Singularities of generic linkage of algebraic varieties

TL;DR

This paper investigates the singularities of generic linkages of algebraic varieties, providing criteria for rational singularities, analyzing log canonical thresholds, and offering a liaison method to extend regularity bounds.

Contribution

It introduces a new description of the Grauert-Riemenschneider sheaf in terms of multiplier ideals and applies it to study singularities and regularity in algebraic geometry.

Findings

Criteria for rational singularities in generic linkages

Log canonical thresholds increase under generic linkage

A simple liaison method to extend regularity bounds

Abstract

Let $Y$ be a generic link of a subvariety $X$ of a nonsingular variety $A$. We give a description of the Grauert-Riemenschneider canonical sheaf of $Y$ in terms of the multiplier ideal sheaves associated to $X$ and use it to study the singularities of $Y$. As the first application, we give a criterion when $Y$ has rational singularities and show that log canonical threshold increases and log canonical pairs are preserved in generic linkage. As another application we give a quick and simple liaison method to generalize the results of de Fernex-Ein and Chardin-Ulrich on the Castelnuovo-Mumford regularity bound for a projective variety.