### Talks

Note: For an extended abstract click on the title of the talk.

Alex Massarenti, On the automorphisms groups of Hassett moduli spaces

The stack \bar\mathcal{M}_{g,n}, parametrizing Deligne-Mumford n-pointed genus g stable curves, and its coarse moduli space \bar M_{g,n} have been among the most studied objects in algebraic geometry for several decades. Hassett introduced new compactifications \bar\mathcal{M}_{g,A} of the moduli stack \mathcal{M}_{g,n} and \bar M_{g,A} for the coarse moduli space M_{g,n} by assigning rational weights A = (a_{1},...,a_{n}), 0< a_{i} <= 1 to the markings. In particular the classical Deligne-Mumford compactification arises for a_1 = ... = a_n = 1. These spaces appear as intermediate steps of the blow-up construction of \bar M_{0,n} developed by M. Kapranov, while in higher genus may be related to the Log Minimal Model Program on \bar M_{g,n}. We deal with fibrations and automorphisms of Hassett spaces. Our approach consists in extending some techniques already used to tackle the same kind of problems for the Deligne-Mumford compactification of M_{g,n}. As special cases we will recover known results on the automorphisms groups of \bar\mathcal{M}_{g,n} and \bar M_{g,n}. Namely Aut(\bar\mathcal{M}_{g,n})\cong Aut(\bar M_{g,n})\cong S_n for any g,n such that 2g-2+n\geq 3.

Sonia L. Rueda, J. Rafael Sendra, and Juana Sendra, On the Approximate Parametrization Problem of Algebraic Curves

The problem of parametrizing approximately algebraic curves and surfaces is an active research field, with many implications in practical applications. The problem can be treated locally or globally. We formally state the problem, in its global version for the case of algebraic curves (planar or spatial), and we report on some algorithms approaching it, as well as on the associated error distance analysis.

We determine the Weierstrass points of weight $q$ for genus 3 hyperelliptic curves in terms of their dihedral invariants. Possible generalizations of this method will be discussed for higher $g$.

D. Sevilla and J. R. Sendra, Radical parametrization of algebraic curves and surfaces

Parametrization of algebraic curves and surfaces is a fundamental topic in CAGD (intersections; offsets and conchoids; etc.) There are many results on rational parametrization, in particular in the curve case, but the class of such objects is relatively small. If we allow root extraction, the class of parametrizable objetcs is greatly enlarged (for example, elliptic curves can be parametrized with one square root). In this talk I will describe the basics and the state of the art of the problem of parametrization of curves and surfaces by radicals.

L. Beshaj and T. Shaska, On superelliptic curves and their Jacobians

We discuss superelliptic curves and their loci in the moduli space of curves. Moreover, for a fixed $g \geq 2$ all the automorphisms groups $G$ are determined, their signatures $sig$, and the irreducibility of the Hurwitz space $H (g, G, sig)$. For small $g$ an algebraic description of  $H (g, G, sig)$ is determined in terms of the invariants of binary forms. The complete decomposition of Jacobians is done for $g \leq 10$ and an algorithm is given for any $g\geq 2$.
Indefinite quaternion algebras are used to define Fuchsian groups acting on the Poincare half-plane, and the associated Shimura curves.
In the talk we explore how, by using embedding theory, the explicit computation of representations of integers by ternary quadratic forms leads to explicit presentations and fundamental domains of those Fuchsian groups, the computation of CM points, and a rich interpretation of the points in the complex upper half-plane.

N. Pagani, From relations in the moduli spaces of curves, to recursions in Gromov-Witten theory

As discovered by Kontsevich in the ninties, each relation in the cohomology of the moduli space of curves gives rise to recursions for enumerative problems, through Gromov-Witten theory. In this talk, I will focus on a new genus 2 relations, found in collaboration with N.Tarasca, and its consequences. I will try to emphasize the computer-assisted and algorithmic aspects involved in finding these relations.

We present an algorithmic method to e.ectively compute a symplectic representation of any .nite group G, coming from its action on a Riemann Surface M of some fi.xed genus $g \geq . 2$. The action of the group may be speci.ed by its signature or by the explicit group as a permutation group with generators. In particular, we .nd and provide a drawing of a fundamental polygon for M capturing this action of G, a symplectic basis for H1(M;Z) and the action of G represented in such a basis. In many cases we can also explicitly obtain a family of Riemann matrices of principally polarized abelian varieties of dimension g, with the action of G, describing in such a way part of the singular locus of Ag. We implement this procedure over SAGE[2], and we present several examples using it. This work has been published as \Adapted hyperbolic polygons and symplectic representations for group actions on Riemann surfaces" [1] and the SAGE routines are available online at https://sites.google.com/a/u.uchile.cl/polygons/home

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