Related papers: Centers of F-purity

We introduce a mixed characteristic analog of log canonical centers in characteristic $0$ and centers of $F$-purity in positive characteristic, which we call centers of perfectoid purity. We show that their existence detects (the failure…

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…

In this paper, we investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to "F-singularities of pairs." The notions of…

We analyze adjunction and inversion of adjunction for the $F$-purity of divisor pairs in characteristic $p > 0$. In this vein, we give a complete answer for principal divisors under $\mathbb{Q}$-Gorenstein assumptions but without…

Given an ambient variety $X$ and a fixed subvariety $Z$ we give sufficient conditions for the existence of a boundary $\Delta$ such that $Z$ is a log canonical center for the pair $(X, \Delta)$. We also show that under some additional…

We introduce and study a log discrepancy function on the space of semivaluations centered on an integral noetherian scheme of positive characteristic. Our definition shares many properties with the analogue in characteristic zero; we prove…

In this article we give two independent proofs of the positive characteristic analog of the log terminal inversion of adjunction. We show that for a pair $(X, S+B)$ in characteristic $p>0$, if $(S^n, B_{S^n})$ is strongly $F$-regular, then…

We present a new relation between an invariant of singularities in characteristic zero (the log canonical threshold) and an invariant of singularities defined via the Frobenius morphism in positive characteristic (the F-pure threshold). We…

Let $R=k[x_1,\dots,x_n]$ be a polynomial ring over a prefect field of positive characteristic. Let $I$ be an unmixed ideal in $R$ and let $J$ be a generic link of $I$ in $S=R[u_{ij}]_{c \times r}$. We describe the parameter test submodule…

Let $R$ be a commutative (Noetherian) local ring of prime characteristic $p$ that is $F$-pure. This paper is concerned with comparison of three finite sets of radical ideals of $R$, one of which is only defined in the case when $R$ is…

Let $k$ be an $F$-finite field containing an infinite perfect field of positive characteristic. Let $(X, \Delta)$ be a projective log canonical pair over $k$. In this note we show that, for a semi-ample divisor $D$ on $X$, there exists an…

In this article, we consider the conjectured relationship between F-purity and log canonicity for polynomials over the complex numbers. We associate to a collection M of n monomials a rational polytope P contained in [0,1]^n. Using P and…

Consider a pair $(R, \ba^t)$ where $R$ is a ring of positive characteristic, $\ba$ is an ideal such that $a \cap $R^{\circ} \neq \emptyset$, and $t > 0$ is a real number. In this situation we have the ideal $\tau_R(\ba^t)$, the generalized…

Let $f : (X, \Delta) \to Y$ be a flat, projective family of sharply $F$-pure, log-canonically polarized pairs over an algebraically closed field of characteristic $p >0$ such that $p \nmid \ind(K_{X/Y} + \Delta)$. We show that $K_{X/Y} +…

For a normal F-finite variety $X$ and a boundary divisor $\Delta$ we give a uniform description of an ideal which in characteristic zero yields the multiplier ideal, and in positive characteristic the test ideal of the pair $(X,\Delta)$.…

We prove that the finitistic test ideal $\tau_{\rm fg}(R, \Delta, \mathfrak{a}^t)$ coincides with the big test ideal $\tau_{\rm b}(R, \Delta, \mathfrak{a}^t)$ if the pair $(R,\Delta)$ is numerically log $\mathbb{Q}$-Gorenstein.

The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic p. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via…

In this note, we consider a corollary of the ACC conjecture for F-pure thresholds. Specifically, we show that the F-pure threshold (and more generally, the test ideals) associated to a polynomial with an isolated singularity are locally…

We introduce a new variant of tight closure and give an interpretation of adjoint ideals via this tight closure. As a corollary, we prove that a log pair $(X,\Delta)$ is plt if and only if the modulo $p$ reduction of $(X,\Delta)$ is…

Suppose $(X, \Delta)$ is a log-$\bQ$-Gorenstein pair. Recent work of M. Blickle and the first two authors gives a uniform description of the multiplier ideal $\mJ(X;\Delta)$ (in characteristic zero) and the test ideal $\tau(X;\Delta)$ (in…