Arithmetic and geometric deformations of $F$-pure and $F$-regular singularities

Abstract

Given a normal Q-Gorenstein complex variety X, we prove that if one spreads it out to a normal Q-Gorenstein scheme X of mixed characteristic whose reduction X p modulo p has normal F -pure singularities for a single prime p, then X has log canonical singularities. In addition, we show its analog for log terminal singularities, without assuming that X is Q-Gorenstein, which is a generalization of a result of Ma-Schwede. We also prove that two-dimensional strongly F -regular singularities are stable under equal characteristic deformations. Our results give an affirmative answer to a conjecture of Liedtke-Martin-Matsumoto on deformations of linearly reductive quotient singularities.

Arithmetic and geometric deformations of threefolds

“…For a more general version, where the normalisation of the central fibre is log canonical we refer to Corollary 3.5. Analogous statements have been recently proven for deformations of 𝐹-regular and 𝐹-pure singularities of arbitrary dimensions in [35,40].…”

“…For completeness, we include the 2-dimensional case (for the case of strongly 𝐹-regular, cf. [40]). Theorem 4.3.…”

Big Cohen–macaulay Test Ideals in Equal Characteristic Zero via Ultraproducts

“…Here, we briefly review the definition of multiplier ideals and refer the reader to [16], [21] for more details. Throughout this subsection, we assume that X is a normal integral scheme essentially of finite type over a field of characteristic zero or X = Spec R, where (R, m) is a normal local domain essentially of finite type over a field of characteristic zero and R is its m-adic completion.…”

Global generation of test ideals in mixed characteristic and applications

“…This is not the only natural way to define τ + (ω X , a t ). For instance, in [MS18a] and [ST21], when X = Spec R, τ + (a t ) was essentially defined as the sum of τ + ( t m div(f )) where f ∈ a m . In fact we use a variant of this below in Section 7.…”