Related papers: Theta-duality on Prym varieties and a Torelli Theo…
Let us consider the locus in the moduli space of curves of genus 2k defined by curves with a pencil of degree k. Since the Brill-Noether number is equal to -2, such a locus has codimension two. Using the method of test surfaces, we compute…
The classical Prym construction associates to a smooth, genus $g$ complex curve $X$ equipped with a nonzero cohomology class $\theta \in H^1(X,\mathbb{Z}/2\mathbb{Z})$, a principally polarized abelian variety (PPAV) $\mbox{Prym}(X,\theta)$.…
We give an inductive proof that the generalized Severi varieties -- the varieties which parametrize (irreducible) plane curves of given degree and genus, with a fixed tangency profile to a given line at several general fixed points and…
A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local-global principle for the…
We discuss Weber's formula which gives the quotient of two Thetanullwerte for a plane smooth quartic in terms of the bitangents. In particular, we show how it can easily be derived from the Riemann-Jacobi formula.
The Prym-Green Conjecture predicts that the resolution of a generic n-torsion paracanonical curve of every genus is natural. The conjecture has mostly been studied so far for level 2, that is, for Prym-canonical curves. Using a construction…
We study the geometry and arithmetic of the curves $C \colon y^3 = x^4 + ax^2 + b$ and their associated Prym abelian surfaces $P$. We prove a Torelli theorem in this context and give a geometric proof of the fact that $P$ has quaternionic…
In this paper we consider the Prym variety $P(\widetilde{C}/C)$ associated to a Galois coverings of curves $f:\widetilde{C}\to C$ branched at $r$ points. We discuss some properties and equivalent definitions and then consider the Prym map…
A new codimension 2 relation among descendent strata in the moduli space of stable, 3-pointed, genus 2 curves is found. The space of pointed admissible double covers is used in the calculation. The resulting differential equations satisfied…
We study the Torelli morphism from the moduli space of stable curves to the moduli space of principally polarized stable semi-abelic pairs. We give two characterizations of its fibers, describe its injectivity locus, and give a sharp upper…
We completely describe the degree of the Gauss map of the theta divisor of bielliptic Prym varieties. We characterize bielliptic Prym varieties whose Gauss degree is the same as Jacobians. We also construct bielliptic Prym varieties with a…
Let $\M_g$ be the course moduli space of complex projective nonsingular curves of genus $g$. We prove that when the Brill-Noether number $\rho(g,1,n)$ is non-negative the Petri locus $P^1_{g,n}\subset \M_g$ has a divisorial component whose…
We present a Prym construction which associates abelian varieties to vertex-transitive strongly regular graphs. As an application we construct Prym-Tyurin varieties of arbitrary exponent $\geq 3$, generalizing a result by Lange, Recillas…
We establish a correspondence between toroidal compactifications of M-theory and del Pezzo surfaces. M-theory on T^k corresponds to P^2 blown up at k generic points; Type IIB corresponds to P^1\times P^1. The moduli of compactifications of…
We prove the analogue of the Riemann-Roch formula for the noncommutative two torus $ A_{\theta} = C(\mathbb{T}_{\theta}^2)$ equipped with an arbitrary translation invariant complex structure and a Weyl factor represented by a positive…
We recast the Galois cohomology of the variety $V$ over a number field $k$ in terms of the K-theory of a $C^*$-algebra $\mathscr{A}_V$ connected to $V$. It is proved that $V$ is isomorphic to $V'$ over $k$ (algebraic closure of $k$, resp.)…
We prove a generalization of the Brill-Noether theorem for the variety of special divisors $W^r_d(C)$ on a general curve $C$ of prescribed gonality. Our main theorem gives a closed formula for the dimension of $W^r_d(C)$. We build on…
The goal of this paper is the study of simple rank 2 parabolic vector bundles over a $2$-punctured elliptic curve $C$. We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to $\mathbb{P}^1 \times…
We survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible…
We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert…