Transverse linear subspaces to hypersurfaces over finite fields

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 $q\ge d(d-1{)}^{\dim X}$. 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.

19 pages; this version contains stronger results and a new theorem for Frobenius classical hypersurfaces. It has the same contents as the published version