  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.
