Base change behavior of the relative canonical sheaf related to higher dimensional moduli

Patakfalvi, Zsolt

doi:10.2140/ant.2013.7.353

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Base change behavior of the relative canonical sheaf related to higher dimensional moduli

Zsolt Patakfalvi

Vol. 7 (2013), No. 2, 353–378

DOI: 10.2140/ant.2013.7.353

Abstract

We show that the compatibility of the relative canonical sheaf with base change fails generally in families of normal varieties. Furthermore, it always fails if the general fiber of a family of pure dimension $n$ is Cohen–Macaulay and the special fiber contains a strictly $S_{n - 1}$ point. In particular, in moduli spaces with functorial relative canonical sheaves Cohen–Macaulay schemes can not degenerate to $S_{n - 1}$ schemes. Another, less immediate consequence is that the canonical sheaf of an $S_{n - 1}$ , $G_{2}$ scheme of pure dimension $n$ is not $S_{3}$ .

Keywords

canonical sheaf, relative canonical sheaf, dualizing complex, relative dualizing complex, base change, depth, moduli of stable varieties

Mathematical Subject Classification 2010

Primary: 14J10

Secondary: 14D06, 14F10, 14E30

Milestones

Received: 9 June 2011

Revised: 26 January 2012

Accepted: 9 March 2012

Published: 25 April 2013

Authors

Zsolt Patakfalvi
Department of Mathematics Princeton University Fine Hall, Washington Road Princeton, NJ 08540 United States
http://www.math.princeton.edu/~pzs

Vol. 7, No. 2, 2013