A polyptych of multi-centered deformation spaces

Abstract

Extending Verdier's deformation space to the normal cone of a closed subscheme and Rost's double deformation space of a pair of nested closed subschemes, we introduce a notion of deformation spaces attached to chains of immersions of arbitrary lengths $n$. One main result, which builds on the formalism of multi-centered dilatations of schemes, is the existence of so-called panelization isomorphisms, which produce under suitable regularity conditions several canonical isomorphisms between a given deformation space of length $n$ and some deformation spaces of smaller lengths. Having these panelization isomorphisms also allows to give geometric descriptions of the strata -- certain restrictions of special interest -- of deformation spaces. \tableofcontents