Dual F-signature of Cohen-Macaulay modules over rational double points

TL;DR

This paper computes the dual F-signature for Cohen-Macaulay modules over rational double points, providing new explicit values and methods that also apply to Hilbert-Kunz multiplicity in positive characteristic.

Contribution

It explicitly determines the dual F-signature for Cohen-Macaulay modules over rational double points, a previously unknown case, and introduces a method applicable to Hilbert-Kunz multiplicity.

Findings

Dual F-signature values for rational double points computed

Method applicable to Hilbert-Kunz multiplicity discussed

Enhances understanding of singularity invariants in positive characteristic

Abstract

The dual $F$-signature is a numerical invariant defined via the Frobenius morphism in positive characteristic. It is known that the dual $F$-signature characterizes some singularities. However, the value of the dual $F$-signature is not known except only a few cases. In this paper, we determine the dual $F$-signature of Cohen-Macaulay modules over two-dimensional rational double points. The method for determining the dual $F$-signature is also valid for determining the Hilbert-Kunz multiplicity. We discuss it in appendix.