Families of invariant divisors on rational complexity-one T-varieties

We study invariant divisors on the total spaces of the homogeneous deformations of rational complexity-one T-varieties constructed by Ilten and Vollmert (2012) [10]. 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.