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Samuel Grushevsky

Publications and source records attributed to Samuel Grushevsky.

At least 55 records · Page 3Linked to original sources
arXiv2012

The class of the locus of intermediate Jacobians of cubic threefolds

We study the locus of intermediate Jacobians of cubic threefolds within the moduli space of complex principally polarized abelian fivefolds, and its generalization to arbitrary genus - the locus of abelian varieties with a singular odd two-torsion point on the theta divisor. Assuming that this locus has expected codimension (which we show to be true for genus up to 5), we compute the class of this locus, and of is closure in the perfect cone toroidal compactification, in the Chow, homology, and the tautological ring. We work out the cases of genus up to 5 in detail, obtaining explicit expressions for the classes of the closures of the locus of products of an elliptic curve and a hyperelliptic genus 3 curve, in moduli of principally polarized abelian fourfolds, and of the locus of intermediate Jacobians in genus 5. In the course of our computation we also deal with various intersections of boundary divisors of a level toroidal compactification, which is of independent interest in understanding the cohomology and Chow rings of the moduli spaces.

math.AG↗
arXiv2011

The superstring cosmological constant and the Schottky form in genus 5

Combining certain identities for modular forms due to Igusa with Schottky-Jung relations, we study the cosmological constant for the recently proposed ansatz for the chiral superstring measure in genus 5. The vanishing of this cosmological constant turns out to be equivalent to the long-conjectured vanishing of a certain explicit modular form of genus 5 on the moduli of curves M_5, and we disprove this conjecture, thus showing that the cosmological constant for the proposed ansatz does not vanish identically. We exhibit an easy modification of the genus 5 ansatz satisfying factorization constraints and yielding a vanishing cosmological constant. We also give an expression for the cosmological constant for the proposed ansatz that should hold for any genus if certain generalized Schottky-Jung identities hold.

math.AG↗
arXiv2011

The Prym map on divisors, and the slope of A_5

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.

math.AG↗
arXiv2011

Principally polarized semi-abelic varieties of small torus rank, and the Andreotti-Mayer loci

We obtain, by a direct computation, explicit descriptions of all principally polarized semi-abelic varieties of torus rank up to 3. We describe the geometry of their symmetric theta divisors and obtain explicit formulas for the involution and its fixed points. These results allow us to give a new proof of the statements about the dimensions, for small genus, of the loci of ppav with theta divisor containing two-torsion points of multiplicity three. We also prove a result about the closure of this set. Our computations used in our work arXiv:1103.1857 to compute the class of the closure of the locus of intermediate jacobians of cubic threefolds in the Chow ring of the perfect cone compactification of the moduli space of principally polarized abelian fivefolds.

math.AG↗
arXiv2010

The Scorza correspondence in genus 3

In this note we prove the genus 3 case of a conjecture of G. Farkas and A. Verra on the limit of the Scorza correspondence for curves with a theta-null. Specifically, we show that the limit of the Scorza correspondence for a hyperelliptic genus 3 curve C is the union of the curve ${x,σ(x))$ (where $σ$ is the hyperelliptic involution), and twice the diagonal. Our proof uses the geometry of the subsystem Γ_{00} of the linear system 2Θ, and Riemann identities for theta constants.

math.AG↗
arXiv2010

The Schottky Problem

In this survey we discuss some of the classical and modern methods in studying the (Riemann-)Schottky problem, the problem of characterizing Jacobians of curves among principally polarized abelian varieties. We present many of the recent results in this subject, and describe some directions of current research. This paper is based on the talk given at the "Classical algebraic geometry today" workshop at MSRI in January 2009.

math.AG↗
arXiv2010

Geometry of A_g and Its Compactifications

In this survey we give a brief introduction to, and review the progress made in the last decade in understanding the geometry of the moduli spaces A_g of principally polarized abelian varieties and its compactifications. Topics surveyed include: compactifications; birational geometry: nef and effective cones, canonical models; homology, Chow rings and intersection theory; and subvarieties of moduli spaces. We also discuss some open problems and possible further directions. This is an expanded and updated version of the talk given at the 2005 Summer Institute for Algebraic Geometry

math.AG↗
arXiv2010

A special case of the $Γ_{00}$ conjecture

In this paper we prove the $Γ_{00}$ conjecture of van Geemen and van der Geer, under the additional assumption that the matrix of coefficients of the tangent has rank at most 2. This assumption is satisfied by Jacobians, and thus our result gives a characterization of the locus of Jacobians among all principally polarized abelian varieties. The proof is by reduction to the (stronger version of the) characterization of Jacobians by semidegenerate trisecants, i.e. by the existence of lines tangent to the Kummer variety at one point and intersecting it in another, proven by Krichever in the course of his proof of the Welters' trisecant conjecture.

math.AG↗
arXiv2009

The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces

We show that certain structures and constructions of the Whitham theory, an essential part of the perturbation theory of soliton equations, can be instrumental in understanding the geometry of the moduli spaces of Riemann surfaces with marked points. We use the ideas of the Whitham theory to define local coordinates and construct foliations on the moduli spaces. We use these constructions to give a new proof of the Diaz' bound on the dimension of complete subvarieties of the moduli spaces. Geometrically, we study the properties of meromorphic differentials with real periods, and their degenerations.

math.AG↗
arXiv2008

The vanishing of two-point functions for three-loop superstring scattering amplitudes

In this paper we show that the two-point function for the three-loop chiral superstring measure ansatz proposed by Cacciatori, Dalla Piazza, and van Geemen vanishes. Our proof uses the reformulation of ansatz in terms of even cosets, theta functions, and specifically the theory of the $Γ_{00}$ linear system on Jacobians introduced by van Geemen and van der Geer. At the two-loop level, where the amplitudes were computed by D'Hoker and Phong, we give a new proof of the vanishing of the two-point function (which was proven by them). We also discuss the possible approaches to proving the vanishing of the two-point function for the proposed ansatz in higher genera.

hep-th↗
arXiv2008

The loci of abelian varieties with points of high multiplicity on the theta divisor

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.

math.AG↗
arXiv2008

Superstring scattering amplitudes in higher genus

In this paper we continue the program pioneered by D'Hoker and Phong, and recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the chiral superstring measure by constructing modular forms satisfying certain factorization constraints. We give new expressions for their proposed ansätze in genera 2 and 3, respectively, which admit a straightforward generalization. We then propose an ansatz in genus 4 and verify that it satisfies the factorization constraints and gives a vanishing cosmological constant. We further conjecture a possible formula for the superstring amplitudes in any genus, subject to the condition that certain modular forms admit holomorphic roots.

hep-th↗
arXiv2007

Some intersections in the Poincare bundle, and the universal theta divisor on the moduli space of (semi)abelian varieties

We compute all the top intersection numbers of divisors on the total space of the Poincare bundle restricted to the product of a curve and the abelian variety. We use these computations to find the class of the universal theta divisor and $m$-theta divisor inside the universal corank 1 semiabelian variety -- the boundary of the partial toroidal compactification of the moduli space of abelian varieties. We give two computational examples: we compute the boundary coefficient of the Andreotti-Mayer divisor (computed by Mumford but in a much harder and ad hoc way), and the analog of this for the universal $m$-theta divisor.

math.AG↗
arXiv2007

Some intersection numbers of divisors on toroidal compactifications of A_g

We study the top intersection numbers of the boundary and Hodge class divisors on toroidal compactifications of the moduli space $A_g$ of principally polarized abelian varieties and compute those numbers that live away from the stratum which lies over the closure of $A_{g-3}$ in the Satake compactification.

math.AG↗
arXiv2007

Singularities of the theta divisor at points of order two

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 Andreotti-Mayer divisor N_0, and describe by using the geometry of the universal scheme of singularities of the theta divisor the components of this intersection that are contained in this locus. Some of the intermediate results obtained along the way of our proof were concurrently obtained independently by C. Ciliberto and G. van der Geer, and by R. de Jong.

math.AG↗
arXiv2006

Jacobians with a vanishing theta-null in genus 4

In this paper we prove a conjecture of Hershel Farkas that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the hessian of the theta function at the corresponding point of order two is degenerate, the abelian variety is a Jacobian. We also discuss possible generalizations to higher genera, and an interpretation of this condition as an infinitesimal version of Andreotti and Mayer's local characterization of Jacobians by the dimension of the singular locus of the theta divisor.

math.AG↗
arXiv2005

Two generalizations of Jacobi's derivative formula

In this paper we generalize the famous Jacobi's triple product identity, considered as an identity for theta functions with characteristics and their derivatives, to higher genus/dimension. By applying the results and methods developed in our previous paper math.AG/0310085 several generalizations to Siegel modular forms are obtained. These generalizations are identities satisfied by theta functions with characteristics and their derivatives at zero. Equating the coefficients of the Fourier expansion of these relations to zero yields non-trivial combinatorial identities.

math.NT↗