We introduce notions of combinatorial blowups, building sets, and nested sets
for arbitrary meet-semilattices. This gives a common abstract framework for the
incidence combinatorics occurring in the context of De Concini-Procesi models
of subspace arrangements and resolutions of singularities in toric varieties.
Our main theorem states that a sequence of combinatorial blowups, prescribed by
a building set in linear extension compatible order, gives the face poset of
the corresponding simplicial complex of nested sets. As applications we trace
the incidence combinatorics through every step of the De Concini-Procesi model
construction, and we introduce the notions of building sets and nested sets to
the context of toric varieties.
There are several other instances, such as models of stratified manifolds and
certain graded algebras associated with finite lattices, where our
combinatorial framework has been put to work; we present an outline in the end
of this paper.Comment: 20 pages; this is a revised version of our preprint dated Nov 2000
and May 2003; to appear in Selecta Mathematica (N.S.
