Cox ring of the generic fiber

TL;DR

This paper develops a method to derive the Cox ring of a generic fiber from that of the total space in a surjective morphism of normal varieties, with applications to toric fiber spaces.

Contribution

It introduces a technique to recover Cox rings of fibers from the Cox ring of the total space, extending understanding of fiber structures in algebraic geometry.

Findings

Cox ring of the generic fiber can be recovered from the Cox ring of the total space.

In some cases, Cox rings of very general fibers can also be recovered.

Application to the blow-up of a toric fiber space demonstrates the method's utility.

Abstract

Given a surjective morphism $\pi\colon X\to Y$ of normal varieties satisfying some regularity hypotheses we prove how to recover a Cox ring of the generic fiber of $\pi$ from the Cox ring of $X$. As a corollary we show that in some cases it is also possible to recover the Cox ring of a very general fiber, and finally we give an application in the case of the blowing-up of a toric fiber space.