The geometry of families of singular curves. (Q2754589)

Let \(\Sigma\) be a smooth projective surface and \(C\) a reduced curve on it. In the Hilbert scheme of all curves on \(\Sigma\) with given Hilbert polynomial \(h\), one considers the subset \(V=V_h(S_1,\dots,S_r)\) of all integral curves having precisely singularities of analytic or topological type \(S_1,\dots,S_r\).NEWLINENEWLINE Among the questions that can be considered, the most important are: 1) when is \(V\neq \emptyset\)? 2) when is \(V\) irreducible? 3) when is \(V\) smooth of the expected dimension?NEWLINENEWLINEAs is well known Severy dealt with the case \(\sum=\mathbb{P}^2\), considering curves of degree \(d\) with \(r\leq (d-1)(d-2)/2\) distinct nodes he proved non-emptiness, smoothness and the expected dimension. He claimed also the irreducibility but this was completely proved by \textit{J. Harris} [Invent. Math. 84, 445--461 (1986; Zbl 0596.14017)]. The authors have written many papers concerning these questions, especially for the case \(\Sigma=\mathbb{P}^2\), and this survey summarizes their results.NEWLINENEWLINE At the end they present some open problems.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00034].