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

Robin Guilbot (1)

1 IMPA - Instituto Nacional de Matemática Pura e Aplicada

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 $\PP_K^n$ 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 toric varieties. Our results are based on the fact that the toric varieties taken as ambient spaces are Mori Dream Spaces : they are naturally endowed with homogeneous coordinates and their Minimal Model Program is quite simple to describe explicitly and works in all cases. As an application we show that any smooth separably rationnally connected variety that can be embedded as a hypersurface in a projective toric variety over a $C_1$ field $K$ has a rational point over $K$, as expected by more general conjectures attributed to Kollár, Lang and Manin.

Mots clés: Algebraic Geometry; Arithmetic Geometry; Toric Varieties; Rational points; Minimal Model Program

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]

Robin Guilbot. Low degree hypersurfaces of projective toric varieties defined over a $C_1$ field have a rational point. 2014. ⟨hal-00986831v1⟩