On the Coble quartic

We review and extend the known constructions relating Kummer threefolds, G¨opel systems, theta constants and their derivatives, and the GIT quotient for 7 points in P^2 to obtain an explicit expression for the Coble quartic. The Coble quartic was recently determined completely in [RSSS12], where it was computed completely explicitly, as a polynomial with 372060 monomials of bidegree (28, 4) in theta constants of the second order and theta functions of the second order, respectively. Our expression is in terms of products of theta constants with characteristics corresponding to G¨opel systems, and is a polynomial with 134 terms. Our approach is based on the beautiful geometry studied by Coble and further investigated by Dolgachev and Ortland and highlights the geometry and combinatorics of syzygetic octets of characteristics, and the corresponding representations of Sp(6, F_2). One new ingredient is the relationship of G¨opel systems and Jacobian determinants of theta functions. In genus 2, we similarly obtain a short explicit equation for the universal Kummer surface, and relate modular forms of level two to binary invariants of six points on P^

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 $\Gamma_{00}$ 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

An explicit solution to the weak Schottky problem

We give an explicit weak solution to the Schottky problem, in the spirit of Riemann and Schottky. For any genus g, we write down a collection of polynomials in genus g theta constants such that their common zero locus contains the locus of Jacobians of genus g curves as an irreducible component. These polynomials arise by applying a specific Schottky-Jung proportionality to an explicit collection of quartic identities for genus g - 1 theta constants

On some differential-geometric aspects of the Torelli map

In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside $\mathsf{A}_g$. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic subvarieties and the Jacobian locus.Comment: To appear on Boll. UMI, special volume in memory of Paolo de Bartolomei

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.Comment: v2: new section 9 on the geometry of the boundary of the locus of intermediate Jacobians of cubic threefolds. Final version to appear in Invent. Mat

Eluding SUSY at every genus on stable closed string vacua

In closed string vacua, ergodicity of unipotent flows provide a key for relating vacuum stability to the UV behavior of spectra and interactions. Infrared finiteness at all genera in perturbation theory can be rephrased in terms of cancelations involving only tree-level closed strings scattering amplitudes. This provides quantitative results on the allowed deviations from supersymmetry on perturbative stable vacua. From a mathematical perspective, diagrammatic relations involving closed string amplitudes suggest a relevance of unipotent flows dynamics for the Schottky problem and for the construction of the superstring measure.Comment: v2, 17 pages, 8 figures, typos corrected, new figure added with 3 modular images of long horocycles,(obtained with Mathematica

Complete moduli of cubic threefolds and their intermediate Jacobians

The intermediate Jacobian map, which associates to a smooth cubic threefold its intermediate Jacobian, does not extend to the GIT compactification of the space of cubic threefolds, not even as a map to the Satake compactification of the moduli space of principally polarized abelian fivefolds. A much better "wonderful" compactification of the space of cubic threefolds was constructed by the first and fourth authors --- it has a modular interpretation, and divisorial normal crossing boundary. We prove that the intermediate Jacobian map extends to a morphism from the wonderful compactification to the second Voronoi toroidal compactification of the moduli of principally polarized abelian fivefolds --- the first and fourth author previously showed that it extends to the Satake compactification. Since the second Voronoi compactification has a modular interpretation, our extended intermediate Jacobian map encodes all of the geometric information about the degenerations of intermediate Jacobians, and allows for the study of the geometry of cubic threefolds via degeneration techniques. As one application we give a complete classification of all degenerations of intermediate Jacobians of cubic threefolds of torus rank 1 and 2.Comment: 56 pages; v2: multiple updates and clarification in response to detailed referee's comment

Extending the Belavin-Knizhnik "wonderful formula" by the characterization of the Jacobian

A long-standing question in string theory is to find the explicit expression of the bosonic measure, a crucial issue also in determining the superstring measure. Such a measure was known up to genus three. Belavin and Knizhnik conjectured an expression for genus four which has been proved in the framework of the recently introduced vector-valued Teichmueller modular forms. It turns out that for g>3 the bosonic measure is expressed in terms of such forms. In particular, the genus four Belavin-Knizhnik "wonderful formula" has a remarkable extension to arbitrary genus whose structure is deeply related to the characterization of the Jacobian locus. Furthermore, it turns out that the bosonic string measure has an elegant geometrical interpretation as generating the quadrics in P^{g-1} characterizing the Riemann surface. All this leads to identify forms on the Siegel upper half-space that, if certain conditions related to the characterization of the Jacobian are satisfied, express the bosonic measure as a multiresidue in the Siegel upper half-space. We also suggest that it may exist a super analog on the super Siegel half-space.Comment: 15 pages. Typos corrected, refs. and comments adde

Triangulated Surfaces in Twistor Space: A Kinematical Set up for Open/Closed String Duality

We exploit the properties of the three-dimensional hyperbolic space to discuss a simplicial setting for open/closed string duality based on (random) Regge triangulations decorated with null twistorial fields. We explicitly show that the twistorial N-points function, describing Dirichlet correlations over the moduli space of open N-bordered genus g surfaces, is naturally mapped into the Witten-Kontsevich intersection theory over the moduli space of N-pointed closed Riemann surfaces of the same genus. We also discuss various aspects of the geometrical setting which connects this model to PSL(2,C) Chern-Simons theory.Comment: 35 pages, references added, slightly revised introductio

Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities

We characterize genus g canonical curves by the vanishing of combinatorial products of g+1 determinants of Brill-Noether matrices. This also implies the characterization of canonical curves in terms of (g-2)(g-3)/2 theta identities. A remarkable mechanism, based on a basis of H^0(K_C) expressed in terms of Szego kernels, reduces such identities to a simple rank condition for matrices whose entries are logarithmic derivatives of theta functions. Such a basis, together with the Fay trisecant identity, also leads to the solution of the question of expressing the determinant of Brill-Noether matrices in terms of theta functions, without using the problematic Klein-Fay section sigma.Comment: 35 pages. New results, presentation improved, clarifications added. Accepted for publication in Math. An