Incidence combinatorics of resolutions

By Eva Maria Feichtner, Dmitry N. Kozlov. 2003

81 citations recorded by the source corpus.

Abstract 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 at the end of this paper.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/40afca95ec7ca3bb3b5ced250684cca870fcc68c

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Tropical Discriminants (2005, methodology): “The latter encode the structure of wonderful compactifications of hyperplane arrangement complements in the work of De Concini and Procesi [5], and they were later studied from a combinatorial point of view in [8, 9, 11].”
The topology of spaces of knots: cosimplicial models (2002, background): “In [20] Feitchner and Kozlov view the posets Ψn+1 as a combinatorial blowups of Pν(n).”
On the topology of nested set complexes (2003, methodology): “In joint work of the first author with Kozlov [3], purely order-theoretic definitions of building sets and nested set complexes for arbitrary meet-semilattices were given.”
Nested complexes and their polyhedral realizations (2005, unknown): “This construction has appeared in [6, 4, 5].”
Matroid polytopes, nested sets and Bergman fans (2004, methodology): “1 from their result using the techniques of combinatorial blow-ups developed in [10].”