for the fibration in quadrics is then defined by the determinantdet M :(2.9) D = (det M = 0)and the degree 2 discriminant divisor Q over which the associated conics in S meet ℓ with multi-plicity 2 at a point is given by determinant of the (3 , M , = (cid:18) l l l l (cid:19) (2.11) Q = (det M , = 0) = ( l l − l = 0) . Projecting a cubic surface with an elliptic curve from a line.
Suppose now we have a triple(
S, E, ℓ ) consisting of a cubic surface S , a smooth hyperplane section E ⊂ S and a line ℓ ⊂ S . Inthis case we obtain another discriminant divisor B ⊂ P , namely the degree 4 branch divisor ofthe the restriction π ℓ | E : E → P . Note that since E is smooth, the branch locus B consists of 4distinct points.In coordinates, again, we may assume ℓ ⊂ P is cut out by x = x = 0. Moreover, since E is irreducible (in fact smooth), it does not contain ℓ , so the linear equation ( l ( x , x , x , x ) = 0)cutting E on S must contain x or x with non-zero coefficient. Interchanging x and x , we mayassume it is x . Then after a change of coordinates, x l , x i x i , i = 1 , ,
3, we may assume hat l = x . Thus the equation for S in these coordinates is given by (2.7), and the equation for E on S is given by ( x = 0):(2.12) E = ( x = 0) ∩ S. Consequently, the branch divisor B is obtained by setting x = 0 in (2.7), and considering thediscriminant, it is therefore given by the determinant of the minor:(2.13) M , = (cid:18) l q q c (cid:19) (2.14) B = (det M , = 0) = ( l c − q = 0) . Condition 2.15 (Generic Eckardt cubic threefold) . Let ( X, p, ℓ ) be a triple consisting of an Eckartcubic threefold ( X, p ) with associated cubic surface pair ( S, E ) , together with a line ℓ ⊂ S . Theassociated discriminant divisors Q (2.11) and B (2.14) associated to the triple ( S, E, ℓ ) have disjointsupport. We say that an Eckardt cubic (
X, p ) satisfies Condition 2.15, if there exists a line ℓ ⊂ S suchthat ( X, p ) and ℓ satisfy Condition 2.15. Remark . We see, for instance from (2.11) and (2.14), that the locus of Eckardt cubics threefolds(
X, p ) that satisfy Condition 2.15 is open in moduli.
Remark . Although we do not need this, one can show that if (
X, p, ℓ ) satisfies Condition 2.15,then the supports of Q and B are both reduced. Moreover, one can show that if ( X, p ) satisfiesCondition 2.15, then for every ℓ ⊂ S , we have that ( X, p, ℓ ) satisfies Condition 2.15.2.2.
Cubic threefolds as fibrations in conics.
Let (
X, ℓ ) be a cubic threefold X ⊂ P togetherwith a line ℓ ⊂ X . Projecting X from the line ℓ determines a rational map X P , which isresolved by blowing up the line ℓ , yielding a fibration in conics:(2.18) π ℓ : Bl ℓ X −→ P . The discriminant is a plane quintic D ⊂ P with at worst nodal singularities, and there is anassociated pseudo-double cover e D → D determined by interchanging the lines in the fiber of π ℓ over the points of D .Recall that a double cover of stable curves e D → D with associated involution ι : e D → e D is called admissible if the only fixed points of the involution are nodes, and at each node of e D fixed by ι , thelocal branches of e D are not interchanged by ι . An admissible double cover is said to be allowable ([DS81, Don92], [Bea77a, p.173,(**)]) if the associated Prym is compact, and an allowable doublecover is said to be a pseudo-double cover ([Bea77b, Def. 0.3.1], [Bea77a, (*), p.157]) if, moreover,the fixed points of the associated involution ι on e D are exactly the singular points of e D .For general ℓ , one has that D is smooth and e D → D is ´etale. Associated to the double cover e D → D is a rank-1 torsion-free sheaf η D on D with H om ( η D , O D ) ∼ = η D , and therefore a thetacharacteristic κ D = η D ⊗ O D (1) ( H om ( κ D , ω D ) ∼ = κ D ), which is odd ( h ( D, κ D ) = 1). One hasthat η D (and therefore κ D ) is a line bundle at a point d ∈ D if and only if e D → D is ´etaleover d . In particular, pairs ( X, ℓ ) consisting of cubic threefold with a line, up to projective lineartransformations, are in bijection with pairs (
D, κ D ), where D is a stable plane quintic, and κ D isan odd theta characteristic, up to projective linear transformations. These results are all due toBeauville [Bea77b]; we refer the reader to [CMF05] where this is discussed further. In particular,we are using [CMF05, Thm. 4.1, Prop. 4.2]. We will explain this in more detail below, where weexpress everything in coordinates. A second degeneracy locus Q ⊂ P can be defined as the degree2 divisor in P over which the associated conics in X meet ℓ with multiplicity 2 at a point (we canput a well-defined scheme structure on Q ; see the description below in coordinates). e now describe this in coordinates. We may assume ℓ ⊂ P is cut out by x = x = x = 0, sothat the equation of X is of the form(2.19) l ( x , x , x ) x + 2 l ( x , x , x ) x x + l ( x , x , x ) x +2 q ( x , x , x ) x + 2 q ( x , x , x ) x + c ( x , x , x ) = 0where l i , q j and c are homogeneous polynomials in x , x , x of degree 1, 2 and 3 respectively.We note the similarity with Equation (2.7); namely, that equation is given by setting x = 0 in(2.19) above. Geometrically, this corresponds to restricting the fibration in conics here to the line( x = 0) ⊂ P x ,x ,x , or, equivalently, to projecting the cubic surface S = X ∩ ( x = 0) from thepoint p ′ = ℓ ∩ S . We will use similar notation to facilitate the translation from one case to theother.Let M be the matrix(2.20) M = l l q l l q q q c . The degree 5 discriminant D ⊂ P for the fibration in conics is then defined by the determinantdet M :(2.21) D = (det M = 0) . The fact that D has at worst nodes follows from the fact that X is smooth (e.g., [Bea77b,Prop. 1.2(iii)]), and the fact that D is stable can then be deduced from Bezout’s theorem. In fact,since the total space Bl ℓ X is smooth, we have that at any point of P , the corank of M is at most2, and d ∈ D is a smooth point of D if and only if the corank of M at d is equal to 1.The fact that e D → D is a pseudo-double cover is shown in [Bea77b, Prop. 1.5], and it is shownin [CG72] (see also [Mur72, Lem. 1.5]) that for a general line, i.e., outside of a divisor in the Fanosurface of lines on X , the discriminant D is smooth and the double cover is connected and ´etale.Moreover, Beauville has shown there is a short exact sequence (see [CMF05, Thm. 4.1])(2.22) 0 / / O P ( − ⊕ ⊕ O P ( − M / / O P ( − ⊕ ⊕ O P / / κ D / / . Conversely, given a pair (
D, κ D ) with D a stable plane quintic and η D an odd theta characteristic,one obtains a presentation of κ D as in (2.22), for a matrix M as in (2.20). The cubic threefold X defined by Equation (2.19) using the entries of M is smooth, and the projection from the line ℓ = ( x = x = x = 0) gives the discriminant D and odd theta characteristic η D up to projectivelinear transformations (see [CMF05, Prop. 4.2]).The degree 2 discriminant divisor Q over which the associated conics in S meet ℓ with multiplicity2 at a point is given by the determinant of the (3 , M , = (cid:18) l l l l (cid:19) (2.24) Q = (det M , = 0) = ( l l − l = 0) . The odd theta characteristic η D has yet another description, in terms of Q . Assuming that Q ∩ D meet at smooth points of Q and D , then one has that the intersection multiplicity at each point ofthe intersection is even, and we have that κ D = q (det M , = 0) = q ( l l − l = 0)where by the square root we mean the unique effective divisor such that twice the divisor is thedivisor ( l l − l ) = 0 on D . This is explained in the proof [CMF05, Prop. 4.2], where in this part of he proof it is assumed that Q and D are smooth; however, the computation is local, and requiresonly that Q and D meet at smooth points.Finally we recall [Bea77b, Thm. 2.1(iii)]: there is a canonical isomorphism of principally polarizedabelian varieties(2.25) P ( e D, D ) ∼ = J X. Eckardt cubic threefolds as fibrations in conics 1: lines through the Eckardtpoint
Let X be a cubic threefold with an Eckardt point p cut out by Equation (1.3). As in Section1, we denote by τ the involution associated with the Eckardt point p (see (1.6)) and set ( S, Π) tobe the corresponding cubic surface pair (see (1.8)). In this section, we give another description ofthe anti-invariant part
J X − τ of the intermediate Jacobian J X (see (1.11)) by projecting X froma τ -invariant line ℓ ′ ⊂ X passing through the Eckardt point p (note that the line ℓ ′ belongs to thecone X ∩ T p X over the elliptic curve E = S ∩ Π; see Lemma 1.16).We now revisit § X, p ) be an Eckardt cubic threefold, and let ℓ ′ ⊂ X be a line passingthrough the Eckardt point p . Let us choose coordinates so that X is given by (1.3) f ( x , . . . , x ) + l ( x , . . . , x ) x = 0, with Eckardt point p = [0 , , , , . Without loss of generality, we assume that ℓ ′ intersects the plane ( x = 0) at the point p ′ = [0 , , , , . Because ℓ ′ passes through the Eckardt point p = [0 , , , , ℓ ′ is ℓ ′ = ( x = x = x = 0) . We may now write the equation of X in the form αx + k ( x , x , x ) x + 2 q ( x , x , x ) x + c ( x , x , x ) + l ( x , x , x , x ) x = 0, where k , q and c are polynomials of degree 1, 2 and 3 respec-tively. Since X contains the line ℓ ′ = { [0 , , , x , x ] } , one finds that α = 0, and the coefficient of x in l ( x , . . . , x ) is zero.Thus we may write the equation of X as(3.1) k ( x , x , x ) x + 2 q ( x , x , x ) x + c ( x , x , x ) | {z } f ( x ,...,x ) + l ( x , x , x ) x = 0The cubic surface S is given by S = ( f = x = 0)and the hyperplane section E ⊂ S is given by E = ( l = 0) ∩ S = ( l = f = x = 0) . We now project X from the line ℓ ′ to the complementary plane P x ,x ,x = ( x = x = 0), andobtain a conic bundle π ℓ ′ : Bl ℓ ′ X → P x ,x ,x . Considering the form of (3.1) and (2.19), we findthat the associated matrix M (2.20) is M = k q l q c nd therefore, from (2.21), the discriminant plane quintic D of the conic bundle Bl ℓ ′ X → P isgiven by(3.2) det k q l q c = l ( kc − q ) = 0 . Clearly, the (nodal) discriminant plane quintic D ⊂ P consists of two components: a line L (3.3) L = ( l = 0)and a (possibly reducible) plane quartic C (3.4) C = ( kc − q = 0) . Geometrically, the plane quartic C is the discriminant for the projection of the cubic surface S from the point p ′ ∈ S . In fact, given the form of f in (3.1), and comparing with (2.1), then for themap π p ′ : Bl p ′ S → P , the associated matrix N (2.2) is given as N = (cid:18) k qq c (cid:19) so that following (2.3), the discriminant plane quartic for Bl p ′ S → P is given as in (3.4). Recallthat the odd theta characteristic κ C determining the pair ( S, p ′ ), described in (2.4), is given by(3.5) κ C = p ( k = 0)i.e., the pair ( S, p ′ ) is determined by the bitangent to C cut by ( k = 0). Proposition 3.6 (Projecting from a line with p ∈ ℓ ′ ⊂ X ) . Let ( X, p ) be an Eckardt cubic threefold,and let ℓ ′ ⊂ X be a line passing through p . The following are equivalent: (1) The line ℓ ′ does not intersect any lines on S . (From Lemma 1.16, the lines through p areparameterized by their intersection with E ⊂ S , so the general line through p satisfies thiscondition.) (2) The discriminant plane quintic D is a union D = C ∪ L of a smooth plane quartic C anda transverse line L , and the associated double cover e D → D is a pseudo-double cover.Proof. As explained in § e D → D is always a pseudo-double cover, and as explainedabove, in our situation, the discriminant D is always the union of a possibly reducible nodalplane quartic with a transverse line. In other words, the content of the proposition is to establishexactly when the plane quartic C is smooth. From the discussion above, the plane quartic C is thediscriminant for the projection of S from p ′ . We saw in § C is smoothif and only if p ′ does not lie on a line of S . (cid:3) Lemma 3.7.
Let ( X, p ) be an Eckardt cubic threefold, and let ℓ ′ ⊂ X be a line passing through p .We project X from ℓ ′ as in Proposition 3.6, and suppose that the quartic component C is smooth.Let e C → C be the associated double cover coming from the conic bundle Bl ℓ ′ X → P (in otherwords, we restrict the double cover e D → D in Proposition 3.6 to C ). The double cover e C → C determines the odd theta characteristic κ C , and conversely.Proof. The data of the double cover e C → C can be described equivalently as a triple ( C, β, L , s ),where β = C ∩ L is the branch divisor on C , L is a line bundle on C with an isomorphism L ⊗ ∼ = O C ( β ), and s is section of O C ( β ) vanishing on β . The claim is that L ∼ = κ C ; i.e., that( C, β, L , s ) gives the same cover as ( C, β, κ C , s ). Indeed, the cover associated to ( C, β, κ C , s ) hasfunction field K ( C )( √ k ) (3.5); here we are working on the open subset of C given by ( l = 0). At thesame time, [Bea77b, Lem. 1.6] states that the cover of the full discriminant e D → D , when restricted o the open subset of D given by ( l = 0) (which is exactly the cover e C → C restricted to the opensubset ( l = 0)) has function field K ( C )( √ k ); i.e., considering (3.2), the minor det M , = kl , and wecan choose coordinates on the plane so that local affine coordinates are given by setting l = 1. (cid:3) Just as Beauville (see § X, ℓ ) consisting of a smooth cubicthreefold X and a line ℓ ⊂ X is equivalent to the data ( D, η D ) consisting of a nodal plane quintic D and an odd theta characteristic κ D , we have a similar result for Eckardt cubic threefolds. Theorem 3.8 (Reconstructing an Eckardt cubic from a plane quartic) . Given a triple ( C, κ C , L ) consisting of a smooth plane quartic, an odd theta characteristic (bitangent) κ C on C , and a line L meeting C transversely, one can associate a triple ( X, p, ℓ ′ ) consisting of an Eckardt cubic threefoldand a line ℓ ′ on X through the Eckardt point p and not meeting any lines on the cubic surface S ⊂ X , such that ( C, κ C , L ) are associated to projection from ℓ ′ . In other words, the data of atriple ( X, p, ℓ ′ ) is equivalent to the data ( C, κ C , L ) . More precisely, a triple ( X ⊂ P , p, ℓ ′ ) recordingthe embedding of X in P determines a triple ( C ⊂ P , / / O P ( − ⊕ O P ( − N / / O P ( − ⊕ O P / / κ C / / , L ⊂ P ) recording the embedding of C ∪ L in P , and a presentation of κ C , and conversely.Proof. Assume we are given (
C, κ C , L ) as in the theorem. Recall from § C, κ C )determines a pair ( S, p ′ ) consisting of a smooth cubic surface S ⊂ P , and a point p ′ not lyingon any lines in S . Let f ( x , x , x , x ) = 0 be the equation of S , and let l ( x , x , x ) = 0 be theequation of L . As explained in § P such that f = k ( x , x , x ) x +2 q ( x , x , x ) x + c ( x , x , x ). Now consider the cubic hypersurface X ⊂ P defined by the equation(3.1); set ℓ ′ = ( x = x = x = 0) and p = [0 , , , , X is smooth, it is an Eckardtcubic threefold with Eckardt point p , since it admits the involution x → − x fixing the hyperplanesection S ⊂ X and the point p ∈ X − S (Proposition 1.2). Moreover, projection from ℓ ′ gives thetriple ( C, κ C , L ).We now show that X is smooth. This will follow from the fact that L meets C transversally.Let Π ⊂ P be defined by ( l = 0). We have seen that to show X is smooth it suffices to show that S and Π meet transversally ( § p ′ ∈ Π. To investigate the intersection S ∩ Π, let C ′ be the closure of the set of points s in S − p ′ such that the line p ′ s meets S tangentially. Forthe fibration in quadrics Bl p ′ S → P , one can check that the proper transform of C ′ is isomorphicto C (this is a special case of a more general statement about smooth fibrations in quadrics wherethe corank of the fibers is at most 1). Now suppose that Π does not meet S transversally at somepoint s ∈ S − p ′ . Then since p ′ ∈ Π, we would have that s ∈ C ′ , and it would follow that Π wasthe tangent plane to S at this point. But then the Zariski tangent space to C ′ ⊂ S at this pointwould be contained in Π, so that projecting from p ′ , we would have the tangent space to C beingcontained in L , which we have assumed is not true. Next let us rule out Π meeting S tangentiallyat p ′ ; i.e., Π = T p ′ S . In this case, the intersection E := S ∩ Π would be a singular plane cubic; sincewe have assumed that p ′ does not lie on a line of S , then E would have to be an irreducible nodalplane cubic, with node at p ′ . But then the two lines in Π forming the tangent cone C p ′ E wouldshow that C ′ passed through p ′ . Consequently, again, we would have that the Zariski tangent spaceto C ′ ⊂ S at p ′ would be contained in Π, so that projecting from p ′ , we would have the tangentspace to C being contained in L , which we have assumed is not true.Thus S and Π meet transversally, and so X is smooth, completing the proof. (cid:3) Let us now assume that ℓ ′ ⊂ X does not meet any lines on S , and let us describe the pseudo-double cover π : e D → D in more detail. First, Proposition 3.6 shows that D = L ∪ C , and thereforesince π is a pseudo-double cover it follows that it can be described as π : e D = e L ∪ e C −→ L ∪ C = D , here e L → L is a connected double cover of the line L branched at the four points L ∩ C , while e C → C is a connected double cover of the smooth plane quartic C branched also at the four points C ∩ L . Considering the equation (3.1) for X , and the fact that L is defined by ( l = 0) (3.3), we seethat the two lines in X lying over a point in L via the projection π ℓ ′ are parameterized by the twopoints in S lying over the point in ℓ via the projection π p ′ . In other words, all together we obtainthe intersection S ∩ Π = E ; i.e., e L ∼ = E . Thus the pseudo-double cover can be described as:(3.9) π : e D = E ∪ e C −→ L ∪ C = D. We now give an explicit description of the Prym variety P ( E ∪ e C, L ∪ C ) following [Bea77b, § e ν : E ∐ e C → E ∪ e C and ν : L ∐ C → L ∪ C be the normalizations of E ∪ e C and L ∪ C respectively.Let π ′ : E ∐ e C → L ∐ C be the double covering map induced by π . Denote the ramification pointsof e C → C (respectively, E → L ) by e c , e c , e c , e c ∈ e C (respectively, e , e , e , e ∈ E ). Note that e ν ( e c i ) = e ν ( e i ) for 1 ≤ i ≤
4. Set H ′ to be the subgroup of Pic( e C ∐ E ) generated by O ( e c i − e i )(1 ≤ i ≤ H be the image of H := H ′ ∩ J ( e C ∐ E ) in the quotient abelian variety J ( E ∐ e C ) /π ′∗ J ( L ∐ C ) ∼ = E × ( J ( e C ) /π ∗ J ( C )). By [Bea77b, Exer. 0.3.5], the group G consists of2-torsion elements and is isomorphic to ( Z / Z ) . More explicitly, H = { , ( O E ( e − e ) , O e C ( e c − e c )) , ( O E ( e − e ) , O e C ( e c − e c )) , ( O E ( e − e ) , O e C ( e c − e c )) } . Recall that J ( e C ) /π ∗ J ( C ) is the dual abelian variety to P ( e C, C ) and therefore comes with a dualpolarization (Remark 1.14). We equip E with its canonical principal polarization. Note also thatas in (7.6), the involution τ on X induces an involution σ on e D . Proposition 3.10.
Let ( P ( e D, D ) , Ξ) be the principally polarized Prym variety. There is an isogenyof polarized abelian varieties φ : E × ( J ( e C ) /π ∗ J ( C )) −→ P ( E ∪ e C, L ∪ C ) = P ( e D, D ) with kernel H ∼ = ( Z / Z ) .Moreover, with respect to the action of σ on ( P ( e D, D ) , Ξ) , the isogeny φ induces isomorphismsof polarized abelian varieties P ( e D, D ) σ ∼ = E and P ( e D, D ) − σ ∼ = J ( e C ) /π ∗ J ( C ) .Proof. The existence of the isogeny φ and the description of the kernel is [Bea77b, Prop. 0.3.3] Wenow explain the assertion regarding P ( e D, D ) σ and P ( e D, D ) − σ . Let ι be the covering involutionassociated with the double cover π : E ∪ e C → L ∪ C . We claim that the action of σ on E is trivial,while the action of σ on e C coincides with ι . Recall that E ∪ e C parameterizes the residual linesto ℓ ′ in a degenerate fiber of the conic bundle π ℓ ′ : Bl ℓ ′ X → P x ,x ,x . We describe the involution σ on E and on e C respectively using Equation (3.1). To this end, let us consider a point x on P x ,x ,x = ( x = x = 0) ⊂ P . The plane it parameterizes is the span h ℓ ′ , x i . Since ℓ ′ and x arepreserved by τ ( x
7→ − x ), then so is the span h ℓ ′ , x i . Assume now that x is in L or C , but notboth. Then the intersection of h ℓ ′ , x i with X consists of three distinct lines, ℓ ′ ∪ m ∪ m ′ , with m and m ′ corresponding to points of E ∪ e C . Since h ℓ ′ , x i is preserved by τ , it follows that either τ interchanges m and m ′ , or it takes m to m and m ′ to m ′ . In other words, we see that σ either actsas the identity on a point of E ∪ e C , or by ι . Now, since the only fixed points of the action of τ on ℓ ′ are the points p = [0 , , , ,
1] and p ′ = [0 , , , , m and m ′ are not interchanged isif one of them passes through p (which forces the other to pass through p ′ ). It is easy to see fromEquation (3.1) m or m ′ passes through p if and only if l ( x ) = 0. Thus, the induced involution σ on e C (respectively, E ) coincides with the covering involution ι (respectively, the identity map), asclaimed. t follows that the isogeny φ is equivariant with respect to the involution = (1 , ι ) on the product E × ( J ( e C ) /π ∗ J ( C )). Since π ∗ J ( C ) = Im(1 + ι ) on J ( e C ), it follows that Im(1 + ) = E × { } .Similarly, Im(1 − ) = { } × J ( e C ) /π ∗ J ( C ). From a dimension count, we have then that φ inducesisogenies E × { } → P ( e D, D ) ι and { } × ( J ( e C ) /π ∗ J ( C )) → P ( e D, D ) − ι .Since H ∩ ( E × { } ) = H ∩ ( { } × ( J ( e C ) /π ∗ J ( C ))) = { (0 , } , these isogenies are isomorphisms.That these are isomorphisms of polarized abelian varieties follows from the fact that φ is a morphismof polarized abelian varieties, and the fact that we are taking the product polarization on E × ( J ( e C ) /π ∗ J ( C )). In other words, if we take Ξ, pull it back to E × ( J ( e C ) /π ∗ J ( C )), and then restrictto a component, we get the same thing as if we take Ξ, restrict it to an invariant (respectively,anti-invariant) part, and then pull-back via this isomorphism. (cid:3) We summarize the situation for Eckardt cubic threefolds in the following theorem.
Theorem 3.11 (Projecting from a line p ∈ ℓ ′ ⊂ X ) . Let ( X, p ) be an Eckardt cubic threefold,and let ℓ ′ ⊂ X be a line passing through p such that ℓ ′ does not meet any lines in the associatedcubic surface S . Let E ⊂ S be the hyperplane section, and let π : e C → C be the restriction ofthe descriminant cover to the irreducible component C that is a smooth plane quartic. We identify J ( e C ) /π ∗ J ( C ) as the dual abelian variety to P ( e C, C ) , with the dual polarization. There is an isogeny φ : E × ( J ( e C ) /π ∗ J ( C )) −→ J X with ker φ ∼ = ( Z / Z ) .Moreover, with respect to the action of τ on ( J X, Θ X ) , the isogeny φ induces isomorphisms ofpolarized abelian varieties J X τ ∼ = E and J X − τ ∼ = J ( e C ) /π ∗ J ( C ) .Proof. Since
J X ∼ = P ( e D, D ), and the action of τ on J X is identified with the action of σ on P ( e D, D ), this is an immediate consequence of Proposition 3.10. (cid:3) Recalling the fiber of the Prym map over the cubic threefold locus
In this section, we recall Donagi–Smith’s result on the fiber of the Prym map over the inter-mediate Jacobian locus, as some details in various steps in that proof are central to our proof ofTheorem 5.1 on the fiber of a related Prym map for Eckardt cubic threefolds.Recall that an admissible double cover of curves is called an allowable double cover if the as-sociated Prym is compact (e.g., [DS81, § I.1.3] and [Bea77a, (**), p.173]). The first result we willreview is:
Theorem 4.1 ([DS81, Thm. V.1.1]) . Let X be a smooth cubic threefold. The fiber of the Prymmap for allowable double covers of genus curves P : R a −→ A over the intermediate Jacobian J X ∈ A is isomorphic to the Fano surface of lines F ( X ) : P − ( J X ) ∼ = F ( X ) . Remark . We state Theorem 4.1 for the moduli stacks. For the coarse moduli spaces, we have P − ( J X ) ∼ = F ( X ) / Aut( X ).Let us note here that from the proof of Theorem 4.1, one obtains: Corollary 4.3 ([DS81, Thm. I.2.1]) . The Prym map P : R a −→ A has degree , corresponding to the lines contained in a general hyperplane section of a smoothcubic threefold X . emark . The proof of Theorem 4.1 use the fact that the degree of the Prym map P is equalto 27, so that Corollary 4.3 does not provide a new derivation of the degree. The point is that onerecovers the degree of the Prym map in terms of the geometry of cubic threefolds.4.1. Strategy for computing the fiber.
We start by reviewing Donagi–Smith’s strategy forcomputing the degree of a generically finite morphism by studying the differential along positivedimensional fibers. We provide here a slightly different presentation, which is perhaps more stream-lined for our purposes.Suppose that f : Y ′ → Y is a proper, surjective, generically finite morphism of integral separatedschemes of finite type over a field. For any closed subscheme Z ⊂ Y , we have the fibered productdiagram.(4.5) Z ′ (cid:31) (cid:127) / / f ′ (cid:15) (cid:15) Y ′ f (cid:15) (cid:15) Z (cid:31) (cid:127) / / Y Let e Y → Y (respectively, e Y ′ → Y ′ ) be the blow-up of Y along Z (respectively, Y ′ along Z ′ ),with exceptional divisor e Z = P C Z Y (respectively, e Z ′ = P C Z ′ Y ′ ). Setting e f : e Y ′ → e Y be theinduced morphism on blow-ups, then the induced morphism of exceptional divisors e f | e Z ′ : e Z ′ → e Z is generically finite, and(4.6) deg f = deg( e f | e Z ′ : e Z ′ → e Z );i.e., the degree of the map f is equal to the degree of the blow-up when restricted to the exceptionaldivisors. Indeed, since deg f = deg( e f : e Y ′ → e Y ), we can immediately reduce to the case where Z and Z ′ in (4.5) are Cartier divisors, in which case we are trying to show deg f = deg f ′ . Then using[Ful98, Prop. 4.2(a), Cor. 4.2.1], we have have f ′∗ ((1 − [ Z ′ ]) ∩ [ Z ′ ]) = f ′∗ s ( Z ′ , Y ′ ) = (deg f ) s ( Z, Y ) =(deg f )((1 − [ Z ]) ∩ [ Z ]). Looking at the component of [ Z ], and the fact that by definition f ∗ [ Z ′ ] =(deg f ′ )[ Z ], we have deg f = deg f ′ .Conversely, once one knows the degree of a generically finite map, one can use this to study thefibers: Lemma 4.7 (Identifying fibers [DS81, § . In the notation above, assume Y is geometricallyunibranch (see [Sta19, Tag 0BQ2, Tag 0BPZ] ; e.g., Y is normal [Sta19, Tag 0BQ3] ), Z is integral,and Z ′ ⊂ Z ′ is a connected component of Z ′ dominating Z , with Z ′ integral, and with the genericfiber of Z ′ → Z connected. Let e Z ′ be the exceptional divisor in the blow-up of Y ′ along Z ′ , andlet e f : e Z ′ e Z ′ → e Z be the associated rational map. If deg e f ≥ deg f , then deg e f = deg f , and Z ′ = Z ′ .Proof. Clearly deg e f ≤ deg( e f | e Z ′ : e Z ′ → e Z ) = deg e f = deg f (4.6), giving the stated equalityof degrees. On the other hand, once we have deg e f = deg f , we see that Z ′ cannot have anyother irreducible components dominating Z (otherwise they would increase the degree count of e f : e Z ′ → e Z ). The assumption that the generic fiber of Z ′ → Z is connected, together with thebasic Lemma 4.10 below, rules out components of Z ′ that do not dominate Z . (cid:3) Remark . The hypothesis that Z ′ be a connected component of Z is necessary (i.e., it is notenough to assume only that Z ′ be a reduced irreducible component of Z ′ ). Indeed, let f : Y ′ → Y be the blow-up of a smooth surface at a point, and let Z ⊂ Y be a smooth irreducible curve throughthe point. Taking Z ′ to be the strict transform of Z , we have that deg( e f : Z ′ → Z ) = 1 ≥ deg f ;however, Z ′ = Z ′ . The hypothesis that Y be geometrically unibranch is also necessary. Indeed, let f : Y ′ → Y be the normalization of an integral nodal curve, let Z ⊂ Y be a node, and let Z ′ beone of the points in Y ′ lying over the node Z . Then deg e f = 1 ≥ deg f , but Z ′ = Z ′ . emark . In this paper we will typically be applying Lemma 4.7 to the quasi-projective coarsemoduli spaces of certain smooth Deligne–Mumford (DM) stacks. The key point is that the coarsemoduli space of a smooth DM stack, locally being the quotient of a smooth space by a finite group,is normal.We now recall a variation on a standard result [DM69, Thm. 4.17(ii)] ([Sta19, Lem. 0BUI]) onconnected components of fibers of maps:
Lemma 4.10.
Let f : X → S be a proper surjective morphism of finite presentation betweenintegral schemes, with S geometrically unibranch. Let n X/S be the function on S counting thenumbers of geometric connected components of fibers of f (e.g., [Sta19, Lem. 055F] ). Then n X/S is lower semi-continuous.Proof.
The proof is essentially identical to that of the standard result [Sta19, Lem. 0BUI]. Forbrevity, the proof is left to the reader. (cid:3)
The Donagi–Smith argument for cubic threefolds.
Suppose now we have a cubic three-fold X ⊂ P together with a line ℓ ⊂ X giving rise to the odd connected ´etale double cover π : e D → D of the smooth plane quintic discriminant D ⊂ P . As in the discussion above, assumethat X = ( F = 0) and D = ( Q = 0).Let R a Q denote the moduli space of allowable odd double covers of plane quintics, and let C denote the moduli space of cubic threefolds. From the commutative diagram (which at this pointwe have not proven is cartesian) R a Q / / (cid:15) (cid:15) R a (cid:15) (cid:15) C / / A we obtain a commutative diagram of codifferentials:(4.11) T ∨ π R a Q T ∨ π R a o o o o ( T π R a /T π R a Q ) ∨ ? _ o o R Q H ( D, O D (4)) o o o o J Q = C ? _ o o T ∨ X C O O T ∨ JX A o o o o O O ( T JX A /T X C ) ∨ ? _ o o O O R F O O H ( P , O P (2)) o o o o O O J F = C ? _ o o O O Here R F = C [ x , . . . , x ] / ( ∂F∂x , . . . , ∂F∂x ) (respectively, J F = ( ∂F∂x , . . . , ∂F∂x )) is defined to be theJacobian ring (respectively, Jacobian ideal), with the superscript denoting the vector space offorms of a given degree; we define R Q and J Q similarly.In fact, we will need to consider the case where π : e D → D is a pseudo-double cover of a nodalplane quintic. We would like to have a diagram as above, but this requires a bit more work. Recallthat for a nodal curve D there is a natural map j : Ω D → ω D with torsion kernel and cokernel(e.g., [DS81, § j : Ω D ⊗ ω D → ω ⊗ D . Lemma 4.12.
Suppose X = ( F = 0) ⊂ P is a smooth cubic threefold, ℓ ⊂ X is a line, and π : e D → D is the associated odd pseudo-double cover of the discriminant plane quintic D = ( Q = ) ⊂ P . There is a commutative diagram of codifferentials: (4.13) R Q H ( D, O D (4)) o o o o J Q = C ? _ o o T ∨ π R a Q T ∨ π R a o o o o ( T π R a /T π R a Q ) ∨ ? _ o o ( R Q ) ∨ O O H ( D, Ω D ⊗ O D (2)) o o o o j ∗ O O K = C ? _ o o O O T ∨ X C O O T ∨ JX A o o o o O O ( T JX A /T X C ) ∨ ? _ o o O O R F O O H ( P , O P (2)) o o o o O O J F = C ? _ o o O O where K is defined as the kernel of the map H ( D, Ω D ⊗ O D (2)) → ( R Q ) ∨ , and the morphism K → J Q is an isomorphism.Proof. We direct the reader to [DS81, § (cid:3) For convenience, we now recall [DS81, Lem. V.1.5, p.88] describing the map J F → J Q in (4.11)and (4.13). Lemma 4.14 ([DS81, Lem. V.1.5, p.88]) . Let X ⊂ PC ∼ = P be a smooth cubic threefold givenby F = 0 , and let ℓ ⊂ X be a line giving rise to the odd pseudo-double cover π : e D → D of thediscriminant plane quintic D ⊂ PC ∼ = P given by Q = 0 . Take coordinates on C such that ℓ =( x = x = x = 0) , so that projection from ℓ gives the map π ℓ : PC PC , [ x , x , x , x , x ] [ x , x , x ] . Under the identifications C ∼ / / J Q C ∼ / / J F ( a , a , a ) ✤ / / P i =2 a i ∂Q∂x i ( a , a , a , a , a ) ✤ / / P a i ∂F∂x i then projectivization of the morphism J F → J Q of (4.13) is the rational map π ℓ . The projectiviza-tion of the dual map ( J Q ) ∨ → ( J F ) ∨ is therefore the inclusion ( P ℓ ) ∨ ⊂ ( P ) ∨ , where we are denotingby ( P ℓ ) ∨ the -plane in ( P ) ∨ corresponding to hyperplanes containing ℓ . (cid:3) We now recall the completion of the proof of Theorem 4.1:
Donagi–Smith’s proof of Theorem 4.1.
We apply the strategy of (4.5) and Lemma 4.7 to the dia-gram(4.15) R a Q / / $ $ ■■■■■■■■■■■ C × A R a / / (cid:15) (cid:15) R a P (cid:15) (cid:15) C / / A where R a Q , the moduli space of odd pseudo-double covers of nodal plane quintics, plays the role of Z ′ .Lemma 4.14 implies that the differential of the Prym map R a → A at a cover ( π : e D → D ) ∈ R a Q has kernel exactly of dimension 2. Thus at π , the fiber of P over P ( e D, D ) =
J X is exactly theFano surface of lines F ( X ). Since this holds at every cover π , this implies that R a Q is a connectedcomponent of the fiber over C , with generic fiber of R a Q over C connected.At the same time, let us use Lemma 4.14 to compute the degree of the map of exceptional divisors g R a Q → e C ; here g R a Q (respectively e C ) is the exceptional divisor in the blow-up of R a (respectively A )along the locus R a Q (respectively C ). For this, we can pick a general cubic threefold X , and a generalpoint in the projectivization of the fiber of the normal bundle, corresponding, by Lemma 4.14, to general hyperplane H ⊂ P . Lemma 4.14 says that the points lying above this chosen pointcorrespond to those lines ℓ ⊂ X that lie in the hyperplane section H . Since X ∩ H is a smoothcubic surface, there are 27 such lines.Since the degree of the Prym map is 27 [DS81, Thm. I.2.1], we can conclude from Lemma 4.7that R a Q = C × A R a , and we are done. (cid:3) Donagi–Smith’s proof of Corollary 4.3.
This now follows immediately from (4.6), and the proof ofTheorem 4.1. (cid:3) Fiber of the Prym map R , → A (1 , , , over the Eckardt cubic threefold locus Our goal is to describe the fiber of the generically finite degree 3 Prym map P , : R , → A (1 , , , over the Eckardt cubic locus; recall that R , is the moduli space of double covers of smooth genus3 curves branched at 4 points, and A (1 , , , is the moduli space of abelian varieties of dimension 4with a polarization of type (1 , , , P , , it is convenient to consider a partial compactification R a , of R , , over which the Prym map extends to a proper map P , : R a , → A (1 , , , . As mentioned in § branchedadmissible covers ), and in fact, we prefer to use the Abramovich–Corti–Vistoli description [ACV03].More precisely, we denote by R g,r the moduli space of branched admissible double covers where thebase curve is a stable curve of genus g with r unordered marked points, and the branch points ofthe cover are given by those r marked points; in the language of [ACV03, p.3560 and Prop. 4.2.2],the space R g,r is the quotient of B bal g,r ( S ) by the symmetric group S r acting by permuting thelabeling of the r points. For any branched admissible double cover, the connected component ofthe kernel of the norm map defines a semi-abelian Prym variety [Bea77a, §
5] [ABH02, § branched allowable doublecovers . From the deformation theory of R g,r [ACV03, § R ag,r ⊂ R g,r of branched allowable double covers. The Prym construction then gives a period map P g,r : R ag,r −→ A Dg − r where D = (1 , . . . ,
1) if r = 0 ,
2, and otherwise D = (1 , . . . , , . . . ,
2) with r − P , , recall that given an Eckart cubic threefold ( X, p ), there is anassociated cubic surface S , together with a smooth hyperplane section E ⊂ S . This elliptic curveis naturally contained in the Fano surface F ( X ) of lines, with E parameterizing the τ -invariantlines in X through the Eckardt point p ; i.e., we may view E as the 1-dimensional component ofthe τ -fixed locus of the Fano surface F ( X ), and E is isomorphic to the elliptic curve S ∩ Π; seeLemma 1.16.
Theorem 5.1.
Let ( X, p ) be an Eckardt cubic threefold with involution τ . The fiber of the Prymmap for allowable double covers of genus curves branched at points P , : R a , −→ A (1 , , , ver the dual abelian variety ( J X − τ ) ∨ ∈ A (1 , , , of the anti-invariant part of the intermediateJacobian J X is isomorphic to the elliptic curve E : P − , (cid:0) ( J X − τ ) ∨ (cid:1) ∼ = E ⊂ F ( X ) . Remark . We state this theorem for the moduli stacks. For the coarse moduli spaces, we have P − , (( J X − τ ) ∨ ) ∼ = E/ Aut(
X, p ); note that the Eckardt automorphism τ ∈ Aut(
X, p ) acts triviallyon E .We prove this theorem over the course of this section. Let us note here that we will also obtainthe following corollary: Corollary 5.3 ([BCV95, NR95], [NO19, Thm. 0.3]) . The Prym map P , : R a , −→ A (1 , , , has degree , corresponding to the three lines through the Eckardt point p contained in a generalhyperplane section of an Eckardt cubic threefold ( X, p ) through the Eckardt point.Remark . The proof of Theorem 5.1 uses the fact that the degree of the Prym map P , is equalto 3, which is due to [BCV95, NR95, NO19], so that Corollary 5.3 does not provide a new derivationof the degree. The new result is that one recovers the degree of the Prym map in terms of thegeometry of Eckardt cubic threefolds.To prove Theorem 5.1, we use the same strategy as in the previous section. Namely, we applythe strategy of (4.5) and Lemma 4.7 to the diagram(5.5) R a Q , E / / & & ▼▼▼▼▼▼▼▼▼▼▼▼▼▼ C E × A (1 , , , R a , / / (cid:15) (cid:15) R a , P , (cid:15) (cid:15) C E / / A (1 , , , where the image of C E plays the role of Z , and the image of R a Q , E plays the role of Z ′ . Here C E isthe moduli space of Eckardt cubic threefolds ( X, p ), and R Q , E is the moduli space of double covers π : e C → C over smooth plane quartics branched at the four points of intersection of a transverseline, determined by an odd theta characteristic (see Lemma 3.7). In other words, R a Q , E is the spaceof covers arising from projecting an Eckardt cubic threefold from a line through the Eckardt point,and restricting the cover to the plane quartic in the discriminant.We therefore start by computing the differentials of the morphisms above.5.1. Differential to the period map for Eckardt cubic threefolds.
Let C E be the modulispace of Eckardt cubic threefolds ( X, p ). Assuming that (
X, p ) is a smooth Eckardt cubic threefolddefined by F = 0 as in (1.3), and R F is defined to be the Jacobian ring, we saw in the proof ofLemma 1.12 that H , ( X ) − τ = ( R F ) − τ and H , ( X ) − τ = ( R F ) − τ , so that we also have T ( JX − τ ) ∨ A (1 , , , = SymHom( H , ( X ) − τ , H , ( X ) − τ ) = SymHom(( R F ) − τ , ( R F ) − τ ) . At the same time, we have T ( X,p ) C E = ( T X C ) τ = ( R F ) τ . Therefore, with regards to the period map C E −→ A (1 , , , we obtain a map of tangent spaces T ( X,p ) C E −→ T ( JX − τ ) ∨ A (1 , , , R F ) τ −→ SymHom(( R F ) − τ , ( R F ) − τ ) . Using Macaulay’s Theorem to identify R F ∼ = ( R F ) ∨ , our differential is( R F ) τ −→ Sym (( R F ) − τ ) ∨ , and the dual map, the codifferential, is canonically H ( P , O P (2)) = Sym (( R F ) τ ) −→ ( R F ) τ . Here we are using the identification R F = C h x , . . . , x i , where the action of τ is given by x
7→ − x ,so that ( R F ) τ = C h x , . . . , x i . We see that the codifferential is the map that takes a quadric in P = ( x = 0), views it as an invariant quadric on P , and then sends it to its class in the invariantJacobian ring. Algebraically, the kernel of this map is clearly J F ∩ Sym C h x , . . . , x i . Usingthe equation (1.3) for F , we see that J F consists of quadrics of the form a ( ∂f∂x + ∂l∂x x ) + · · · + a ( ∂f∂x + ∂l∂x x ) + 2 a lx . In order for this not to contain x , we must have a = 0, and also0 = P a i ∂l∂x i = l ( a , . . . , a ). We therefore have:(5.7) ( T ( JX − τ ) ∨ A (1 , , , /T ( X,p ) C E ) ∨ ∼ = J F ∩ Sym C h x , . . . , x i ∼ = ( l = x = 0) ⊂ C . We identify J F ∩ Sym C h x , . . . , x i with ( l = x = 0) ⊂ C via ( a , . . . , a ) P i =0 a i ∂F∂x i , andcall these the quadratics X t polar to points t = ( a , . . . , a ,
0) of ( l = x = 0) ⊂ P with respect to X . Corollary 5.8 (Infinitesimal Torelli theorem) . The differential of the period map C E → A (1 , , , is injective.Proof. We consider (5.6). Since dim( R F ) τ = 7, dim Sym ( R F ) − τ = 10, and the cokernel of (5.6)has dimension 3 (5.7), it follows that (5.6) is injective. (cid:3) Differential of the inclusion of the moduli space of double covers of smooth planequartics branched at four collinear points into the moduli space of double covers ofgenus curves branched at four points. We let R Q , E be the moduli space of double covers π : e C → C over smooth plane quartics branched at the four points of intersection of a transverseline, determined by an odd theta characteristic (see Lemma 3.7), and we consider the inclusion R Q , E −→ R , . Giving a cover π : e C → C in R Q , E is equivalent to giving the plane quartic C , the branch points ofthe cover, which are themselves determined by a line L , and an odd square root of the branch divisor.Thus the space R Q , E has dimension 6 + 2 + 0 = 8. Moreover, the first order deformations of π ascovers in R Q , E are identified with the first order deformations of the plane quintic C ∪ L that remainthe union of a plane quartic and a line. Up to change of coordinates, we may assume that the quniticpolynomial Q defining C ∪ L is Q = gx for some quartic g . Then T π R Q , E ⊂ T e C ∪ e L → C ∪ L R Q ∼ = R Q can be identified with the span of the quintic monomials divisible by x ; indeed, these remain theunion of a plane quartic and a line and then a dimension count gives the equality. Denote thisspace by R Q | x . More canonically, one can see that these deformations are exactly the locally trivialdeformations of the plane quintic C ∪ L , and are then identified with H m ( R Q ) , as in Remark 5.9,with dual H m ( R Q ) . Remark . If D is a singular plane quintic,then the Jacobian ring no longer satisfies the duality of Macaulay’s theorem. However, there is stilla natural map(5.10) ( R Q ) ∨ → R Q nd the image is identified with the dual of the space of locally trivial deformations. More precisely,fix the maximal ideal m = ( x , x , x ) in the Jacobian ring R Q . There is an inclusion of gradedrings [Ser14, (11)] H m ( R Q ) ⊂ R Q and the ring H m ( R Q ) satisfies a similar duality as in Macaulay’stheorem; namely, there is an isomorphism H m ( R Q ) i ∼ = ( H m ( R Q ) − i ) ∨ [Ser14, Thm. 3.4] (we notethat this isomorphism is more subtle than in Macaulay’s theorem; see [Ser14, Rmk. 3.5]). Thespace H m ( R Q ) ⊂ R Q is identified with the first order locally trivial deformations [Ser14, Cor. 2.2].In summary, we have morphisms ( R Q ) ∨ → ( H m ( R Q ) ) ∨ ∼ = H m ( R Q ) ⊂ R Q , and the image of thecomposition ( R Q ) ∨ → R Q is identified with the dual of the space of locally trivial deformations, asclaimed.Another way to verify that H m ( R Q ) = R Q | x is to use the identification H m ( R Q ) = J sat f /J f [Ser14, (5)]. In fact, by [Ser14, (11)] when d is sufficiently large, the graded piece of our Jacobianring J dQ is of dimension 4 (corresponding to the four nodes C ∩ L of the quintic C ∪ L ) and consists ofmonomials which are not divisible by x (corresponding to the infinitesimal deformations smoothingthe four nodes). Using a similar argument, we get that H m ( R Q ) = R Q | x .At the same time, T π R , = H ( C, T C ⊗ O C ( L )) = H ( C, O C (3)) ∨ , and so the differential canbe described as a map T π R Q , E −→ T π R , H m ( R Q ) → H ( C, O C (3)) ∨ . By the above argument, the codifferential sits in a commutative diagram(5.11) H ( C, O C (3)) · x ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ / / H m ( R Q ) (cid:15) (cid:15) R Q where the diagonal map is given by multiplication by x , and the vertical map is defined in Remark5.9.The kernel of the diagonal map is x C [ x , x , x ] ∩ J Q . Since J Q = C h g + x ∂g∂x , x ∂g∂x , x ∂g∂x i ,we see that the kernel is the space consisting of linear combinations a ( g + x ∂g∂x ) + a ( x ∂g∂x ) + a ( x ∂g∂x ), with a = 0. In other words, R Q /H ( C, O C (3)) ∼ = J Q ∩ x C [ x , x , x ] ∼ = C . where the identification with C is given by (0 , a , a ) x P a i ∂g∂x i . Remark . The condition a = 0 is more canonically the condition l ( a , a , a ) = 0.5.2.1. Differential of the inclusion of the moduli space of allowable double covers of nodal planequartics branched at four collinear points into the moduli space of allowable double covers of genus curves branched at points. We have T π R a Q , E −→ T π R a , H m ( R Q ) → H ( C, Ω C ⊗ O C (2)) ∨ with codifferential sitting in a commutative diagram. H ( C, Ω C ⊗ O C (2)) j ∗ (cid:15) (cid:15) / / H m ( R Q ) (cid:15) (cid:15) H ( C, O C (3)) · x / / R Q ommutativity follows from (5.11) and a degeneration argument (see e.g., [DS81, § Differential of the Prym map P , for smooth double covers. The differential of thePrym map P , : R , → A (1 , , , at a branched cover π : e C → C of a smooth plane quartic T π R , −→ T π A (1 , , , is canonically identified with H ( C, T C ⊗ O C (1)) → Sym H ( C, ω C ⊗ η C ) ∨ where η C is the square root of the branch divisor Br determining the cover (i.e., η ⊗ C ∼ = O C ( Br )).The codifferential is given as the cup productSym H ( D, ω C ⊗ η C ) → H ( C, ω ⊗ C ⊗ O C ( Br )) . For the restriction of an odd pseudo-double cover e C ∪ e L → C ∪ L to π : e C → C , this is the canonicalrestriction H ( P , O P (2)) → H ( C, O C (3))of a quadric to the Prym canonical model C ⊂ P . Note that the Prym canonical model of C isalso the image of C under the Prym canonical map of the plane quintic C ∪ L and its associatedpseudo-double cover.5.4. Differential of the Prym map P , for allowable double covers. The differential of thePrym map P , : R a , → A (1 , , , at an allowable double cover π : e C → C of a nodal plane quartic T π R a , −→ T P , ( π ) A (1 , , , is canonically identified with H ( C, Ω C ⊗ ω C ⊗ O C (1)) ∨ → Sym H ( C, ω C ⊗ η C ) ∨ where η ⊗ C ∼ = O C ( Br ) is the square root of the branch divisor determining the cover. (For us, thiswill be the locus where the line meets the plane quartic transversally.) The codifferential is givenas the cup product Sym H ( D, ω C ⊗ η C ) → H ( C, Ω C ⊗ ω C ⊗ O C ( Br )) . For the restriction of an odd pseudo-double cover e C ∪ e L → C ∪ L to π : e C → C , this is the followingmap. H ( P , O P (2)) / / ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ H ( C, Ω C ⊗ O C (2)) j ∗ (cid:15) (cid:15) H ( C, O C (3))5.5. Interlude connecting to the case of cubic threefolds.
Suppose now we have a smoothEckardt cubic threefold (
X, p ) together with a line ℓ ′ ⊂ X passing through the Eckardt point p ,giving rise to a pseudo double cover π : e C ∪ e L → C ∪ L of a plane quartic and a transverse line C ∪ L ⊂ P . As in the discussion above, assume that X = ( F = 0) and C ∪ L = ( Q = 0).For the commutative diagram (which at this point we have not proven is cartesian) R a Q , E / / (cid:15) (cid:15) R a , (cid:15) (cid:15) C E / / A (1 , , , e have: Proposition 5.13.
In the notation above, we have the commutative diagram of codifferentials: (5.14) R Q H ( O C (3)) · x o o J Q ∩ x C [ x , x , x ] = C ? _ o o T ∨ π R a Q , E T ∨ π R a , o o ( T π R a , /T π R a Q , E ) ∨ ? _ o o ( R Q | x ) ∨ O O H (Ω C (2)) o o j ∗ O O K E = C ? _ o o O O T ∨ ( X,p ) C E O O T ∨ J A o o O O ( T J A /T X C ) ∨ ? _ o o O O ( R F ) τ O O H ( O P (2)) o o O O J F ∩ C [ x , . . . , x ] = C ? _ o o O O with the vertical map on the right induced by the natural map from cubic threefolds J F → J Q .Proof. For this we consider the following commutative diagram.(5.15) R a Q (cid:25) y + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ R a Q , E (cid:31) (cid:127) / / (cid:15) (cid:15) (cid:15) (cid:15) $ (cid:4) ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ R a , (cid:31) (cid:127) / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) R τ (cid:31) (cid:127) / / (cid:15) (cid:15) (cid:15) (cid:15) R a (cid:15) (cid:15) (cid:15) (cid:15) C (cid:25) y , , ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ C E / / $ (cid:4) ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ A (1 , , , A τ o o (cid:31) (cid:127) / / A Here A τ is the moduli of principally polarized abelian varieties admitting an involution τ so that theinvariant part has dimension 1 and the anti-invariant part has dimension 4. By [Rod14, Cor. 5.4], A τ is an irreducible subvariety of A of dimension 11. The map A τ → A (1 , , , is the map taking aprincipally polarized abelian variety to the dual of the anti-invariant part. We set R τ := A τ × A R a to be the space of allowable covers admitting an involution τ so that the invariant part of the Prymvariety has dimension 1 and the anti-invariant part has dimension 4. The map R a , → R a is themap that attaches a P at the branch points of the base curve, and takes the branched cover of this P at the attaching points and glues this to the cover curve at the branch points. Such a cover hasthe extra involution τ by taking ι on the cover of the plane quartic, and the identity on the othercomponent of the cover. Thus the map just defined has image contained in R τ as indicated in thediagram. The map R Q , E → R a Q is given by attaching a line to the plane quartic at the markedpoints, and then taking the cover as indicated above.To analyze the diagram, we describe a few more differentials. In the bottom row, given ( X, p ) ∈C E , then we have T ( JX − τ ) ∨ A (1 , , , T JX A τ = ( T JX A ) τ o o / T JX A SymHom(( R F ) − τ , ( R F ) − τ ) SymHom τ ( R F , R F ) o o / SymHom( R F , R F )( J F ∩ C [ x , . . . , x ] ) ∨ (( J F ) ∨ ) τ o o / ( J F ) ∨ where the map on the left is the one induced by the decomposition R iF = ( R iF ) τ ⊕ ( R iF ) − τ . Notethat on the left, we are also choosing coordinates for F as in § n the top row, given π C : e C → C in R a Q , E , with associated cover π D : e D → D in R τa , then wehave T π C R a , / / T π D R τ = ( T π D R a ) τ / T π D R a H (Ω C ⊗ ω C ( p + · · · + p )) ∨ j ∗ (cid:15) (cid:15) / / ( H (Ω D ⊗ ω D ) ∨ ) τ / j ∗ (cid:15) (cid:15) H (Ω D ⊗ ω D ) ∨ j ∗ (cid:15) (cid:15) H ( ω ⊗ C ( p + · · · + p )) ∨ / / ( H ( ω ⊗ D ) ∨ ) τ / H ( ω ⊗ D ) ∨ H ( O C (3)) ∨ / / ( H ( O D (4)) ∨ ) τ / H ( O D (4)) ∨ Now tracing through the diagram and the given differentials, one obtains the stated result. (cid:3)
Next we use Proposition 5.13, and Lemmas 4.14 and 4.12 to give the following corollary:
Corollary 5.16.
Let ( X, p ) ⊂ PC ∼ = P be a smooth Eckardt cubic threefold given by F = 0 ,and let ℓ ⊂ X be a line through the Eckardt point p giving rise to an odd pseudo-double cover π : e C ∪ e L → C ∪ L of the union of a plane quartic C and a transverse line L in PC ∼ = P given by Q = ( kc − q ) x = 0 (NB l = x ). Take coordinates on C such that ℓ = ( x = x = x = 0) , sothat projection from ℓ gives the map π ℓ : PC PC , [ x , x , x , x , x ] [ x , x , x ] . Under theidentifications C ∼ / / J Q C ∼ / / J F ( a , a , a ) ✤ / / P i =2 a i ∂Q∂x i ( a , a , a , a , a ) ✤ / / P a i ∂F∂x i then projectivization of the morphism J F → J Q of (4.11) is the rational map π ℓ . The projectiviza-tion of the dual map ( J Q ) ∨ → ( J F ) ∨ is therefore the inclusion ( P ℓ ) ∨ ⊂ ( P ) ∨ corresponding tohyperplanes containing ℓ .The map J F ∩ C [ x , . . . , x ] → J Q ∩ x C [ x , x , x ] in (5.14) is the restriction of the map J F → J Q defined above. Therefore, the projectivization of the dual map takes L ∨ → H ∨ ⊂ ( P ) ∨ isthe inclusion corresponding to hyperplanes P a i x i = 0 containing ℓ and such that l ( a , . . . , a ) = 0 .Proof. This follows immediately from Proposition 5.13, and Lemmas 4.14 and 4.12. (cid:3)
Proof of Theorem 5.1.
Proof of Theorem 5.1.
We apply the strategy of (4.5) and Lemma 4.7 to the diagram (5.5), wherethe image of C E plays the role of Z , and the image of R a Q , E plays the role of Z ′ .Corollary 5.16 implies that the differential of the Prym map R a , → A (1 , , , at a cover ( π : e C → C ) ∈ R a Q , E has kernel exactly of dimension 1. Thus at π , the fiber of P , over P ( e C, C ) = (
J X − τ ) ∨ is exactly E , the curve in the Fano surface of lines F ( X ) consisting of lines through the Eckardtpoint p . Since this holds at every cover π , this implies that the image of R a Q , E is an irreducible andconnected component of the fiber over the image of C E , with generic fiber of the image of R a Q , E overthe image of C E connected.At the same time, let us use Corollary 5.16 to compute the degree of the map of exceptionaldivisors ] R a Q , E → e C E after blowing up along the corresponding loci (see § X, p ), and a general point in the projectivization of the fiber ofthe normal bundle of the image at the image of (
X, p ), corresponding by Corollary 5.16 to a generalhyperplane H ⊂ P passing through the Eckardt point. The lemma says that the points lying abovethis chosen point correspond to those lines ℓ ′ ⊂ X that pass through the Eckardt point p and lie in he hyperplane section H . Since the lines through p on X are parameterized by E ⊂ S ⊂ X , and E ∩ H consists of three points, there are three such lines.Since the degree of the Prym map is 3, we can conclude from Lemma 4.7 that R a Q , E = C E × A (1 , , , R a , , and we are done. (cid:3) Proof of Corollary 4.3.
This now follows immediately from (4.6), and the proof of Theorem 4.1. (cid:3) Global Torelli theorem for Eckardt cubic threefolds
Applying Theorem 5.1, we prove the global Torelli theorem for the period map P : M → A (1 , , , of cubic surface pairs defined in Section 1. Theorem 6.1 (Global Torelli) . The period map P : M → A (1 , , , for cubic surface pairs isinjective.Proof. Using the isomorphism
M ∼ = C E , we prove that the period map C E → A (1 , , , is injective.In other words, let ( X , p ) and ( X , p ) be Eckardt cubic threefolds coming from cubic surfacepairs ( S i , Π i ). Denoting the involution associated with p i by τ i for i = 0 ,
1, we will prove that ifthe anti-invariant parts are isomorphic to each other
J X − τ ∼ = J X − τ , then ( X , p ) is isomorphicto ( X , p ).First recall that from Theorem 3.11, for i = 0 ,
1, there exist a τ i -invariant line ℓ ′ i ⊂ X i through p i ,such that when projecting X i from ℓ ′ i we get a covering π i : e C i → C i in R , such that P ( e C i , C i ) ∼ =( J X − τ i i ) ∨ as polarized abelian varieties. More precisely, the double cover e C i → C i is obtained byprojecting X i from ℓ ′ i and restricting the discriminant pseudo-double cover π i : e C i ∪ E i → C i ∪ L i tothe smooth quartic component C i . Recall that e C i → C i ( i = 0 ,
1) and E i → L i are branched at thefour intersection points C i ∩ L i , and therefore the branched double cover e C i → C i determines theentire discriminant pseudo-double cover. Recall also that the cover π i : e C i → C i is determined by anodd theta characteristic (i.e., bi-tangent) κ C i on C i (Lemma 3.7), and that this theta characteristicalso determines the cubic surface S i associated to ( X i , p i ) (see (3.5) and § i isdetermined by L i ; it is the pre-image of L i under the projection of P = ( x = 0) from p ′ i = ℓ ′ i ∩ S ′ i .In particular, the data ( X i , p i , ℓ i ), up to projective linear transformations, is equivalent to the data( C i , κ C i , L i ) (see Theorem 3.8).Now since P ( e C , C ) ∼ = P ( e C , C ), we have from Theorem 5.1 that there exists a τ -invariant line ℓ ′′ ⊂ X through p , such that when projecting X from ℓ ′′ we get the covering π : e C → C . Infact, as described above, projecting from ℓ ′′ , we get the full discriminant e C ∪ E → C ∪ L . But thenthe triple ( X , p , ℓ ′ ) is projectively equivalent to ( X , p , ℓ ′′ ). In other words, ( X , p ) ∼ = ( X , p ). (cid:3) Eckardt cubic threefolds as fibrations in conics 2: point-wise invariant lines
Let X be a cubic threefold with an Eckardt point p (see Equation (1.3)). Denote the associatedinvolution in (1.6) by τ . In Section 7, we study the τ -decomposition of J X (under a genericityassumption; see Condition 2.15) via the linear projection of X from a line point-wise fixed by τ (i.e., from one of the 27 lines on the cubic surface S ⊂ X ). As an application, we prove that theperiod map P for cubic surface pairs (defined in Section 1) is generically finite onto its image. Asthe results in this section are not used in the proofs of our main theorems, and the proofs of theresult in this section are similar to those is §
3, we give more brief treatment of the proofs in thissection. .1. Projecting Eckardt cubic threefolds from point-wise invariant lines.
We now revisit § X, p )be an Eckardt cubic threefold and let ℓ ⊂ S be a line contained in the associated cubic surface (1.7).Let us choose coordinates so that X is given by Equation (1.3) f ( x , . . . , x ) + l ( x , . . . , x ) x = 0,with the Eckardt point p = [0 , , , , . Let L , L , L be linear forms on P with ℓ = ( L = L = L = 0). Since ℓ ⊂ S = X ∩ ( x = 0) = 0,we can assume that L = x . As a consequence, we can assume that L , L do not include x .Therefore, by a change of coordinates in x , x , x , x , we can assume that the line ℓ is cut out by x = x = x = 0: ℓ = ( x = x = x = 0) . Because ℓ is not contained in the cone X ∩ T p X , the linear polynomial l in Equation (1.3) containseither x or x (note that T p X = ( l = 0)). Interchanging x and x , we assume it is x . After achange of coordinates (namely, x l and x i x i for 1 ≤ i ≤ X is given by(7.1) f ( x , . . . , x ) + x x = 0 . Then we have S = ( f ( x , . . . , x ) = 0) ⊂ P x ,x ,x ,x and the hyperplane section is(7.2) E = ( x = 0) ∩ S = ( x = f = 0) . Now we project X from the line ℓ to a complementary plane P x ,x ,x = ( x = x = 0) andobtain a conic bundle π ℓ : Bl ℓ X → P x ,x ,x . Following (2.19), we write the equation of X as(7.3) l ( x , x ) x + 2 l ( x , x ) x x + l ( x , x ) x +2( q ( x , x ) + 12 x ) x + 2 q ( x , x ) x + c ( x , x ) = 0where l i , q j and c are homogeneous polynomials in x , x of degree 1, 2 and 3 respectively. (In(2.19) these were polynomials in x , x , x , but here we have x appearing in (7.1) only in themonomial x x and so we have modified the notation slightly to reflect this.)We now consider the associated matrix (2.20):(7.4) M = l l q + x l l q q + x q c . The equation of the discriminant D ⊂ P is (2.21):(7.5) det( M ) = − l x + det (cid:18) l q l q (cid:19) x + det l l q l l q q q c = 0 . Note that in our coordinates, the involution τ on X is induced by an involution τ on P , namely x
7→ − x . This induces an involution τ on P x ,x ,x given also by x
7→ − x . Since ℓ is fixed by τ ,there is an induced involution on Bl ℓ X , and from the definitions it is clear that π ℓ : Bl ℓ X → P x ,x ,x is equivariant with respect to the involution τ . Restricting to the discriminant D , we see that τ induces involutions(7.6) σ : e D → e D and σ D : D → D making the cover e D → D equivariant. emark . Fiberwise, we can describe the involutions σ and σ D as follows. The involution τ on X sends a degenerate fiber ℓ ∪ m ∪ m ′ to another degenerate fiber ℓ ∪ τ ( m ) ∪ τ ( m ′ ); this then definesthe action of σ on e D . Let ι : e D → e D be the covering involution associated with e D → D . From theprevious geometric description of σ , we deduce that σι = ισ . Then σ induces an involution on D and one verifies easily that this involution coincides with σ D . Lemma 7.8.
If the discriminant curve D is smooth, then the quotient curve D := D/σ D is smoothof genus .Proof. For brevity, the proof is left to the reader. (cid:3)
Proposition 7.9 (Projecting from ℓ ⊂ S ⊂ X ) . Let ( X, p ) be an Eckardt cubic threefold and let ℓ ⊂ S be a line contained in the associated cubic surface (1.7) . The following are equivalent: (1) ( X, p ) and ℓ satisfy Condition 2.15 (note that such a line ℓ ⊂ S exists on a general Eckardtcubic threefold; see Remark 2.16). (2) The discriminant plane quintic D is smooth and the double cover e D → D is connected and´etale.Proof. For brevity, this is left to the reader. (cid:3)
Klein group towers of coverings.
We are interested in studying the Prym variety P ( e D, D ).The key point is that the covering curve e D admits two commuting involutions, namely σ inducedfrom τ (7.6), and ι induced from the double cover e D → D . It has been clear going back to [Mum74]that one should consider the associated tower of covers (7.11) induced by taking quotients of e D bythe various subgroups of h σ, ι i ⊂ Aut( e D ). While Mumford focused on a particular case involvinghyperelliptic curves, this general approach, including studying more complicated automorphismgroups of the covering curve of a branched double cover, was explored in more depth in [Don92],and then generalized in [RR03] to include the case we study here. We explain this in the contextof double covers of discriminant curves of Eckardt cubic threefolds.We introduce the following notation. For any element g = 1 of the Klein four group h σ, ι i ⊂ Aut( e D ), we denote the quotient curve e D g := e D/ h g i . In particular, e D ι = D . Lemma 7.10.
We have the following commutative diagram: (7.11) e D a σ } } ⑤⑤⑤⑤⑤⑤⑤⑤ a σι ´et (cid:15) (cid:15) a ι ´et ❍❍❍❍❍❍❍❍❍ e D σ b σ ´et ! ! ❈❈❈❈❈❈❈❈❈ e D σιb σι (cid:15) (cid:15) e D ι = D b ι { { ✈✈✈✈✈✈✈✈✈✈ D Moreover, (1)
The map a σ is a double covering map branched at twelve points. The maps a σι and a ι areboth ´etale double covering maps. (2) The map b σ is an ´etale double covering map. Both b σι and b ι are double covering mapsramified at six points. (3) The curves are all smooth and their genera are given as follows: g ( e D ) = 11 , g ( e D σ ) = 3 , g ( e D σι ) = 6 , g ( D ) = 6 and g ( D ) = 2 . roof. This essentially follows from [RR03, Thm. 6.3]. For brevity, the details are left to thereader. (cid:3)
Proposition 7.12 ([RR03, Thm. 6.3]) . In the notation of (7.11) , let ( P ( e D, D ) , Ξ) be the principallypolarized Prym variety. There is an isogeny of polarized abelian varieties φ ι : P ( e D σ , D ) × P ( e D σι , D ) −→ P ( e D, D ) , ( y , y ) a ∗ σ ( y ) + a ∗ σι ( y ) with ker( φ ι ) ∼ = ( Z / Z ) , where a ∗ σ denotes the pull-back between Jacobians, and similarly for a ∗ σι .More explicitly, we have ker( φ ι ) = { ( y , y ) ∈ P ( e D σ , D )[2] × P ( e D σι , D )[2] | a ∗ σ ( y ) = a ∗ σι ( y ) } . Moreover, with respect to the action of σ on ( P ( e D, D ) , Ξ) , the isogeny φ ι induces isomorphismsof polarized abelian varieties P ( e D, D ) σ ∼ = P ( e D σ , D ) and P ( e D, D ) − σ ∼ = P ( e D σι , D ) / h b ∗ σι ǫ i , where ǫ is the -torsion line bundle on D defining the ´etale double cover e D σ → D , and b ∗ σι ǫ is nontrivial.Proof. This essentially follows from [RR03, Thm. 6.3]. For brevity, the details are left to thereader. (cid:3)
Another Klein tower shows up in Mumford’s hyperelliptic construction for the ´etale double cover b σ : e D σ → D . Lemma 7.13 ([Mum74, p.346]) . In the notation of Lemma 7.10, with E the hyperplane section ofthe cubic surface S (1.5) , we have the following commutative diagram (7.14) e D σb σ = b D ´et | | ②②②②②②②②② b E (cid:15) (cid:15) b R % % ❏❏❏❏❏❏❏❏❏❏ D c D ❋❋❋❋❋❋❋❋❋ E c E (cid:15) (cid:15) R ∼ = P c R y y ssssssssss T ∼ = P and we have an isomorphism of principally polarized abelian varieties P ( e D σ , D ) ∼ = E × J ( R ) = E. Proof.
This can be deduced from [BO19, Prop. 4.2, Cor. 4.3] (see also [Mum74, § (cid:3) Putting this together, we obtain the following theorem:
Theorem 7.15 (Projecting from ℓ ⊂ S ⊂ X ) . Let ( X, p ) be an Eckardt cubic threefold, let ℓ ⊂ S be a line contained in the associated cubic surface (1.7) satisfying Condition 2.15 (note that sucha line ℓ ⊂ S exists on a general Eckardt cubic threefold; see Remark 2.16), let E ⊂ S be thehyperplane section (1.5) , and let b σι : e D σι → D be the branched cover of the smooth genus curve D from (7.11) .There is an isogeny of polarized abelian varieties φ : E × P ( e D σι , D ) −→ J X with ker( φ ) ∼ = ( Z / Z ) .Moreover, with respect to the action of τ on ( J X, Θ X ) , the isogeny φ induces isomorphisms ofpolarized abelian varieties J X τ ∼ = E and J X − τ ∼ = P ( e D σι , D ) / h b ∗ σι ǫ i , where ǫ is the -torsion linebundle defining the ´etale double cover e D σ → D from (7.11) , and b ∗ σι ǫ is nontrivial. roof. Since
J X ∼ = P ( e D, D ), and the action of τ on J X is identified with the action of σ on P ( e D, D ), this is an immediate consequence of Proposition 7.12 and Lemma 7.13. (cid:3)
Remark . Using results of [NO20, NO19, LO11], one can use Theorem 7.15 to prove that theperiod map P : M → A (1 , , , for cubic surface pairs is generically finite-to-one onto its image,and that the differential of P is injective at a generic point. Since we can use other methods togive a short proof of a stronger statement (Corollary 5.8), we omit this proof here. References [ABH02] V. Alexeev, Ch. Birkenhake, and K. Hulek,
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Department of Mathematics, University of Colorado, Boulder, Colorado 80309-0395, USA
Email address : [email protected] Institute of Mathematical Sciences, ShanghaiTech University, Shanghai, 201210 China
Email address : [email protected]@shanghaitech.edu.cn