The moduli space of vector bundles of rank 2 on a nonsingular algebraic surface possesses several compactifications constructed by different methods, for example, that of Gieseker. The points of Gieseker's compactification represent the S-equivalence classes of semistable sheaves on S.

The tree-like compactification permits to simplify the nature of sheaves that arise on the boundary, replacing them by vector bundles, but complicates the base: the original surface S should be blown up into a tree of bubbles.

We define a moduli functor of rank 2 vector bundles on the bubble trees with root S and prove that this functor has a coarse moduli space, which is a separated algebraic space of finite type. The construction uses an embedding into the Fulton-McPherson configuration space and a quotient by a proper action of a linear group.

This is a joint work with Tikhomirov and Trautmann.