相关论文: Rational singularities associated to pairs
It is a conjecture of Koll\'ar that a variety $X$ with rational singularities in some open subvariety $U$ has a rationalification; that is, a proper, birational morphism $f: Y \rightarrow X$ such that $Y$ has rational singularities, and…
Let $(Z,o)$ be a three-dimensional terminal singularity of type $cA/r$. We prove that all exceptional divisors over $o$ with discrepancies $\le 1$ are rational.
A set of $m$ distinct nonzero rationals $\{a_1, a_2,\ldots, a_m\}$ such that $a_i a_j+1$ is a perfect square for all $1\le i <j \le m$, is called a rational Diophantine $m$-tuple. If in addition, $a_i^2+1$ is a perfect square for $1\le i\le…
An algebraic isopair is a commuting pair of pure isometries that is annihilated by a polynomial defining a distinguished variety $\mathcal{V}$. The notion of the rank of a pure algebraic isopair with finite bimultiplicity is introduced. For…
This work is concerned with categorical methods for studying singularities. Our focus is on birational derived splinters, which is a notion that extends the definition of rational singularities beyond varieties over fields of characteristic…
The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the…
Inspired by several works on jet schemes and motivic integration, we consider an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. The resulting invariant, which we call Jacobian…
The weighted dual graph of a two-dimensional normal singularity $(X, x)$ represents the topological nature of the exceptional locus of its minimal log resolution. $(X, x)$ and its graph are said to be taut if the singularity can be uniquely…
We extend Hacon--M\textsuperscript{c}Kernan's rational chain connectedness theorem to the complex analytic setting. As a consequence, we prove that the fibers of any resolution of singularities of complex analytic kawamata log terminal…
This work establishes simple criteria for detecting higher rational singularities via the intersection Du Bois complex and the irrationality complex of a normal variety over the complex numbers.
In prime characteristic there are important invariants that allow us to measure singularities. For certain cases, it is known that they are rational numbers. In this article, we show this property for Stanley-Reisner rings in several cases.
In this survey, we explain a version of topological $L^2$-Serre duality for singular complex spaces with arbitrary singularities. This duality can be used to deduce various $L^2$-vanishing theorems for the $\overline\partial$-equation on…
We investigate semiconjugate rational functions, that is rational functions $A,$ $B$ related by the functional equation $A\circ X=X\circ B$, where $X$ is a rational function of degree at least two. We show that if $A$ and $B$ is a pair of…
We prove that hypersurfaces defined by irreducible square-free polynomials have rational singularities. As an easy consequence, we deduce that certain (possibly non-square-free) polynomials associated to pairs of square-free polynomials…
Let k be an algebraically closed field of characteristic zero. We show that the centre of a homologically homogeneous, finitely generated k-algebra has rational singularities. In particular if a finitely generated normal commutative…
The main purpose of this paper is to establish some useful partial resolutions of singularities for pairs from the minimal model theoretic viewpoint. We first establish the existence of log canonical modifications of normal pairs under some…
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…
Consider a projective variety $X \subset \mathbb{P}^n$ (over an algebraically closed field of characteristic zero), together with a (reduced) simple normal crossings divisor $E \subset \mathbb{P}^n$, where the degrees of both $X$ and $E$…
We define a notion of Hodge modules with rational singularities. A variety has rational singularities in the usual sense, if it is normal and the Hodge module related to intersection cohomology has rational singularities in the present…
Higher rational and higher Du Bois singularities have recently been introduced as natural generalizations of the standard definitions of rational and Du Bois singularities. In this note, we discuss these properties for isolated…