## Will Sawin - $\overline{M}_{1,n}$ is usually not uniruled in characteristic $p$

epiga:4134 - Épijournal de Géométrie Algébrique, March 20, 2019, Volume 3 - https://doi.org/10.46298/epiga.2019.volume3.4134
$\overline{M}_{1,n}$ is usually not uniruled in characteristic $p$

Authors: Will Sawin

Using etale cohomology, we define a birational invariant for varieties in characteristic $p$ that serves as an obstruction to uniruledness - a variant on an obstruction to unirationality due to Ekedahl. We apply this to $\overline{M}_{1,n}$ and show that $\overline{M}_{1,n}$ is not uniruled in characteristic $p$ as long as $n \geq p \geq 11$. To do this, we use Deligne's description of the etale cohomology of $\overline{M}_{1,n}$ and apply the theory of congruences between modular forms.

Volume: Volume 3
Published on: March 20, 2019
Accepted on: March 20, 2019
Submitted on: December 11, 2017
Keywords: Mathematics - Algebraic Geometry,14M20