Computing jumping numbers in higher dimensions

TL;DR

This paper extends algorithms for computing jumping numbers from rational surfaces to higher-dimensional varieties, introducing new divisor concepts to efficiently identify relevant jumping numbers.

Contribution

It generalizes the notion of antinef divisors to $ ext{pi}$-antieffective divisors, enabling computation of jumping numbers in dimensions three and above.

Findings

Introduces $ ext{pi}$-antieffective divisors for higher dimensions

Provides a method to identify a small subset of candidate jumping numbers

Many candidate numbers are confirmed as actual jumping numbers

Abstract

The aim of this paper is to generalize the algorithm to compute jumping numbers on rational surfaces described in [AAD14] to varieties of dimension at least 3. Therefore, we introduce the notion of $\pi$-antieffective divisors, generalizing antinef divisors. Using these divisors, we present a way to find a small subset of the `classical' candidate jumping numbers of an ideal, containing all the jumping numbers. Moreover, many of these numbers are automatically jumping numbers, and in many other cases, it can be easily checked.