General hyperplane sections of threefolds in positive characteristic

TL;DR

This paper investigates the singularities of hyperplane sections of threefolds in positive characteristic, establishing conditions under which these sections have rational double points or klt singularities.

Contribution

It proves that hyperplane sections inherit rational double points from canonical singularities and, for p>5, also inherit klt singularities from the threefold.

Findings

Hyperplane sections of threefolds with canonical singularities have rational double points.

For p>5, hyperplane sections of threefolds with klt singularities also have klt singularities.

Results extend understanding of singularity behavior in positive characteristic.

Abstract

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$.