Enumeration of uni-singular algebraic hypersurfaces

Abstract

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 equi-singular strata in the parameter space of hypersurfaces. We suggest an inductive procedure, based on 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 details the generalized Newton-non-degenerate singularities. We give also examples of enumeration in some other cases.

Enumeration of singular hypersurfaces on arbitrary complex manifolds

“…which recovers the formulas obtained in [6] and [1]. For general m, the numbers N (A 2 ) and N (A 3 ) agree with the results of Kerner in [4]. c…”

Enumeration of cones with singular base

“…For hypersurfaces with less than six nodes, cf. [21]; see also [8][9][10][11] for some cases of higher, still isolated (≤ 2) singularities.…”