Topological order complexes and resolutions of discriminant sets

V. A. Vassiliev

Abstract: If elements of a partially ordered set run over a topological space, then the corresponding order complex admits a natural topology, providing that similar interior points of simplices with close vertices are close to one another. Such {\it topological order complexes} appear naturally in the {\it conical resolutions} of many singular algebraic varieties, especially of {\it discriminant varieties}, i.e. the spaces of singular geometric objects. These resolutions generalize the {\it simplicial resolutions} to the case of non-normal varieties. Using these order complexes we study the cohomology rings of many spaces of nonsingular geometrical objects, including the spaces of nondegenerate linear operators in $R^n$, $C^n$ or $H^n$, of homogeneous functions $R^2 \to R^1$ without roots of high multiplicity in $RP^1$, of nonsingular hypersurfaces of a fixed degree in $CP^n$, and of Hermitian matrices with simple spectra.

Classification (MSC2000): 14J17; 55U99

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