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

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 $S_3$.