Related papers: Rational singularities associated to pairs

We prove several results about the behavior Du Bois singularities and Du Bois pairs in families. Some of these generalize existing statements about Du Bois singularities to the pair setting while others are new even in the non-pair setting.…

In this paper, we generalize the notion of rational singularities for any reflexive sheaf of rank $1$, link our notion of rational singularities with the notion of rational singularities in [Kov11], and prove generalizations of standard…

We begin by giving a derived characterization of rational singularities for pairs in the sense of Schwede--Takagi. This characterization extends a characterization of rational singularities due to Lank--Venkatesh to pairs on normal…

Rational pairs generalize the notion of rational singularities to reduced pairs $(X,D)$. In this paper we deal with the problem of determining whether a normal variety $X$ has a rationalizing divisor, i.e. a reduced divisor $D$ such that…

In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being Q-Gorenstein or the pair being log Q-Gorenstein. The main…

Let $Y$ be a generic link of a subvariety $X$ of a nonsingular variety $A$. We give a description of the Grauert-Riemenschneider canonical sheaf of $Y$ in terms of the multiplier ideal sheaves associated to $X$ and use it to study the…

A resolution-free definition of rational singularities is introduced, and it is proved that for a variety admitting a resolution of singularities, so in particular in characteristic zero, this is equivalent to the usual definition. It is…

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 $(X,\Delta)$.…

Rational pairs, recently introduced by Koll\'ar and Kov\'acs, generalize rational singularities to pairs $(X,D)$. Here $X$ is a normal variety and $D$ is a reduced divisor on $X$. Integral to the definition of a rational pair is the notion…

We prove that good quotients of algebraic varieties with 1-rational singularities also have 1-rational singularities. This refines a result of Boutot on rational singularities of good quotients.

The authors review results implicit in their recent paper [2] on the product/quotient representation of rationals by rationals of the type $( an + b )/ ( An+ B )$ and give a detailed account of a particular related non-intuitive…

Let X be a smooth complex variety and Y be a closed subvariety of X, or more generally, a closed subscheme of X. We are interested in invariants attached to the singularities of the pair (X, Y). We discuss various methods to construct such…

In the present paper, we devise a version of topological $L^2$-Serre duality for singular complex spaces with arbitrary singularities. This duality is used to deduce various new $L^2$-vanishing theorems for the $\bar{\partial}$-equation on…

We generalize the notions of F-regular and F-pure rings to pairs $(R,\a^t)$ of rings $R$ and ideals $\a \subset R$ with real exponent $t > 0$, and investigate these properties. These ``F-singularities of pairs'' correspond to singularities…

An algorithm for resolution of singularities in characteristic zero is described. It is expressed in terms of multi-ideals, that essentially are defined as a finite sequence of pairs, each one consiting of a sheaf of ideals and a positive…

By using superisolated surface singularities whose link is a rational homology sphere we give counterexamples to some of the most important conjetures concernig invariants of normal surface singularities.

We study the singularities of the secant variety $\Sigma(X,L)$ associated to a smooth variety $X$ embedded by a sufficiently positive adjoint bundle $L$. We show that $\Sigma(X,L)$ is always Du Bois singular. Examples of secant varieties…

We prove that the $F$-jumping numbers of the test ideal $\tau(X; \Delta, \ba^t)$ are discrete and rational under the assumptions that $X$ is a normal and $F$-finite variety over a field of positive characteristic $p$, $K_X+\Delta$ is…

One of the central aims of the Minimal Model Program is to show that a projective log canonical pair $(X,\Delta)$ with $K_X+\Delta$ pseudoeffective has a good model, i.e.\ a minimal model $(Y,\Delta_Y)$ such that $K_Y+\Delta_Y$ is…

Let $X$ be a nonsingular rational variety. We prove that $X\times \mathbb{C}^2$ is uniformly rational. It follows that nonsingular stably rational varieties are stably uniformly rational.