The F-signature Function on the Ample Cone

TL;DR

This paper introduces an F-signature function on the ample cone of a globally F-regular variety, proving its continuity, extension to the boundary, and relation to volume, revealing new geometric insights in positive characteristic.

Contribution

It defines and analyzes the F-signature function on the ample cone, establishing its Lipschitz continuity, boundary extension, and comparison with volume functions, a novel contribution in positive characteristic geometry.

Findings

F-signature function is locally Lipschitz continuous.

The F-signature function extends to the boundary of the ample cone.

For nef but not big divisors, the F-signature extension is zero.

Abstract

For any fixed globally F-regular projective variety X over an algebraically closed field of positive characteristic, we study the F-signature of section rings of X with respect to the ample Cartier divisors on X. In particular, we define an F-signature function on the ample cone of X and show that it is locally Lipschitz continuous. We further prove that the F-signature function extends to the boundary of the ample cone. We also establish an effective comparison between the F-signature function and the volume function on the ample cone. As a consequence, we show that for divisors that are nef but not big, the extension of the F-signature is zero.