On rationalizing divisors
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On rationalizing divisors Lorenzo Prelli1
© Akadémiai Kiadó, Budapest, Hungary 2017
Abstract 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 (X, D) is a rational pair. We give a criterion for cones to have a rationalizing divisor, and relate the existence of such a divisor to the locus of rational singularities of a variety. Keywords Singularities · Rational pairs · Divisors
1 Introduction The development of the minimal model program showed the importance of working with pairs or the form (X, D), where D is a Weil divisor on X . Since many important classes of singularities of Q-Gorenstein varieties have a natural extension to pairs, one might ask whether there is a way to generalize the notion of rational singularities to pairs. Rational pairs have been recently introduced by Kollár and Kovács [6, Section 2.5], and their deformation theory has been explored in Lindsay Erickson’s PhD thesis [2]. In this paper, we focus on the following: Question Given a normal variety X , when is there a reduced divisor D such that (X, D) is rational? We will call such a divisor a rationalizing divisor. The first result is a condition for cones on rational pairs: Theorem A (Theorem 4.2) Let (X, D) be a rational pair, and let L be an ample line bundle. Then the cone (C(X, L), C(D, L | D )) is a rational pair if and only if H i (X, L m (−D)) = 0 for i > 0, m ≥ 0
B 1
Lorenzo Prelli [email protected] Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA
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L. Prelli
As a consequence of the above theorem, we get the following Corollary B (Corollary 4.5) Let X be a normal variety of dimension n and let L be an ample line bundle on X . If the cone C(X, L) has a rationalizing divisor, then H n (X, L m ) = 0 for m ≥ 0 Then we give a necessary condition for the existence of a rationalizing divisor: Theorem C (5.2) Let (X, D) be a rational pair. Then the non rational locus of X has codimension at least 3. We conclude the paper with an example that shows that the above theorems provide necessary but not sufficient conditions to guarantee the existence of a rationalizing divisor.
2 Background In this section we present the basic definitions and results on rational pairs. We will work on an algebraically closed field k of characteristic zero. The main references are [6, Section 2.5], and [3, Chapter 3]. Definition 2.1 A reduced pair (X, D) consists of the datum of a normal variety X and a reduced Weil divisor D on X . The pair analogue for smoothness is an snc pair. Definition 2.2 A simple normal crossing (or snc) pair (X, D) is a pair such that X is smooth, every component Di of D is smooth and the Di ’s intersect transversely. For any reduced pair (X, D), its snc locus of (X, D) is the largest open set U such that (U, D |U ) is snc. As the reader might expect, we require maps between varieties to respect the pair
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