Jet schemes and singularities of W^r_d(C) loci

TL;DR

This paper investigates the jet schemes of the theta divisor on smooth projective curves, establishing their irreducibility and deriving formulas for log canonical thresholds, thereby providing new insights into the singularities of W^r_d(C) loci.

Contribution

It introduces a novel approach using jet schemes to analyze the singularities of W^r_d(C) loci and recovers Kempf's theorem through this method.

Findings

Jet schemes of the theta divisor are irreducible.

Dimensions of jet schemes are explicitly estimated.

Log canonical thresholds are computed for general curves.

Abstract

Kempf proved that the theta divisor of a smooth projective curve C has rational singularities. In this paper we estimate the dimensions of the jet schemes of the theta divisor and show that all these schemes are irreducible. In particular, we recover Kempf's theorem in this way. For general projective smooth curves, our method also gives a formula for the log canonical threshold of the pair (\pic^d(C), W^r_d(C)).