In this paper we study singularities defined by the action of Frobenius in characteristic p > 0. We prove results analogous to inversion of adjunction along a center of log canonicity. For example, we show that if X is a Gorenstein normal variety then to every normal center of sharp F-purity W ⊆ X such that X is F-pure at the generic point of W , there exists a canonically defined -ޑdivisorFurthermore, the singularities of X near W are "the same" as the singularities of (W, W ). As an application, we show that there are finitely many subschemes of a quasiprojective variety that are compatibly split by a given Frobenius splitting. We also reinterpret Fedder's criterion in this context, which has some surprising implications.
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 Qdivisor ∆ such that −K X − ∆ is ample and (X, ∆) 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 Hacon's 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.
In this paper, we study a positive characteristic analogue of the centers of log canonicity of a pair $(R, \Delta)$. 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 $\Delta = 0$, we show that uniformly $F$-compatible ideals coincide with the annihilators of the $\mathcal{F}(E_R(k))$-submodules of $E_R(k)$ as defined by Smith and Lyubeznik.Comment: Typos corrected. To appear in Math.
We prove that the F-jumping numbers of the test ideal τ (X ; , a t ) are discrete and rational under the assumptions that X is a normal and F-finite scheme over a field of positive characteristic p, K X + is Q-Cartier of index not divisible p, and either X is essentially of finite type over a field or the sheaf of ideals a is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero. Mathematics Subject Classification (2000) 13A35 · 14B05M. Blickle was supported by the Deutsche Forschungsgemeinschaft (DFG) through a Heisenberg fellowship and through the SFB/Transregio 45 Periods, moduli spaces and arithmetic of algebraic varieties. K.
Let (R, m, k) be an excellent local ring of equal characteristic. Let j be a positive integer such that H i m (R) has finite length for every 0 ≤ i < j. We prove that if R is F-injective in characteristic p > 0 or Du Bois in characteristic 0, then the truncated dualizing complex τ >−j ω q R is quasi-isomorphic to a complex of k-vector spaces. As a consequence, F-injective or Du Bois singularities with isolated non-Cohen-Macaulay locus are Buchsbaum. Moreover, when R has F-rational or rational singularities on the punctured spectrum, we obtain stronger results generalizing [Ma15] and [Ish84].
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