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

TL;DR

This paper investigates the behavior of the relative canonical sheaf in families of normal varieties, revealing conditions under which it fails to be compatible with base change, especially in degenerations involving Cohen-Macaulay and S_{n-1} schemes.

Contribution

It demonstrates the failure of base change compatibility of the relative canonical sheaf in certain degenerations of normal varieties, with implications for moduli spaces.

Findings

Compatibility of the relative canonical sheaf with base change generally fails.

Fails specifically when the general fiber is Cohen-Macaulay and the special fiber contains an S_{n-1} point.

Canonical sheaf of an S_{n-1}, G_2 scheme of pure dimension n is not S_3.

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.