Note: For an extended abstract click on the title of the talk. Alex Massarenti, On the automorphisms groups of Hassett moduli spacesThe 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
D. Sevilla and J. R. Sendra, Radical parametrization of algebraic curves and surfaces
L. Beshaj and T. Shaska, On superelliptic curves and their Jacobians
N. Pagani, From relations in the moduli spaces of curves, to recursions in Gromov-Witten theoryAs 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. Antonio Behn, Anita Rojas, Rubi.. Rodri..guez, Symplectic representations for .finite group actions on Riemann surfacesWe 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 |