Orbits of the Automorphism Group of Horospherical Varieties, and Divisor Class Group

In 2013 Bazhov proved a criterion for two points on a complete toric variety to lie in the same orbit of the neutral component of the automorphism group. This criterion is formulated in terms of the divisor class group. The same year Arzhantsev and Bazhov obtained a similar criterion for affine toric varieties. We prove a necessary condition similar to this criterion in the cases of affine and projective horospherical varieties.