Related papers: Real normalized differentials and Arbarello's conj…

A component of the moduli space M_g(Y,b) of stable maps from genus g curves to a variety Y is said to be regular if it is generically smooth and of the expected dimension provided by deformation theory. In this note we prove existence of…

The geometric torsion conjecture asserts that the torsion part of the Mordell--Weil group of a family of abelian varieties over a complex quasiprojective curve is uniformly bounded in terms of the genus of the curve. We prove the conjecture…

We consider two cycles on the moduli space of compact type curves and prove that they coincide. The first is defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized rational curve,…

The main goal of this article is to compute the class of the divisor of $\overline{\mathcal{M}}_3$ obtained by taking the closure of the image of $\Omega\mathcal{M}_3(6;-2)$ by the forgetful map. This is done using Porteous formula and the…

We investigate the viability of defining an intersection product on algebraic cycles on a singular algebraic variety by pushing forward intersection products formed on a resolution of singularities. For varieties with resolutions having a…

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of…

This article treats various aspects of the geometry of the moduli of r-spin curves and its compactification. Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves (algebraic curves with a…

We give a new proof of Faber's intersection number conjecture concerning the top intersections in the tautological ring of the moduli space of curves $\M_g$. The proof is based on a very straightforward geometric and combinatorial…

Let $ R \subset \R $ be a GCD-domain. In this paper, Weinberg's conjecture on the $ n \times n $ matrix algebra $ M_{n}(R) \ (n \geq 2) $ is proved. Moreover, all the lattice orders (up to isomorphisms) on a full $ 2 \times 2 $ matrix…

We prove the existence of complete minimal surfaces of genus g>1 which minimize the total curvature for their genus. Our method is first to identify this (Weierstrass high dimensional period) problem with the problem of finding a particular…

We show that various natural algebro-geometric moduli stacks, including the stack of curves, have the property that every Deligne-Mumford gerbe over a field appears as the residual gerbe of one of their points. These gerbes are universal…

Principally polarized abelian surfaces with prescribed real multiplication (RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus 2 curves correspond to rational points on the Hilbert modular surfaces via their…

We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface $X\subset \bbP^{4}$ of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type…

We discover that tautological intersection numbers on $\bar{\mathcal{M}}_{g, n}$, the moduli space of stable genus $g$ curves with $n$ marked points, are evaluations of Ehrhart polynomials of partial polytopal complexes. In order to prove…

This paper concerns the intersection numbers of tautological classes on moduli spaces of parabolic bundles on a smooth projective curve. We show that such intersection numbers are completely determined by wall-crossing formulas, Hecke…

For any finite abelian group G, we study the moduli space of abelian $G$-covers of elliptic curves, in particular identifying the irreducible components of the moduli space. We prove that, in the totally ramified case, the moduli space has…

We introduce a class of normal complex spaces having only mild sin-gularities (close to quotient singularities) for which we generalize the notion of a (analytic) fundamental class for an analytic cycle and also the notion of a relative…

We describe a very large class of conjectural relations in the tautological ring of the moduli space $\bar{M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points, extending and generalizing the Faber-Zagier relations. These notes…

The moduli space $\M_{g,n}$ of $n-$pointed stable curves of genus $g$ is stratified by the topological type of the curves being parametrized: the closure of the locus of curves with $k$ nodes has codimension $k$. The one dimensional…

The strata of the moduli spaces of Abelian differentials are non-homogenous spaces carrying natural bi-algebraic structures. Partly inspired by the case of homogenous spaces carrying bi-algebraic structures (such as torii, Abelian varieties…