Incidence combinatorics of resolutions

Details Incidence combinatorics of resolutions Incidence combinatorics of resolutions

2003-05-11

arxiv.org/abs/math/0305154v2

Selecta Mathematica 10 (2004), 37-60

Mathematics

Combinatorics

20 pages; this is a revised version of our preprint dated Nov 2000 and May 2003; to appear in Selecta Mathematica (N.S.)

Scientific paper

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.

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