Contribution of jumping numbers by exceptional divisors, with an appendix by Karen E. Smith and Kevin Tucker

TL;DR

This paper explores conditions under which exceptional divisors contribute jumping numbers in algebraic geometry, providing new examples and extending previous results from surfaces to higher dimensions.

Contribution

It establishes necessary and sufficient conditions for exceptional divisors to contribute jumping numbers and constructs a novel example where an exceptional divisor does not contribute any.

Findings

Identifies conditions for contribution of jumping numbers by exceptional divisors

Constructs an example of a non-contributing exceptional divisor in higher dimensions

Extends results from surface cases to arbitrary dimension varieties

Abstract

We investigate some necessary and sufficient conditions for an exceptional divisor to contribute jumping numbers of an effective divisor on a variety of arbitrary dimension, inspired by the results for curves on surfaces by Smith and Thompson and Tucker. In particular, we construct an example of an exceptional divisor that is not contracted in the log canonical model, and does not contribute any jumping numbers.