Eva Maria Feichtner

Chow rings of toric varieties defined by atomic lattices

We study a graded algebra

On the topology of nested set complexes

D=D(\mathcal{L},\mathcal{G})

On cohomology algebras of complex subspace arrangements

over ℤ defined by a finite lattice ℒ and a subset

Abelianizing the real permutation action via blowups

\mathcal{G}

On orbit configuration spaces of spheres

in ℒ, a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi [2]. Our main result is a representation of D, for an arbitrary atomic lattice ℒ, as the Chow ring of a smooth toric variety that we construct from ℒ and

A desingularization of real differentiable actions of finite groups

\mathcal{G}

Rational versus real cohomology algebras of low-dimensional toric varieties

. We describe this variety both by its fan and geometrically by a series of blowups and orbit removal. Also we find a Gröbner basis of the relation ideal of D and a monomial basis of D.

Matroid polytopes, nested sets and Bergman fans.

Nested set complexes appear as the combinatorial core of De Concini-Procesi arrangement models. We show that nested set complexes are homotopy equivalent to the order complexes of the underlying meet-semilattices without their minimal elements. For atomic semilattices, we consider the realization of nested set complexes by simplicial fans proposed by the first author and Yuzvinsky and we strengthen our previous result showing that in this case nested set complexes in fact are homeomorphic to the mentioned order complexes.

Incidence combinatorics of resolutions

The integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement. We give a combinatorial presentation of the cohomology algebra in the spirit of the Orlik-Solomon result on the cohomology algebras of complex hyperplane arrangements. Our methods are elementary: we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space. We describe simplicial cochains that generate the cohomology. Among them we distinguish a linear basis, study cup product multiplication, and derive an algebra presentation in terms of generators and relations.

The Integral Cohomology Algebras of Ordered Configuration Spaces of Spheres

We present an abelianization of the permutation action of the symmetric group S_n on R^n in analogy to the Batyrev abelianization construction for finite group actions on complex manifolds. The abelianization is provided by a particular De Concini-Procesi wonderful model for the braid arrangement. In fact, we show a stronger result, namely that stabilizers of points in the arrangement model are isomorphic to direct products of Z_2. To prove that, we develop a combinatorial framework for explicitly describing the stabilizers in terms of automorphism groups of set diagrams over families of cubes. We observe that the natural nested set stratification on the arrangement model is not stabilizer distinguishing with respect to the S_n-action, that is, stabilizers of points are not in general isomorphic on open strata. Motivated by this structural deficiency, we furnish a new stratification of the De Concini-Procesi arrangement model that distinguishes stabilizers.