On the singularities of effective loci of line bundles

TL;DR

This paper proves that semi-regular loci of effective line bundles on smooth projective varieties have at worst rational singularities, extending known results from smooth curves to higher dimensions and providing a specific example for ruled surfaces.

Contribution

It generalizes Kempf's result on rational singularities from smooth curves to higher-dimensional varieties and analyzes the singularities of semi-regular loci.

Findings

Semi-regular loci have at worst rational singularities.

Extension of Kempf's result to higher dimensions.

Explicit example for a ruled surface.

Abstract

We prove that every irreducible component of semi-regular loci of effective line bundles in the Picard scheme of a smooth projective variety has at worst rational singularities. This generalizes Kempf's result on rational singularities of $W^0_d$ for smooth curves. We also work out an example of such loci for a ruled surface.