Equivalence of families of singular schemes on threefolds and on ruled fourfolds (Q1428938)

Given a smooth projective threefold \(X\) over the complex numbers and a rank-two globally generated vector bundle \(\mathcal{F}\) on \(X\), the Severi variety \({\mathcal{V}}_{\delta} ({\mathcal{F}})\) is the subscheme of \({\mathbb{P}}(H^0({\mathcal{F}}))\) parametrizing global sections \(s\) of \({\mathcal{F}}\) whose zero-loci \(C_s=V(s)\) are irreducible \(\delta\)-nodal curves on \(X\). In an earlier paper [Trans.~Am.~Math.~Soc. 355, No.~12, 4901--4932 (2003; Zbl 1019.14015)], the author presented a cohomological description of the Zariski tangent space at a point of such a Severi variety. Let \({\mathcal P}={\mathbb P}_X({\mathcal F})\rightarrow X\) be the projective space bundle associated to \({\mathcal F}\) with its natural projection \(\pi\) on \(X\), and let \({\mathcal O}_{\mathcal P}(1)\) denote its tautological line bundle. Then the author also showed in his earlier paper that a point \([s]\in {\mathcal{V}}_{\delta} ({\mathcal{F}})\) corresponds to a divisor \(G_s\in | {\mathcal O}_{\mathcal P}(1)|\) that contains the fibers \(\pi^{-1}(p_i)\), where \(p_1,\dots,p_\delta\) are the nodes of \(C_s\). This correspondence is referred to here as the Severi correspondence, and the author studies its geometric significance. Put \({\mathcal R}_\delta({\mathcal O}_{\mathcal P}(1))=\{ G_s\in | {\mathcal O}_{\mathcal P}(1)| : [s]\in {\mathcal{V}}_{\delta} ({\mathcal{F}})\}\). Then the author shows that the point \([G_s]\in {\mathcal R}_\delta({\mathcal O}_{\mathcal P}(1))\) is smooth and of the expected codimension (which the author calls ``regular'') if and only if the same is true of the point \([s]\in {\mathcal{V}}_{\delta} ({\mathcal{F}})\). As a consequence, results from his earlier paper about the regularity of \({\mathcal{V}}_{\delta} ({\mathcal{F}})\) now yield theorems about the regularity of \({\mathcal R}_\delta({\mathcal O}_{\mathcal P}(1))\). Some of the results here are related to results of \textit{E.~Ballico} and \textit{L.~Chiantini} [Collect.~Math.~49, 191--201 (1998; Zbl 0959.14011)], who considered the case \(X={\mathbb P}^3\).