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H 1 The rationality problem for forms of moduli spaces of stable marked curves

Auteurs : Reichstein, Zinovy (Auteur de la Conférence)
CIRM (Editeur )

Résumé : 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

Codes MSC :
14H10 - Families, moduli (algebraic)
20G15 - Linear algebraic groups over arbitrary fields
14E08 - Rationality questions

 Informations sur la Vidéo Langue : Anglais Date de publication : 25/09/2015 Date de captation : 01/09/2015 Collection : Research talks ; Algebraic and Complex Geometry ; Topology Format : quicktime ; audio/x-aac Durée : 01:00:51 Domaine : Algebraic & Complex Geometry ; Topology Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2015-09-01_Reichstein.mp4 Informations sur la rencontre Nom de la rencontre : Cohomological Methods in the Theory of Algebraic GroupsOrganisateurs de la rencontre : Calmes, Baptiste ; Chernousov, Vladimir ; Karpenko, NikitaDates : 31/08/2015 - 04/09/2015 Année de la rencontre : 2015 URL Congrès : http://conferences.cirm-math.fr/1001.htmlCitation Data DOI : 10.24350/CIRM.V.18823503 Cite this video as: Reichstein, Zinovy (2015). The rationality problem for forms of moduli spaces of stable marked curves. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18823503 URI : http://dx.doi.org/10.24350/CIRM.V.18823503

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