Eva-Maria Feichtner

Abstract: Resolutions of spaces that are prescribed by the combinatorial incidence structure of natural stratifications have appeared at many places. Most recent examples include the Fulton-MacPherson compactifications of classical configuration spaces of complex algebraic varieties, DeConcini-Procesi ``wonderful'' models for arrangement complements, and compactifications of mixed subspace and half-space arrangements in work of Kontsevich.

We present an abstract framework for the incidence combinatorics of strata in these situations: Inspired by the combinatorial notions used by DeConcini & Procesi, we define building sets and nested sets for arbitrary meet-semilattices on purely order-theoretic level. We define combinatorial blowups of meet-semilattices and show that a sequence of such combinatorial blowups, prescribed by a building set, transforms the original meet-semilattice into the face poset of the simplicial complex of nested sets. 

Specializing to the context of arrangement models, our combinatorial blowups serve to trace the incidence combinatorics of strata through every step of the DeConcini-Procesi model construction. More general, and even going beyond the context mentioned above, they provide a common abstract framework for the incidence combinatorics occurring in other situations in algebraic geometry, e.g., simplicial resolutions of toric varieties.