Frobenius on the cohomology of thickenings

TL;DR

This paper studies the Frobenius map's injectivity on thickenings of smooth projective varieties in positive characteristic, providing uniform bounds that are sharp for hypersurfaces and elliptic curves.

Contribution

It establishes characteristic-independent bounds for Frobenius injectivity on thickenings of smooth varieties, including complete intersections and elliptic curves.

Findings

Bounds are independent of the characteristic.

Bounds are sharp for hypersurfaces.

Bounds are sharp for elliptic curves.

Abstract

We investigate the injectivity of the Frobenius map on thickenings of smooth varieties in projective space over a field of positive characteristic. We obtain uniform bounds -- i.e., independent of the characteristic -- on the thickening that ensures an injective Frobenius map when the projective variety is a smooth complete intersection or an arbitrary projective embedding of an elliptic curve. Our bounds are sharp in the case of hypersurfaces, and in the case of elliptic curves.