On fibrations whose geometric fibers are nonreduced

TL;DR

This paper establishes bounds on the embedding dimensions of geometric generic fibers in fibrations over positive characteristic bases, extending known results about reduced fibers over curves and illustrating with specific algebraic varieties.

Contribution

It provides a new bound on the embedding dimensions of geometric generic fibers in positive characteristic fibrations, generalizing classical results.

Findings

Bound on embedding dimensions in terms of base dimension

Extension of reduced fiber results to higher dimensions

Illustrations with Fermat hypersurfaces and genus one curves

Abstract

We give a bound on embedding dimensions of geometric generic fibers in terms of the dimension of the base, for fibrations in positive characteristic. This generalizes the well-known fact that for fibrations over curves, the geometric generic fiber is reduced. We illustrate our results with Fermat hypersurfaces and genus one curves.