Premium
Enumeration of uni‐singular algebraic hypersurfaces
Author(s) -
Kerner D.
Publication year - 2008
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdm036
Subject(s) - mathematics , gravitational singularity , singularity , enumeration , pure mathematics , homology (biology) , complete intersection , algebraic number , degenerate energy levels , isolated singularity , singular point of a curve , singularity theory , algebraic geometry , algebraic variety , class (philosophy) , intersection homology , mathematical analysis , combinatorics , cohomology , biochemistry , chemistry , physics , quantum mechanics , artificial intelligence , computer science , gene
We enumerate complex algebraic hypersurfaces in ℙ n , of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equisingular strata in the parameter space of hypersurfaces. We suggest an inductive procedure, based on an intersection theory combined with liftings and degenerations. The procedure computes the (co)homology class in question, whenever a given singularity type is properly defined and the stratum possesses good geometric properties. We consider in detail the generalized Newton‐non‐degenerate singularities. We also give examples of enumeration in some other cases.