Projective varieties covered by isotrivial families

TL;DR

This paper proves that in a family of projective schemes over a discrete valuation ring, if the total space is isotrivial, then the generic fiber of the base is also isotrivial, revealing a relation between total space and base.

Contribution

It establishes a new result linking the isotriviality of a family of projective schemes to the isotriviality of its base's generic fiber over a discrete valuation ring.

Findings

If the family X over R is isotrivial, then the generic fiber of Y over R is also isotrivial.

The result applies to projective schemes over discrete valuation rings with certain smoothness conditions.

Provides insight into the structure of families of projective varieties over valuation rings.

Abstract

Let X,Y be projective schemes over a discrete valuation ring R, where Y is generically smooth and g:X \to Y a surjective R-morphism such that g_*\mathcal{O}_X = \mathcal{O}_Y. We show that if the family X \to Spec(R) is isotrivial, then the generic fiber of the family Y \to Spec(R) is isotrivial.