A look into the Severi varieties of curves in higher codimension
The authors consider curves in \({\mathbb P^n}\) of degree \(d\) with arithmetic genus \(p_a\) and only \(\delta\) nodes as singularities (nodal curves). In the first part of the paper using smoothing results [see \textit{E. Sernesi}, Invent. Math. 75, 25-57 (1984; Zbl 0541.14024)], the authors produce (when \(n \geq 3\), \(\delta \geq {n+1}\) and \(0 \leq \delta \leq p_a \leq(n(d-n)-1/(n-1)\)) inductively examples of nodal curves in \({\mathbb P^n}\) and obtain some non-emptiness results for the corresponding Severi variety, i.e. for the subset of the corresponding component of the Hilbert scheme, which parameterizes nodal curves of geometric genus \(p_a- {\delta}\). In the second part they consider the curves in \({\mathbb P^3}\) which are the \(0\)-locus of the global sections of a rank \(2\) vector bundle \(E\) over \({\mathbb P^3}\) ; they define the Severi variety \({\mathcal V}_{\delta}(E)\) as the subvariety of \({\mathbb P}(H^0(E))\) parameterizing sections whose \(0\)-locus is a nodal curve with \(\delta\) nodes and they develop a local deformation theory for \({\mathcal V}_{\delta}(E)\). Using these results they show that, if \(E\) is a rank \(2\) vector bundle over \({\mathbb P^3}\) with \(E(-1)\) generated by global sections, the Severi variety \({\mathcal V}_{\delta}(E(m))\), for \(m \gg 0\) and \(\delta \leq {m+1}\), is either empty or smooth of the (expected) dimension \(h^0(E(m))-1-{\delta}\) and that \({\mathcal V}_{\delta}(E(m))\) is non empty for \(m \gg 0\) and \(\delta \leq m\). The authors also give some examples which measure the sharpness of their result.
scientific article; zbMATH DE number 1383018