Geodesic
Topological Order Complexes and Resolutions of Discriminant Sets
Publications de l'Institut Mathématique, _N_S_66 (1999) no. 80, p. 165 .

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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 topological order complexes appear naturally in the conical resolutions of many singular algebraic varieties, especially of discriminant varieties, i.e. the spaces of singular geometric objects. These resolutions generalize the 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 : 14J17 55U99 
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     author = {V. A. Vassiliev},
     title = {Topological {Order} {Complexes} and {Resolutions} of {Discriminant} {Sets}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {165 },
     publisher = {mathdoc},
     volume = {_N_S_66},
     number = {80},
     year = {1999},
     zbl = {0953.55011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1999_N_S_66_80_a9/}
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V. A. Vassiliev. Topological Order Complexes and Resolutions of Discriminant Sets. Publications de l'Institut Mathématique, _N_S_66 (1999) no. 80, p. 165 . http://geodesic.mathdoc.fr/item/PIM_1999_N_S_66_80_a9/