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

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

2010-05-28

arxiv.org/abs/1005.5207v3

Mathematics

Algebraic Geometry

Third, updated version. Minor changes, the non-standard usage of S_d at a point replaced by the adequate depth conditions. Com

Scientific paper

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.

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