On positivity of the limit F-signature

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 $p\to \mathrm{\infty }$. 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.