Transverse linear subspaces to hypersurfaces over finite fields

TL;DR

This paper refines criteria for the existence of linear sections on hypersurfaces over finite fields, extending previous results to higher codimensions and special classes of hypersurfaces.

Contribution

It provides new bounds and conditions for the existence of smooth and reduced linear sections on hypersurfaces over finite fields, generalizing earlier work by Ballico.

Findings

Refined bounds for higher codimensional linear sections

Criteria for hyperplane sections on Frobenius classical hypersurfaces

Existence results for reduced hyperplane sections

Abstract

Ballico proved that a smooth projective variety $X$ of degree $d$ over a finite field of $q$ elements admits a smooth hyperplane section if $q\geq d(d-1)^{\dim X}$. In this paper, we refine this criterion for higher codimensional linear sections on smooth hypersurfaces and for hyperplane sections on Frobenius classical hypersurfaces. We also prove a similar result for the existence of reduced hyperplane sections on reduced hypersurfaces.