$F$-signature under birational morphisms

TL;DR

This paper investigates how the F-signature, an invariant in algebraic geometry, behaves under proper birational morphisms, revealing conditions for increase, decrease, or doubling of the invariant.

Contribution

It establishes conditions under which F-signature increases or doubles under birational maps, and provides examples of its decrease and Hilbert-Kunz multiplicity increase.

Findings

F-signature increases under small morphisms or when $K_Y geq \, \, \\pi^*K_X$

F-signature can be at least twice as large after certain birational maps

Examples show F-signature can decrease and Hilbert-Kunz multiplicity can increase without these conditions

Abstract

We study $F$-signature under proper birational morphisms $\pi : Y \to X$, showing that $F$-signature strictly increases for small morphisms or if $ K_Y \geq \pi ^*K_X$. In certain cases, we can even show that the $F$-signature of $Y$ is at least twice as that of $X$. We also provide examples of $F$-signature dropping and Hilbert-Kunz multiplicity increasing under birational maps without these hypotheses.