Related papers: Explicit Moduli of Superelliptic Curves with Level…
Given a lattice polygon $P$ with $g$ interior lattice points, we associate to it the moduli space of tropical curves of genus $g$ with Newton polygon $P$. We completely classify the possible dimensions such a moduli space can have. For…
We are interested in studying moduli spaces of rank 2 logarithmic connections on elliptic curves having two poles. To do so, we investigate certain logarithmic rank 2 connections defined on the Riemann sphere and a transformation rule to…
This article generalizes the operad of moduli spaces of curves to SUSY curves. SUSY curves are algebraic curves with additional supersymmetric or supergeometric structure. Here, we focus on the description of the relevant category of graphs…
We consider modular properties of nodal curves on general $K3$ surfaces. Let $\mathcal{K}_p$ be the moduli space of primitively polarized $K3$ surfaces $(S,L)$ of genus $p\geqslant 3$ and $\mathcal{V}_{p,m,\delta}\to \mathcal{K}_p$ be the…
This is a survey paper dealing with moduli aspects of curves over finite fields. It discusses counting points of moduli spaces, relations with modular forms and stratifications on moduli spaces.
We give a complete description of the two-dimensional moduli spaces of stable Higgs bundles of rank 2 over complex projective line with one irregular singular point, having a regular leading-order term, and endowed with a generic compatible…
We give an algebraic construction of the moduli space of irregular singular connections of generic ramified type on a smooth projective curve. We prove that the moduli space is smooth and give its dimension. Under the assumption that the…
In this note we discuss a class of hyperelliptic curves introduced by Abel in a 1826 paper. After some indications of the context in which he introduced them and a description of his main result we give some results on the moduli space of…
In this long survey article we show that the theory of elliptic and hyperelliptic curves can be extended naturally to all superelliptic curves. We focus on automorphism groups, stratification of the moduli space $\mathcal{M}_g$, binary…
Let $\mathcal{M}_{g,2}$ be the moduli space of curves of genus $g$ with a level-2 structure. We prove here that there is always a non hyperelliptic element in the intersection of four thetanull divisors in $\mathcal{M}_{6,2}$. We prove also…
A 10-dimensional symplectic moduli space of torsion sheaves on the cubic 4-fold is constructed. It parametrizes the stable rank 2 vector bundles on the hypeplane sections of the cubic 4-fold which are obtained by Serre's construction from…
The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus $\L_g$ of such genus $g$ hyperelliptic curves is a $g$-dimensional subvariety of the moduli space of hyperelliptic curves $\H_g$. We discover a…
In this paper, we add the information of level structure to supersingular elliptic curves and study these objects with the motivation of isogeny-based cryptography. Supersingular elliptic curves with level structure map to Eichler orders in…
We study the symplectic geometry of the moduli space of closed n-gons with fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces have a symplectic structure coming from Poisson Lie theory. We construct completely…
We describe a framework to construct tropical moduli spaces of rational stable maps to a smooth tropical hypersurface or curve. These moduli spaces will be tropical cycles of the expected dimension, corresponding to virtual fundamental…
This is a note on calculating intersection numbers on moduli spaces of curves. A codimension 3 relation among tautological classes on the moduli space of genus 4 curves is given.
Let $M$ be the moduli space of rank 3 stable bundles with fixed determinant of degree 1 on a smooth projective curve of genus $g\geq 2$. When $C$ is generic, we show that any essential elliptic curve on $M$ has degree (respect to…
Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with…
In these lectures we cover basics of the theory of heights starting with the heights in the projective space, heights of polynomials, and heights of the algebraic curves. We define the minimal height of binary forms and moduli height for…
Moduli spaces of quadratic differentials with prescribed singularities are not necessarily connected. We describe here all cases when they have a special hyperelliptic connected component. We announce the general classification theorem: up…