In moduli theory, it happens often that a moduli space is constructed as a quotient. This is a powerful tool, in fact from the quotient structure one can infer several interesting properties about the properties of the moduli space itself. In this talk, I will recall briefly a construction of the moduli stack of trigonal curves as a quotient stack, that I gave in a joint work with Vistoli a few years ago. Then I will move to a recent work with Loenne, where we draw from this construction some surprising results on the fundamental group of the moduli space, that reveals to be of completely different nature from the space of hyperelliptic curve.[-]

Let $\overline{M_{g,n}}$ be the moduli space of stable curves of genus $g$ with $n$ marked points. It is a classical problem in algebraic geometry to determine which of these spaces are rational over $\mathbb{C}$. In this talk, based on joint work with Mathieu Florence, I will address the rationality problem for twisted forms of $\overline{M_{g,n}}$ . Twisted forms of $\overline{M_{g,n}}$ are of interest because they shed light on the arithmetic geometry of $\overline{M_{g,n}}$, and because they are coarse moduli spaces for natural moduli problems in their own right. A classical result of Yu. I. Manin and P. Swinnerton-Dyer asserts that every form of $\overline{M_{0,5}}$ is rational. (Recall that the $F$-forms $\overline{M_{0,5}}$ are precisely the del Pezzo surfaces of degree 5 defined over $F$.) Mathieu Florence and I have proved the following generalization of this result.
Let $ n\geq 5$ is an integer, and $F$ is an infinite field of characteristic $\neq$ 2.
(a) If $ n$ is odd, then every twisted $F$-form of $\overline{M_{0,n}}$ is rational over $F$.
(b) If $n$ is even, there exists a field extension $F/k$ and a twisted $F$-form of $\overline{M_{0,n}}$ which is unirational but not retract rational over $F$.
We also have similar results for forms of $\overline{M_{g,n}}$ , where $g \leq 5$ (for small $n$ ). In the talk, I will survey the geometric results we need about $\overline{M_{g,n}}$ , explain how our problem reduces to the Noether problem for certain twisted goups, and how this Noether problem can (sometimes) be solved.

Keywords: rationality - moduli spaces of marked curves - Galois cohomology - Noether's problem[-]

In this talk I will discuss about the unirationality of the Hurwitz spaces $H_{g,d}$ parametrizing d-sheeted branched simple covers of the projective line by smooth curves of genus $g$. I will summarize what is already known and formulate some questions and speculations on the general behaviour. I will then present a proof of the unirationality of $H_{12,8}$ and $H_{13,7}$, obtained via liaison and matrix factorizations. This is part of two joint works with Frank-Olaf Schreyer.[-]

A meromorphic differential on a Riemann surface is said to be real-normalized if all its periods are real. Real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles form real orbifolds whose topology is closely related to that of moduli spaces of Riemann surfaces with marked points. We propose a combinatorial model for the real normalized differentials with a single order 2 pole and use it to analyze certain ergodic properties of the corresponding absolute period foliation. It is a joint work with Igor Krichever and Sergey Lando.[-]

In the past several decades, it has been established that numerous fundamental invariants in physics and geometry can be expressed in terms of the so-called Witten-Kontsevich intersection numbers. In this talk, I will present a novel approach for calculating their large genus asymptotics. Our technique is based on a resurgent analysis of the n-point functions of such intersection numbers, which are computed using determinantal formulae and depend significantly on the presence of an underlying ODE. I will show how, with this approach, we are able to extend the recent results of Aggarwal with the computation of all subleading corrections. If time permits, I will also explain how the same technique can be applied to address other enumerative problems.
Based on a joint work with B. Eynard, E. Garcia-Failde, P. Gregori, D. Lewanski.[-]

The general principally polarized abelian variety of dimension at most five is known to be a Prym variety. This reduces the study of abelian varieties of small dimension to the beautifully concrete theory of algebraic curves. I will discuss recent breakthrough on finding a structure theorem for principally polarized abelian varieties of dimension six as Prym-Tyurin varieties associated to covers with $E_6$-monodromy, and the implications this uniformization result has on the geometry of the moduli space $A_6$. This is joint work with Alexeev, Donagi, Izadi and Ortega.[-]