Enumeração de hipersuperfícies com subesquemas singulares

Abstract

This thesis investigates the locus of hypersurfaces with nonisolated singularities. More precisely, given a closed, irreducibe subvariety of a Hilbert scheme, (...), we define a subvariety (...), formed by the hypersurfaces of degree d in (...) which are singular along some (variable) member (...). Assuming that a general member (...) is smooth, irreducible and positive dimensional, we show that the degree of (...) is expressed by a polinomial (...) for all (...). The polynomial (...) is made explicit for a few families W, distinguished by the existence of an adequate description in the literature. Notably, we study the cases (...). The method consists in describing a desingularization (...) such that (...) parameterizes a flat family of subschemes of (...) the general member of which is defined by an ideal of the form (...), square of the ideal of a general member (...). The variety (...) comes equipped, for (...), with a vector subbundle (...) of the trivial bundle (...), with fiber over a general member (...) formed by the (...) such that the gradient vanishes along W. Moreover, the map (...) induced in the projectivization has image the variety (...) and is generically injective for (...). Polynomiality follows using Grothendieck-Riemann-Roch. In the cases above displayed Botts localization at fixed points is employed to derive explicit formula for the degree of (...).