Documents  14H10 | enregistrements trouvés : 8

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

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.

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.