Families of Invariant Divisors on Rational Complexity-One T-Varieties

TL;DR

This paper investigates invariant divisors on rational complexity-one T-varieties, establishing a canonical isomorphism of Picard groups between general and special fibers that preserves key geometric invariants.

Contribution

It introduces a natural subgroup of the Picard group for general fibers and proves its isomorphism with the Picard group of the special fiber, preserving important invariants.

Findings

Canonical isomorphism between Picard groups of fibers

Preservation of Euler characteristic, intersection numbers, and canonical class

Identification of a natural subgroup of the Picard group

Abstract

We study invariant divisors on the total spaces of the homogeneous deformations of rational complexity-one T-varieties constructed by Ilten and Vollmert. In particular, we identify a natural subgroup of the Picard group for any general fiber of such a deformation, which is canonically isomorphic to the Picard group of the special fiber. This isomorphism preserves Euler characteristic, intersection numbers, and the canonical class.