Enumeration of cones with singular base (Q634716)

The paper under review deals with the problem of enumeration of hypersurfaces with a singular locus of positive dimension. Particularly cones in the three dimensional projective space over plane curves with \(\delta \leq 6\) nodes. This same setup allows to enumerate cones in \(\mathbb{P}^n\) with vertex of dimension \(v\) and base variety a hypersurface of dimension \(b\) with at most six double points and \(b+v+1=n-1\). For a fixed \(v\) there are polynomials \(p_{\delta,n}^v(d)\) (explicitly computed for the first few values of \(v,n\)) computing the degree of the subvariety of hypersurfaces of degree \(d\) described above. As a byproduct, how to impose Schubert incidence conditions on the vertex and some results on a few cases of higher multiplicities are also provided.