Sebastian Casalaina-Martin
For the Jacobian of a curve, the Riemann singularity theorem gives a geometric interpretation of the singularities of the theta divisor in terms of special linear series on the curve. This paper proves an analogous theorem for Prym varieties. Applications of this theorem to cubic threefolds, and Prym varieties of dimension five, are also considered.
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Refereed version to appear on Journal of Algebraic Geometry. Unchanged content. Strongly ameliorated presentation thanks to referee's action.
We prove canonical isomorphisms between Spin Verlinde spaces, i.e, spaces of global sections of a determinant line bundle over the moduli space of semistable Spin-bundles over a smooth projective curve C, and the dual spaces of theta functions over Prym varieties of unramified double covers of C.
Algebraic curves in Hilbert modular surfaces that are totally geodesic for the Kobayashi metric have very interesting geometric and arithmetic properties, e.g. they are rigid. There are very few methods known to construct such algebraic geodesics that we call Kobayashi curves. We give an explicit way of constructing Kobayashi curves using determinants of derivatives of theta functions. This construction also allows to calculate the Euler characteristics of the Teichmueller curves constructed by McMullen using Prym covers.
Let p:C' -> C be an unramified double covering of irreducible smooth curves and let P be the attached Prym variety. We prove the schematic theta-dual equalities in the Prym variety T(C')=V^2 and T(V^2)=C', where V^2 is the Brill-Noether locus of P associated to p considered by Welters. As an application we prove a Torelli Theorem analogous to the fact that the g-th symmetric product of a curve D of genus g determines the curve.
In this paper we compute the pullback of divisor classes under the Prym map (extended to the boundary), and apply this result to get a lower bound on the slope of effective divisors on the perfect cone compactification of the moduli space of principally polarized abelian fivefolds. In the appendix by Klaus Hulek, the notion of slope for arbitrary toroidal compactifications is discussed, and the slope bound is shown to hold in general.
Let $P \cup P'$ be the two component Prym variety associated to an étale double cover $\tilde{C} \to C$ of a non-hyperelliptic curve of genus $g \geq 6$ and let $|2Ξ_0|$ and $|2Ξ_0'|$ be the linear systems of second order theta divisors on $P$ and $P'$ respectively. The component $P'$ contains canonically the Prym curve $\tilde{C}$. We show that the base locus of the subseries of divisors containing $\tilde{C} \subset P'$ is scheme-theoretically the curve $\tilde{C}$. We also prove canonical isomorphisms between some subseries of $|2Ξ_0|$ and $|2Ξ_0'|$ and some subseries of second order theta …
If the theta-null divisor $Θ_{\rm null}$ is moved to the Prym moduli space through the diagram $\mathcal{S}_{g}^{+}\rightarrow\mathcal{M}_{g}\leftarrow\mathcal{R}_{g}$, it splits into two irreducible components $\mathcal{P}_{\rm\! null}^{+}$ and $\mathcal{P}_{\rm\! null}^{-}$. Using test curve techniques, we compute the class of the closure of each of these components in terms of the generating classes of $\mathrm{Pic}(\overline{\mathcal{R}}_{g})_{\mathbb{Q}}$.
In the moduli space $\mathcal{R}_g$ of double étale covers of curves of a fixed genus $g$, the locus of covers of curves with a semicanonical pencil decomposes as the union of two divisors $\mathcal{T}^e_g$ and $\mathcal{T}^o_g$. Adapting arguments of Teixidor for the divisor of curves having a semicanonical pencil, we prove that both divisors are irreducible and compute their divisor classes in the Deligne-Mumford compactification $\overline{\mathcal{R}}_g$.
For $i\geq2$, we compute the first coefficients of the class $[\overline{D}(μ;3)]$ in the rational Picard group of the moduli of Prym curves $\overline{\mathcal{R}}_{2i}$, where $D(μ;3)$ is the divisor parametrizing pairs $[C,η]$ for which there exists a degree $2i$ map $π\colon C\rightarrow \mathbb{P}^1$ having ramification profile $(2,\ldots,2)$ above two points $q_1, q_2$, a triple ramification somewhere else and satisfying $\mathcal{O}_C(\frac{π^{*}(q_1)-π^{*}(q_2)}{2})\cong η$. Furthermore, we provide several new Prym enumerative results related to this situation.
Lifting Prym Divisors on a Bielliptic Loop of Loops
本文针对双椭圆曲线的平展二重覆盖，证明其环之环上的普赖姆除子可提升为原曲线的同秩普赖姆除子，进而确定普赖姆-布里尔-诺特簇的维数、既约不可约性及上同调类。
The boundary of the bielliptic Prym locus
本研究通过分析奇异双椭圆曲线的余法几何，应用于双椭圆Prym轨迹边界，获得了Gauss映射结果并计算了Chern-Mather类与特征循环。
We study the behavior of theta characteristics on an algebraic curve under the specialization map to a tropical curve. We show that each effective theta characteristic on the tropical curve is the specialization of $2^{g-1}$ even theta characteristics and $2^{g-1}$ odd theta characteristics. We then study the relationship between unramified double covers of a tropical curve and its theta characteristics, and use this to define the tropical Prym variety.
Theta functions of level n on the principally polarised Prym varieties of an algebraic curve are dual to sections of the orthogonal theta line bundle on the moduli space of Spin(n)-bundles over the curve. As a by-product of our computations we also note that when n is odd the pfaffian line bundle on moduli space has a basis of sections labelled by the even theta characteristics of the curve.
Van Geemen and van der Geer, Donagi, Beauville and Debarre proposed characterizations of the locus of jacobians which use the linear system of $2Θ$-divisors. We give new evidence for these conjectures in the case of Prym varieties.
The class of the Prym-Brill-Noether divisor
本文研究了Prym-Brill-Noether分割的类别，计算了其在模空间中的系数，并证明了相关模空间的泛型性。
For an Abel-Prym curve contained in a Prym variety, we determine the cohomological support loci of its twisted ideal sheaves and the dimension of its theta-dual.
Theta-duality and Prym-Torelli for cyclic covers of hyperelliptic curves
本文通过theta对偶性重构方法，研究超椭圆曲线奇素数度循环覆盖的Prym映射的通用单射性，识别出额外剩余轨迹，并在不同度下给出单射性条件。
We study moduli spaces of K3 surfaces endowed with a Nikulin involution and their image in the moduli space R_g of Prym curves of genus g. We observe a striking analogy with Mukai's well-known work on ordinary K3 surfaces. Many of Mukai's results have a very precise Prym-Nikulin analogue, for instance: (1) A general Prym curve from R_g is a section of a Nikulin surface if and only if g\leq 7, g\neq 6. (2) R_7 has the structure of a Mori fibre space over the corresponding moduli space of polarized Nikulin surfaces. (3) In the case of genus next to maximal (g=6), the Prym-Nikulin locus in R_6 is…
The Prym map of type (g,n,r) associates to every cyclic covering of degree n of a curve of genus g, ramified at a reduced divisor of degree r, the corresponding Prym variety. We show that the corresponding map of moduli spaces is generically finite in most cases. From this we deduce the dimension of the image of the Prym map.
A Hurwitz divisor on the Moduli of Prym curves
本文针对亏格 $g=2i\geq4$ 的情形，计算了 Prym 曲线模空间上特定 Hurwitz 除子类的首项系数，并给出了相关的计数结果。
Monodromy of the Prym map and semicanonical pencils in genus 6
本文证明了亏格6中限制于奇半典范铅笔除子的Prym映射是双有理的，其单值群为Weyl群$WD_5$，并揭示了模空间中其他除子的次数及几何性质。
It is shown that the map from the Jacobian of the spectral curve to the moduli of stable bundles of rank 2 is generically simply branched along an irreducible divisor. This observation falsifies the key step in the "abelianization of the SU(2) WZW connection" presented in a recent paper [Yoshida, Annals 2006]
Geometry of Prym semicanonical pencils and an application to cubic threefolds
本研究探讨模空间中半典范线束对应除子上的Prym映射，揭示其差异与低亏格几何特性，并通过对特定除子的分析得到三次三维簇直线的计数结果。
We study the rational Picard group of the projectivized moduli space of holomorphic n-differentials on complex genus g stable curves. We define (n - 1) natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differentials with a double zero. We give two different proofs of this result, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the univer…
The Gauss map for bielliptic Prym varieties
本文研究双椭圆 Prym 簇 theta 除子的 Gauss 映射，完整刻画其次数、识别与 Jacobian 次数相同的情形，并构造低次数例子及 5 维 Andreotti-Mayer 轨迹的完整结果。
We prove that the Jacobian of a general curve C of genus g=2a+1, with g>4, can be realized as a Prym-Tyurin variety for the Brill-Noether curve W^1_{a+2}(C). As a consequence of this result we are able to compute the class of the sum of the secant divisors for the curve C, embedded with a complete linear series g^{a-1}_{3a-2}
We show that the Verlinde formula for moduli spaces of spin bundles on an algebraic curve gives dimensions of direct sums of spaces of theta functions over the finite set of Prym varieties of unramified double covers of the curve. We then construct a natural duality (though we do not show it is nondegenerate) between the latter spaces and the spaces of sections of the theta bundles on the moduli spaces of spin bundles corresponding to the orthogonal representation. Finally, it is remarked that this should be viewed as a special case of a conjectured `strange duality' for spin bundles.
With every smooth, projective algebraic curve $\tilde{C}$ with involution $σ:\tilde{C}\longrightarrow \tilde{C}$ without fixed points is associated the Prym data which consists of the Prym variety $P:=(1-σ)J(\tilde{C})$ with principal polarization $Ξ$ such that $2Ξ$ is algebraically equivalent to the restriction on $P$ of the canonical polarization $Θ$ of the Jacobian $J(\tilde{C})$. In contrast to the classical Torelli theorem the Prym data does not always determine uniquely the pair $(\tilde{C},σ)$ up to isomorphism. In this paper we introduce an extension of the Prym data as follows. We con…
We introduce the notion of an effective Arakelov divisor for a number field and the arithmetical analogue of the dimension of the space of sections of a line bundle. We study the analogue of the theta divisor for a number field.
We study the enumerative geometry of the moduli space R_g of Prym varieties of dimension g-1 (also known as the space of admissible double covers). Our main result is that the compactification of R_g is of general type as soon as g>13. We achieve this by computing the class of two types of cycles on R_g: one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym-Green conjecture on syzygies of Prym-canonical curves. In the appendix we show that even though R_g has non-canonic…
Prym loci of branched double coverings and generalized Andreotti-Mayer loci
本文将安德烈奥蒂-迈耶轨迹推广至非主极化阿贝尔簇模空间，证明分支二重覆盖的普里姆轨迹是广义安德烈奥蒂-迈耶轨迹的不可约分支。
Secant loci on moduli of Prym varieties
本文将 Lazarsfeld 关于一般极化 K3 曲面曲线无意外割线的结果推广到 Prym 情形，证明标准与非标准一般 Nikulin 曲面上曲线的 Prym-典范嵌入无意外割线，并据此确定奇数亏格稳定 Prym 曲线模空间中泛差除子的类。
The classes of two theta divisors on an abelian variety in the naive Grothendieck ring of varieties need not be congruent modulo the class of the affine line.
We show that analytic torsion of smooth theta divisor is represented by a Siegel modular form characterizing the Andreotti-Mayer locus if its dimension is positive.
We show that the locus of stable rank four vector bundles without theta divisor over a smooth projective curve of genus two is in canonical bijection with the set of theta-characteristics. We give several descriptions of these bundles and compute the degree of the rational theta map.
This is an expository paper about the topics listed in the title.
In this note we study the geometry of principally polarized abelian varieties (ppavs) with a vanishing theta-null (i.e. with a singular point of order two and even multiplicity lying on the theta divisor). We describe the locus within the theta-null divisor where this singularity is not an ordinary double point. By using theta function methods we first show that this locus does not equal the entire theta-null divisor (this was shown previously by O. Debarre). We then show that this locus is contained in the intersection of the theta-null divisor with the other irreducible components of the And…
The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. In this paper we use the Prym map to show that this version of the Hodge conjecture is true for the theta divisor of a general abelian fivefold.
We show that the gradient and the hessian of the Riemann theta function in dimension n can be combined to give a theta function of order n+1 and modular weight (n+5)/2 defined on the theta divisor. It can be seen that the zero locus of this theta function essentially gives the ramification locus of the Gauss map. For jacobians this leads to a description in terms of theta functions and their derivatives of the Weierstrass point locus on the associated Riemann surface.
The rationality of ineffective spin genus-4 thetanull loci
本文证明了由亏格4且thetanull消失的曲线C与无效thetacharacteristic θ构成的对[C,θ]所给出的除子是有理簇，并得出Prym模空间中的类似除子也是有理的。
We prove that Prym varieties of algebraic curves with two smooth fixed points of involution are exactly the indecomposable principally polarized abelian varieties whose theta-functions provide explicit formulae for integrable 2D Schrödinger equation.
A generic strictly semistable bundle of degree zero over a curve X has a reducible theta divisor, given by the sum of the theta divisors of the stable summands of the associated graded bundle. The converse is not true: Beauville and Raynaud have each constructed stable bundles with reducible theta divisors. For X of genus at least 5, we construct stable vector bundles over X of rank $r$ for all $r \geq 5$, with reducible and nonreduced theta divisors. We also adapt the construction to symplectic bundles. In the appendix, Raynaud's original example of a stable rank 2 vector bundle with reducibl…
We discuss variations of Hodge structures on abelian varieties that arise from intersecting translates of theta divisors with a special focus on the case of abelian varieties of dimension 4
We prove that Prym varieties are characterized geometrically by the existence of a symmetric pair of quadrisecant planes of the associated Kummer variety. We also show that Prym varieties are characterized by certain (new) theta-functional equations. For this purpose we construct and study a difference-differential analog of the Novikov-Veselov hierarchy.
In this paper we prove a strengthening of the generic vanishing result in characteristic $p>0$ given in [HP16]. As a consequence of this result, we show that irreducible $Θ$ divisors are strongly F-regular and we prove a related result for pluri-theta divisors.
We present a new class of examples of base points for the generalized theta divisor on the moduli space of semistable vector bundles of trivial determinant on a compact Riemann surface and we prove that for sufficiently large rank the base locus is positive dimensional.
In this paper we show that semistable vector bundles on a Castelnuovo curve of genus g >= 2 have theta divisors. As a corollary, we deduce that semistable vector bundles on a smooth, general curve of genus g >= 2 which extend to semistable vector bundles on any Castelnuovo degeneration of the general curve admit a theta divisor.
In this paper we give a birational model for the theta divisor of the intermediate Jacobian of a generic cubic threefold $X$. We use the standard realization of $X$ as a conic bundle and a $4-$dimensional family of plane quartics which are totally tangent to the discriminant quintic curve of such a conic bundle structure. The additional data of an even theta characteristic on the curves in the family gives us a model for the theta divisor.
We prove a motivic version of the Lefschetz hyperplane theorem for a smooth ample divisor $Θ$ on an Abelian variety. We use this to construct a motive $P$ that realizes the primitive cohomology of $Θ$.
Let $(A,Θ)$ be a complex principally polarized abelian variety of dimension $g\geq 4$. Based on vanishing theorems, differentiation techniques and intersection theory, we show that whenever the theta divisor $Θ$ is irreducible, its multiplicity at any point is at most $g-2$. This improves work of Kollár, Smith-Varley, and Ein-Lazarsfeld. We also introduce some new ideas to study the same type of questions for pluri-theta divisors.
We study the loci of principally polarized abelian varieties with points of high multiplicity on the theta divisor. Using the heat equation and degeneration techniques, we relate these loci and their closures to each other, as well as to the singular set of the universal theta divisor. We obtain bounds on the dimensions of these loci and relations among their dimensions, and make further conjectures about their structure.
We produce a lower bound for the dimension of the base locus of the generalized theta divisor on the moduli space SU_C(r) of semistable vector bundles of rank r and trivial determinant on a smooth curve C of genus g > 1.
We study the theta divisor of the compactified jacobian of a nodal, possibly reducible, curve. We compute its irreducible components and give it a geometric interpretation consistent with the classical Brill-Noether theory of smooth curves. Some applications on hyperelliptic stable curves are appended.
We prove that the cotangent bundle of a complete intersection of two general translates of the theta divisor of the jacobian of a general curve of genus 4 is ample. From this the same result for a general principally polarized abelian variety of dimension 4 follows.
The singularities of theta divisors have played an important role in the study of algebraic varieties. This paper surveys some of the recent progress in this subject, using as motivation some well known results, especially those for Jacobians.
We give a bound on the number of points of order two on the theta divisor of a principally polarized abelian variety A. When A is the Jacobian of a curve C the result can be applied in estimating the number of effective square roots of a fixed line bundle on C.
We construct families of principally polarized abelian varieties whose theta divisor is irreducible and contains an abelian subvariety. These families are used to construct examples when the Gauss map of the theta divisor is only generically finite and not finite. That is, the Gauss map in these cases has at least one positive-dimensional fiber. We also obtain lower-bounds on the dimension of Andreotti-Mayer loci.
The Jacobian varieties of smooth curves fit together to form a family, the universal Jacobian, over the moduli space of smooth marked curves, and the theta divisors of these curves form a divisor in the universal Jacobian. In this paper we describe how to extend these families over the moduli space of stable marked curves (or rather an open subset thereof) using a stability parameter. We then prove a wall-crossing formula describing how the theta divisor varies with the stability parameter. We use that result to analyze a divisor on the moduli space of smooth marked curves that has recently be…
Raynaud has shown that over a general curve of genus $g \ge 2$, every semistable bundle of rank three and integral slope admits a theta divisor. We show that this can fail for special curves: Over any bielliptic curve of genus $g \ge 5$, we construct a stable rank three bundle of trivial determinant with no theta divisor. This gives a partial answer to a question of Beauville.
Given a smooth projective curve X, we give effective very ampleness bounds for generalized theta divisors on the moduli spaces $SU_X(r,d)$ and $U_X(r,d)$ of semistable vector bundles of rank r and degree d on X with fixed, respectively arbitrary, determinant.
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