On the uniform positivity of $F$-signature under reduction modulo $p$

TL;DR

This paper proves that for certain algebraic singularities, the mod p reductions have uniformly positive F-signature for almost all primes p, advancing understanding of singularity invariants in positive characteristic.

Contribution

It confirms the positivity conjecture for pure subrings of regular local rings and reduces the problem to the Gorenstein case, connecting to F-alpha invariants.

Findings

Affirmative proof for pure subrings of regular local rings

Reduction of the conjecture to the Gorenstein case

Discussion of connections with F-alpha invariants

Abstract

Carvajal-Rojas, Schwede and Tucker asked whether the mod $p$ reductions of a complex klt type singularity have uniformly positive $F$-signature for almost all primes $p$. In this paper, we give an affirmative answer to this conjecture in the case of pure subrings of regular local rings--for example, reductive quotient singularities. We also show that the conjecture can be reduced to the Gorenstein case. Finally, we discuss the connection with $F$-alpha invariants--a characteristic $p$ analog of Tian's alpha invariants introduced by Pande--for log Fano pairs.