Enumera\c{c}\~ao de hipersuperf\'icies com subesquemas singulares

Let $\mathbb{W}$ be an irreducible subvariety a Hilbert scheme $Hilb_{p_W} (t) ($\mathbb{P}$^n )$. We show that under mild hypothesis there are polynomial formulas for the degrees of the loci of hypersurfaces in ${\mathbb{P}}^n$ with singular subschemes containing some member of the family $\mathbb{W}$. The formulas are made explicit in a number of cases.