Rational and Semi-Rational Singularities

TL;DR

This paper investigates Kollár's conjecture on rational singularities and extends the results to semi-rational singularities, providing new insights and conditions for their behavior in various dimensions.

Contribution

It proves that a stronger version of Kollár's conjecture implies the analogous conjecture for Gorenstein semi-rational singularities, with special considerations for surfaces and higher dimensions.

Findings

Conditions under which Cohen-Macaulay property is preserved during normalization.

The proof differs between surfaces and higher-dimensional varieties.

Auxiliary results on Cohen-Macaulay properties are established.

Abstract

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 which is an isomorphism over $U$. Whether this is true is already unknown in the case of a (normal) threefold with rational singularities along a curve $C$ except at a single point $p \in C$. There is an analogous conjecture for demi-normal varieties $X$, where we must insist that $Y$ has only semi-rational singularities. Our main result is that if a stronger version of Koll\'ar's conjecture is true for rational (normal) singularities, then the analogous conjecture is true for Gorenstein semi-rational (non-normal) singularities. We illustrate this first for surfaces. As the same procedure does not carry over directly to higher dimensions, we must use a slightly different proof for that case. We include conditions under which the Cohen-Macaulay property (slightly weaker than that of having rational singularities) is preserved when passing between a demi-normal variety and its normalization. It is this condition that is automatic for surfaces and which makes a proof slightly different in higher dimensions. In fact, it is not the case that the normalization of a Cohen-Macaulay variety is Cohen-Macaulay (or vice-versa, as is already apparent with surfaces). These auxiliary results are needed for the proof of the general result, but they are interesting in their own right. The proof of the general statement is at the end of the paper.