Vertex and source determine the block variety of an indecomposable module

The block variety VG,b(M) of a finitely generated indecomposable module M over the block algebra of a p-block b of a finite group G, introduced in (J. Algebra 215 (1999) 460), can be computed in terms of a vertex and a source of M. We use this to show that VG,b(M) is connected, and that every closed homogeneous subvariety of the affine variety VG,b defined by block cohomology H*(G,b) (cf. Algebras Rep. Theory 2 (1999) 107) is the variety of a module over the block algebra. This is analogous to the corresponding statements on Carlsonʹs cohomology varieties in (Invent. Math. 77 (1984) 291).