On positivity of the limit F-signature

TL;DR

This paper proves a conjecture about the positivity of F-signatures in certain singularities and smooth hypersurfaces, using degenerations, birational geometry, and K-stability inspired techniques.

Contribution

It establishes the conjecture for three-dimensional non-weakly exceptional singularities and low-degree smooth hypersurfaces, advancing understanding of F-signature positivity.

Findings

F-signatures remain bounded away from zero for specific singularities.

The conjecture holds for three-dimensional non-weakly exceptional singularities.

It is verified for smooth hypersurfaces of very low degree.

Abstract

We study a conjecture of Carvajal-Rojas, Schwede and Tucker which states that for a complex KLT singularity $(R, \mathfrak{m})$, the F-signatures of the reductions of $R$ to characteristic $p \gg 0$ remain bounded away from zero as $p \to \infty$. We prove that this conjecture holds for three-dimensional non-weakly exceptional singularities by an inductive argument. We also prove that the conjecture holds for smooth hypersurfaces of very low degree by constructing isotrivial normal toric degenerations. By considering the version of this conjecture for the Frobenius-alpha invariant, our techniques are inspired by K-stability theory and involve using degenerations and birational geometry.