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jeunes en arithmétique et variétés algébriques

jeunes en arithmétique et variétés algébriques

IRRATIONALITY, UNRAMIFIED COHOMOLOGY AND HYPERSURFACES

program by Stefan Schreieder

Maison Clément (France), 8-12 July 2024

One fundamental problem in classical algebraic geometry is understanding whether a smooth hypersurface $X\subseteq \mathbb{P}^{n+1}$ of degree $d$ is rational, i.e. birational to $\mathbb{P}^n$. Easy geometric constructions permit to show that such an hypersurface is rational if $d\leq 2$; on the other hand, the existence of global differentials obstructs rationality if $d\geq n+2$. This settles the case $n = 1$ and, together with the rationality of cubic surfaces, the case $n=2$. In higher dimension the situation becomes more complicated. For instance, only in the seventies Clemens-Griffiths and Iskovskikh-Manin showed the irrationality of $3$-dimensional hypersurfaces of degree $d=3$ and $d= 4$ respectively.

Later on, new tools to prove irrationality have emerged: in the nineties, Kollár showed that a general hypersurface of degree $d\geq 2(n+3)/3$ is irrational by using differential forms and degenerations to characteristic $2$; around ten years ago, Voisin introduced the decomposition of the diagonal, a technique to prove that irrationality via degeneration to mildly singular varieties. Combining Kollár's arguments with a refinement of Voisin's method due to Colliot-Thélène-Pirutka, Totaro proved that a general hypersurface $X$ of degree $d\geq 2(n +2)/3$ is stably irrational, i.e. $X\times \mathbb{P}^m$ is irrational for every $m$.

The ultimate goal of the meeting will be to understand the proof of the following recent logarithmic bound of Schreieder: a general hypersurface of degree $d\geq \log_2(n)+2$ is stably irrational. Needless to say, for $n \to \infty$ this logarithmic bound is way stronger than the previously known linear bounds. It will be the occasion to learn the techniques cited above as well as unramified cohomology, which all play a crucial role in Schreieder's proof.

Arbeitsgemeinschaft à la Française

We plan to revive the tradition of an annual series of conferences in arithmetic geometry based on the model of Oberwolfach's Arbeitsgemeinschaft. They were previously organized by Jean-Benoît Bost and François Loeser from 1995 to 2002 at Luminy. The first editions covered topics such as Euler systems, higher class field theory or modular forms and Galois representations.

The public we have in mind consists mainly of PhD students and early postdocs, with the aim of offering a friendly ambience to learn mathematical subjects that do not necessarily belong to one's own research area. At the end of each edition, the topic for the next one will be voted, and a scientific program will be written in close collaboration with a leading expert. About six months before the conference, participants will apply and the talks will be distributed among them.

Organizers

Emiliano Ambrosi (Université de Strasbourg) /

Giuseppe Ancona (Université de Strasbourg) /

Mattia Cavicchi (Université Paris Saclay) /

Marco Maculan (Institut Mathématique de Jussieu) /

Scientific Committee

Stefan Schreieder (Leibniz Universität Hannover)