Enumeração de superfícies em P³ singulares ao longo de curvas redutíveis

Abstract

This thesis investigates families of singular surfaces in \(\mathbb{P}^3\) along certain reducible curves. Let \(\mathbb{W}_1,\mathbb{W}_2,\dots ,\mathbb{W}_l\) denote closed, irreducible subvarieties of a Hilbert scheme, \(Hilb_{P_{\mathbb{W}_i}(t)}(\mathbb{P}^3)\), of curves in \(\mathbb{P}^3\) with Hilbert polynomial \(P_{\mathbb{W}_i}(t),i=1,\dots,l\).

Let \(\mathbb{W}'\) be the closure in \(Hilb_{P_{\mathbb{W}'}(t)}(\mathbb{P}^3)\) of the family of subschemes of \(\mathbb{P}^3\) defined by ideals of the form \(\mathcal{I}_{\underline{W}}:=(\mathcal{I}_{W_1})^2\cap\dots\cap(\mathcal{I}_{W_l})^2\), where \(W_i\) is a generic member of \(\mathbb{W}_i\), for all \(i=1,\dots ,l\).

We show that for \(d\gg0\) the general member of \(\mathbb{P}(H^0(\mathcal{I}_{\underline{W}}(d)))\) is a surface of degree \(d\) in \(\mathbb{P}^{3}\) singular along \(W':=W_1\cup \dots \cup W_l\).

We consider the subvariety \(\Sigma(\mathbb{W}', d) \subset \mathbb{P}^{N_d} = \mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^{3}}(d)))\), formed by surfaces of degree \(d\) in \(\mathbb{P}^{3}\), which are singular along some \(W'\) as above.

We show that the degree of \(\Sigma(\mathbb{W}', d)\) is given by a polynomial \(p^{\mathbb{W}'}(d)\) for \(d\gg0\). We are able to compute explicitly \(p^{\mathbb{W}'}(d)\) for certain \(\mathbb{W}'\). We state a conjecture about the degree of \(p^{\mathbb{W}'}(d)\), verified for the families worked out here.