Karl Schwede

Globally F-regular and log Fano varieties☆

Abstract We prove that every globally F-regular variety is log Fano. In other words, if a prime characteristic variety X is globally F-regular, then it admits an effective Q -divisor Δ such that − K X − Δ is ample and ( X , Δ ) has controlled (Kawamata log terminal, in fact globally F-regular) singularities. A weak form of this result can be viewed as a prime characteristic analog of de Fernex and Hacons new point of view on Kawamata log terminal singularities in the non- Q -Gorenstein case. We also prove a converse statement in characteristic zero: every log Fano variety has globally F-regular type. Our techniques apply also to F-split varieties, which we show to satisfy a “log Calabi–Yau” condition. We also prove a Kawamata–Viehweg vanishing theorem for globally F-regular pairs.

CENTERS OF F-PURITY

In this paper, we study a positive characteristic analogue of the centers of log canonicity of a pair (R, Δ). We call these analogues centers of F-purity. We prove positive characteristic analogues of subadjunction-like results, prove new stronger subadjunction-like results, and in some cases, lift these new results to characteristic zero. Using a generalization of centers of F-purity which we call uniformly F-compatible ideals, we give a characterization of the test ideal (which unifies several previous characterizations). Finally, in the case that Δ = 0, we show that uniformly F-compatible ideals coincide with the annihilators of the

The canonical sheaf of Du Bois singularities

{\mathcal{F}(E_R(k))}

F-singularities via alterations

-submodules of ER(k) as defined by Lyubeznik and Smith.

p−1-Linear Maps in Algebra and Geometry

In this paper, we prove that singularities of

Bertini theorems for F-singularities

F

Rings of Frobenius operators

-injective type are Du~Bois. This extends the correspondence between singularities associated to the minimal model program and singularities defined by the action of Frobenius in positive characteristic.

Depth of

We prove that a Cohen–Macaulay normal variety X has Du Bois singularities if and only if π∗ωX′(G)≃ωX for a log resolution π:X′→X, where G is the reduced exceptional divisor of π. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen–Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proof that (generalizations of) semi-log-canonical singularities are Du Bois, in the Cohen–Macaulay case. It also follows that the Kodaira vanishing theorem holds for semi-log-canonical varieties and that Cohen–Macaulay semi-log-canonical singularities are cohomologically insignificant in the sense of Dolgachev.