Transverse linear subspaces to hypersurfaces over finite fields

Abstract

Ballico proved that a smooth projective variety X of degree d and dimension m over a finite field of q elements admits a smooth hyperplane section if $q\ge d{(d-1)}^m$. 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.