Related papers: Weierstrass Prym eigenforms in genus four

In this paper, we construct spines, i.e., $\Mod_g$-equivariant deformation retracts, of the Teichm\"uller space $\T_g$ of compact Riemann surfaces of genus $g$. Specifically, we define a $\Mod_g$-stable subspace $S$ of positive codimension…

We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for…

The punctured solenoid $\S$ is an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichm\"uller space of $\S$ is introduced, studied, and found to…

We classify curves in the moduli space of curves that are both Shimura- and Teichmueller curves: Except for the moduli space of genus one curves there is only a single such curve. We start with a Hodge-theoretic description of Shimura…

In this article we study compact Riemann surfaces of genus $g$ with an automorphism of prime order $g+1.$ The main result provides a classification of such surfaces. In addition, we give a description of them as algebraic curves, determine…

We prove that there are no Shimura-Teichm\"uller curves generated by genus five translation surfaces, thereby completing the classification of Shimura-Teichm\"uller curves in general. This was conjectured by M\"oller in his original work…

Let $C$ be a smooth projective curve of genus $g\geq 4$ over the complex numbers and ${\cal SU}^s_C(r,d)$ be the moduli space of stable vector bundles of rank $r$ with a fixed determinant of degree $d$. In the projectivized cotangent space…

We show that any equation from the Davey--Stewartson hierarchy induces an infinite family of geometrically different deformations of tori in $\R^4$ preserving the Willmore functional. We expose a derivation of the Weierstrass representation…

We study how are permuted Weierstrass points of Veech surfaces in $\mathcal{H}(2)$, the stratum of Abelian differentials on Riemann surfaces in genus two with a single zero of order two. These surfaces were classified by McMullen relying on…

In this paper, we show that the divisor given by couples [C,{\theta}] where C is a curve of genus 4 with a vanishing thetanull and {\theta} is an ineffective thetacharacteristic is a rational variety. By our construction, it follows also…

Let $\Tei_{g,n}$ be the Teichm\"uller space of Riemann surfaces of genus $g$ with $n$ punctures. It is conjectured that the Teichm\"uller and Carath\'{e}odory metrics agree on a Teichm\"{u}ller disk if and only if all the zeros of the…

We prove that in characteristic p>0 the locus of stable curves of p-rank at most f is pure of codimension g-f in the moduli space of stable curves. Then we consider the Prym map and analyze it using tautological classes. We study the locus…

We give finite presentations for the fundamental group of moduli stacks of smooth Weierstrass curves over complex projective space P^n which extend the classical result for elliptic curves to positive dimensional base. We thus get natural…

Let $S$ be a closed orientable surface with genus $g\geq 2$. For a sequence $\s_i$ in the Teichm\"uller space of $S$, which converges to a projective measured lamination $[\lam]$ in the Thurston boundary, we obtain a relation between $\lam$…

We classify the connected components of the space of representations of the fundamental group of a closed oriented surface of genus $\geq 2$ in $Sp(4,{\mathbf R})$. We prove that this is equivalent to classifying the connected components of…

We study the induced Ekedahl-Oort stratification on the moduli space $\mathcal{M}_4$ of curves of genus $4$ in characteristic $3$. By constructing families of curves with given Ekedahl-Oort type and by computing the dimension of the…

Given an Enriques surface $T$, its universal K3 cover $f: S\to T$, and a genus $g$ linear system $|C|$ on $T$, we construct the relative Prym variety $P_H=\Prym_{v, H}(\D/\CC)$, where $\CC\to |C|$ and $\D\to |f^*C|$ are the universal…

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…

This is a commentary on Teichm\"ullers' paper "Ver\"anderliche Riemannsche Fl\"achen" (Variable Riemann Surfaces), published in 1944. This paper is the last one that Teichm\"uller wrote on the problem of moduli. At most places the paper…

We show that the moduli space R_9 of Prym curves of genus 9 is uniruled. This is the largest genus g for which R_g has negative Kodaira dimension.