On Semirational Singularities

Abstract

We study semiresolutions of quasi-projective varieties with properties $G_1$, $S_2$, and seminormality. Equivalently, these are varieties $X$ with Serre's $S_2$ property, such that there exists an open subvariety $U$, with complement of codimension at least two, such that the only singularities of $U$ are (analytically) double normal crossings. Such varieties have been called "demi-normal" by Koll\'ar \cite{Kol13}. First, we discuss why these properties are ideal for the study of nonnormal varieties that appear in the birational classification of varieties. We define semiresolutions and provide examples to illustrate the procedure of gluing along the conductor as the fundamental tool in obtaining a semiresolution of $X$ from a resolution of its normalization. As an application of these methods, we discuss semirational singularities. Our main results are a semismooth Grauert-Riemenschneider vanishing theorem (Theorem 4.2) and a proof that the definition of semirational singularities does not depend on the semiresolution chosen (Theorem 4.3) . The smooth version of (4.2) first appeared in \cite{GR70}. We also explain why semirational singularities are, in particular, Cohen-Macaulay and DuBois. Finally, we discuss the role semiresolutions play in forming a nonnormal interpretation of the results of de Fernex and Hacon in \cite{dFH09}, where the $\mathbb{Q}$-Cartier hypothesis was found to be extraneous in the birational classification of normal varieties.