Low degree hypersurfaces of projective toric varieties defined over a $C_1$ field have a rational point

Abstract

Quasi algebraically closed fields, or $C_1$ fields, are defined in terms of a low degree condition. Namely, the field $K$ is $C_1$ if every degree $d$ hypersurface of the projective space $\mathbb{P}_K^n$ contains a $K$-point as soon as $d\leq n$. In this article we define a notion of low toric degree generalizing this condition for hypersurfaces of simplicial projective split toric varieties. This allows us to prove a particular case of the $C_1$ conjecture of Koll\'{a}r, Lang and Manin : any smooth separably rationally connected variety that can be embedded as such a hypersurface over a $C_1$ field has a rational point. Our results are based on the fact that the ambient toric varieties are Mori Dream Spaces : they are naturally endowed with homogeneous coordinates and their Minimal Model Program works in all cases.