We show that a generic principally polarized abelian variety (ppav) is uniquely determined by its theta hyperplanes. These are the non-projectivized version of those studied by Caporaso and Sernesi (see math.AG/0204164), which in a sense are a generalization to ppavs of bitangents of plane quartics, and of hyperplanes tangent to a canonical curves of genus $g$ in $g-1$ points. More precisely, we show that, generically, the set of gradients of all odd theta functions at the point zero uniquely determines a ppav with level (4,8) structure. We also show that our map is an immersion of the moduli space of ppavs.
We compute the rational cohomology of the moduli space of non-singular non-hyperelliptic complex projective curves of genus 3 with an odd theta characteristic.
We show that a general canonical curve is uniquely determined by the finite set of hyperplanes cutting theta-characteristics on it. Geometrical and combinatorial properties of the moduli space of stable spin curves are proved, which play an essential role.
A general canonical curve X determines a finite set T(X) of hyperplanes, which is in bijective correspondence with the set of odd theta-characteristics of X. The definition of T(X) can be extended to certain singular curves, in a way that is compatible with degenerations. If X is a Deligne-Mumford stable curve (or a cuspidal curve), we describe the enumerative and local geometry of T(X); this is applied to show that some singular curves are uniquely determined by T(X). This work generalizes the preliminary steps used in AG/0008239 (with E. Sernesi) to show that a general curve of genus 3 can b…
We study the birational geometry of some moduli spaces of abelian varieties with extra structure: in particular, with a symmetric theta structure and an odd theta characteristic. For a $(d_1,d_2)$-polarized abelian surface, we show how the parities of the $d_i$ influence the relation between canonical level structures and symmetric theta structures. For certain values of $d_1$ and $d_2$, a theta characteristic is needed in order to define Theta-null maps. We use these Theta-null maps and preceding work of other authors on the representations of the Heisenberg group to study the birational geom…
Let M be the moduli space of SO(r)-bundles on a curve, and L the determinant bundle on M. We define an isomorphism of H^0(M,L) onto the dual of the space of r-th order theta functions on the Jacobian of C. This isomorphism identifies the map M -->|L|* defined by the linear system |L| with the map M -->|r Theta| which associates to a quadratic bundle (E,q) the Theta divisor of the vector bundle E . The two components M+ and M- of M are mapped into the subspaces of even and odd theta functions respectively. Finally we discuss the analogous question for Sp(2r)-bundles.
The level moduli space $A_g^{4,8}$ is mapped to the projective space by means of gradients of odd Theta functions, such a map turning out no to be injective in the genus 2 case. In this work a congruence subgroup $Γ$ is located between $Γ_2(4,8)$ and $Γ_2(2,4)$ in such a way the map factors on the related level moduli space $A_Γ$, the new map being injective on $A_Γ$. Satake's compactification $\text{Proj}A(Γ)$ and the desingularization $\text{Proj}S(Γ)$ are also due to be investigated, since the map does not extend to the boundary of the compactification; to aim at this, an algebraic descript…
We give a new method for constructing vector-valued modular forms from singular scalar-valued ones. As an application we prove the identity between two remarkable spaces of vector-valued modular forms which seem to be unrelated at a first look, since they are constructed in two very different ways. If $V_{grad}$ is the vector space generated by vector-valued modular forms constructed with gradients of odd theta functions and $V_Θ$ is the one generated by vector-valued modular forms arising from second order theta constants with our new construction, we will prove that $V_{grad}=V_Θ$. This resu…
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.
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.
Let $C$ be a smooth projective curve of genus $g\geq 3$ and let $η$ be an odd theta characteristic on it such that $h^0(C,η) = 1$. Pick a point $p$ from the support of $η$ and consider the one-dimensional linear system $|η+ p|$. In general this linear system is base-point free and all its ramification points (i.e. ramification points of the corresponding branched cover $C\to\mathbb P^1\simeq \mathbb PH^0(C,η+p)$) are simple. We study the locus in the moduli space of odd spin curves where the linear system $|η+ p|$ fails to have this general behavior. This locus splits into a union of three div…
We investigate a class of odd (ramification) coverings $C \to \mathbb{P}^1$ where $C$ is hyperelliptic, its Weierstrass points maps to one fixed point of $\mathbb{P}^1$ and the covering map makes the hyperelliptic involution of $C$ commute with an involution of $\mathbb{P}^1$. We show that the total number of hyperelliptic odd coverings of minimal degree $4g$ is ${3g \choose g-1} 2^{2g}$ when $C$ is general. Our study is approached from three main perspectives: if a fixed effective theta characteristic is fixed they are described as a solution of a certain class of differential equations; then…
A stream of new theta relations is obtained. They follow from the general Thomae formula, which is a new result giving expressions for theta derivatives (the zero values of the lowest non-vanishing derivatives of theta functions with singular half-period characteristics) in terms of branch points and the period matrix of a hyperelliptic Riemann surface. The new theta relations contain (i) linear relations on the vector space of first order theta derivatives which are arranged in gradients, (ii) relations between second order theta derivatives and symmetric bilinear forms on the vector space of…
We study the geometry of some varieties of sums of powers related to the Klein quartic. This allows us to describe the birational geometry of certain moduli spaces of abelian surfaces. In particular we show that the moduli space $\mathcal{A}_2(1,7)^{-}_{sym}$ of $(1,7)$-polarized abelian surfaces with a symmetric theta structure and an odd theta characteristic is unirational by showing that it admits a dominant morphism from a unirational conic bundle.
The goal of the paper is to give an analytic proof of the formula of G. Farkas for the divisor class of spinors with multiple zeros in the moduli space of odd spin curves. We make use of the technique developed by Korotkin and Zograf that is based on properties of the Bergman tau function. We also show how the Farkas formula for the {\it theta-null} in the rational Picard group of the moduli space of even spin curves can be derived from classical theory of theta functions.
In this paper we establish the relationships between theta functions of arbitrary order and their derivatives. We generalize our previous work math.AG/0310085 and prove that for any n>1 the map sending an abelian variety to the set of Gauss images of its points of order 2n is an embedding into an appropriate Grassmannian (note that for n=1, i.e. points of order 2, we only get generic injectivity). We further discuss the generalizations of Jacobi's derivative formula for any dimension and any order. The series of embedding of the level covers of moduli spaces of principally polarized abelian va…
Let X be a symplectic or odd orthogonal Grassmannian parametrizing isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in H^*(X,Z) as a polynomial in certain special Schubert classes. We study theta polynomials, a family of polynomials defined using raising operators whose algebra agrees with the Schubert calculus on X. Furthermore, we prove that theta polynomials are special cases of Billey-Haiman Schubert polynomials and use this connection to express the former as positive linear …
We use the gradients of theta functions at odd two-torsion points --- thought of as vector-valued modular forms --- to construct holomorphic differential forms on the moduli space of principally polarized abelian varieties, and to characterize the locus of decomposable abelian varieties in terms of the Gauss images of two-torsion points.
We describe an efficient algorithm for the computation of separable isogenies between abelian varieties represented in the coordinate system given by algebraic theta functions. Let $A$ be an abelian variety of dimension $g$ defined over a field of odd characteristic. Our algorithm decomposes in two principal steps. First, given a theta null point for $A$ and a subgroup $K$ isotropic for the Weil pairing, we explain how to compute the theta null point corresponding to the quotient abelian variety $A/K$. Then, from the knowledge of a theta null point of $A/K$, we give an algorithm to obtain a ra…
A general Boltzmann machine with continuous visible and discrete integer valued hidden states is introduced. Under mild assumptions about the connection matrices, the probability density function of the visible units can be solved for analytically, yielding a novel parametric density function involving a ratio of Riemann-Theta functions. The conditional expectation of a hidden state for given visible states can also be calculated analytically, yielding a derivative of the logarithmic Riemann-Theta function. The conditional expectation can be used as activation function in a feedforward neural …
The Weddle surface is classically known to be a birational (partially desingularized) model of the Kummer surface. In this note we go through its relations with moduli spaces of abelian varieties and of rank two vector bundles on a genus 2 curve. First we construct a moduli space A\_2(3)^- parametrizing abelian surfaces with a symmetric theta structure and an odd theta characteristic. Such objects can in fact be seen as Weddle surfaces. We prove that A\_2(3)^- is rational. Then, given a genus 2 curve C, we give an interpretation of the Weddle surface as a moduli space of extensions classes (in…
