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

TL;DR

This paper derives explicit polynomial formulas for the degrees of hypersurfaces in projective space with singular subschemes containing members of a given family, under mild conditions.

Contribution

It provides new explicit formulas for degrees of loci of hypersurfaces with prescribed singular subschemes, extending previous understanding in algebraic geometry.

Findings

Formulas for degrees of hypersurfaces with singular subschemes

Explicit cases computed and presented

General conditions under which formulas hold

Abstract

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.