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Proceedings of the London Mathematical Society Advance Access originally published online on November 23, 2007
Proceedings of the London Mathematical Society 2008 96(3):623-668; doi:10.1112/plms/pdm036
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© 2007 London Mathematical Society

Enumeration of uni-singular algebraic hypersurfaces

D. Kerner

School of Mathematical Sciences
Tel Aviv University
Ramat Aviv
69978 Tel Aviv
Israel

Received 12 September 2005. Revision received 2 April 2007.

We enumerate complex algebraic hypersurfaces in Pn, 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.

2000 Mathematics Subject Classification 14N10, 14J70 (primary), 14C17, 14J17 (secondary).


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