The Chen-Ruan cohomology of moduli of curves of genus 2 with marked points

In this work we describe the Chen-Ruan cohomology of the moduli stacks of smooth and stable genus 2 pointed curves, and its algebraic counterpart: the stringy Chow ring. In the first half of the paper we compute the additive structure of the Chen-Ruan cohomology ring for the moduli stack of stable pointed genus 2 curves, describing it as a rationally graded vector space. In the second part we give generators for the even Chen--Ruan cohomology ring as an algebra on the ordinary cohomology.Comment: 48 pages, 3 Figures. Final version to appear in Advances in Mathematic

The Automorphisms group of \bar{M}_{g,n}

Let \bar{\mathcal{M}}_{g,n}$ be the moduli stack parametrizing Deligne-Mumford stable n-pointed genus g curves and let \bar{M}_{g,n} be its coarse moduli space: the Deligne-Mumford compactification of the moduli space of n-pointed genus g smooth curves. We prove that the automorphisms groups of \bar{\mathcal{M}}_{g,n} and \bar{M}_{g,n} are isomorphic to the symmetric group on n elements S_{n} for any g,n such that 2g-2+n\geq 3, and compute the remaining cases.Comment: 22 page

On the automorphism group of certain algebraic varieties

We study the automorphism groups of two families of varieties. The first is the family of stable curves of low genus. To every such curve, we can associate a combinatorial object, a stable graph, which encode many properties of the curve. Combining the automorphisms of the graph with the known results on the automorphisms of smooth curves, we obtain precise descriptions of the automorphism groups for stable curves with low genera. The second is the family of numerical Godeaux surfaces. We compute in details the automorphism groups of numerical Godeaux surfaces with certain invariants; that is, corresponding to points in some specific connected components of the moduli space; we also give some estimates on the order of the automorphism groups of the other numerical Godeaux surfaces and some characterization on their structures

Reduction of Plane Quartics and Cayley Octads

We give a conjectural characterisation of the stable reduction of plane quartics over local fields in terms of their Cayley octads. This results in p-adic criteria that efficiently give the stable reduction type amongst the 42 possible types, and whether the reduction is hyperelliptic or not. These criteria are in the vein of the machinery of "cluster pictures" for hyperelliptic curves. We also construct explicit families of quartic curves that realise all possible stable types, against which we test these criteria. We give numerical examples that illustrate how to use these criteria in practice.Comment: Comments are welcome, 683 pictures, improved introductio

Arithmetic of Genus Three Curves and Their Jacobians

The Birch–Swinnerton-Dyer Conjecture predicts that, given an abelian variety A over a number field K, its rank, rk(A/K), is equal to the order of vanishing of its L-function L(A/K, s) at s = 1. A consequence of this is the Parity Conjecture; rk(A/K) and the order of vanishing at s=1 of L(A/K, s) are expected to have the same parity. The parity of the latter is given by the root number w(A/K), and so the Parity Conjecture states that (−1)^rk(A/K) = w(A/K). This thesis investigates what can be said about the Parity Conjecture when A is the Jacobian of a curve of genus 3. Part of this requires developing the local theory of non-hyperelliptic genus 3 curves. We introduce a combinatorial object called an octad diagram, which we conjecture to recover the essential data of stable models

Non-perturbative topological recursion and knot invariants

The goal of the present thesis is to study new examples, applications and com- putational aspects of the topological recursion formalism introduced by Eynard and Orantin. We develop efficient methods for the calculation of non-perturbative wave functions associated to spectral curves of genus one. Our results are used to test two conjectures. The first one relates perturbative knot invariants obtained from the AJ Conjecture and a state integral model to the wave function obtained from topological recursion. The second conjecture describes the structure of the quantum curve for the Weierstrass spectral curve. We are able to verify the conjectures up to some order in a formal parameter h and we state a stronger version of the conjecture in the case of the Weierstrass curve. Some of these results are based upon joint work with Greyson Potter

Biregular and Birational Geometry of Algebraic Varieties

Every area of mathematics is characterized by a guiding problem. In algebraic geometry such problem is the classification of algebraic varieties. In its strongest form it means to classify varieties up to biregular morphisms. However, birationally equivalent varieties share many interesting properties. Therefore for any birational equivalence class it is natural to work out a variety, which is the simplest in a suitable sense, and then study these varieties. This is the aim of birational geometry. In the first part of this thesis we deal with the biregular geometry of moduli spaces of curves, and in particular with their biregular automorphisms. However, in doing this we will consider some aspects of their birational geometry. The second part is devoted to the birational geometry of varieties of sums of powers and to some related problems which will lead us to computational geometry and geometric complexity theory