F-signature of pairs: Continuity, p-fractals and minimal log discrepancies

TL;DR

This paper studies the properties of the F-signature of triples, demonstrating its continuity, convexity, and semi-continuity, and relating it to minimal log discrepancies and p-fractals in algebraic geometry.

Contribution

It establishes the continuity and convexity of the F-signature function, explores its semi-continuity on spectra, and links it to minimal log discrepancies and p-fractals.

Findings

F-signature is continuous as a function of t.

F-signature is convex for principal ideals.

Minimal log discrepancy bounds the F-signature from above.

Abstract

This paper contains a number of observations on the {$F$-signature} of triples $(R,\Delta,\ba^t)$ introduced in our previous joint work. We first show that the $F$-signature $s(R,\Delta,\ba^t)$ is continuous as a function of $t$, and for principal ideals $\ba$ even convex. We then further deduce, for fixed $t$, that the $F$-signature is lower semi-continuous as a function on $\Spec R$ when $R$ is regular and $\ba$ is principal. We also point out the close relationship of the signature function in this setting to the works of Monsky and Teixeira on Hilbert-Kunz multiplicity and $p$-fractals. Finally, we conclude by showing that the minimal log discrepancy of an arbitrary triple $(R,\Delta,\ba^t)$ is an upper bound for the $F$-signature.