Irreducibility of the Hilbert scheme of smooth curves in \(\mathbb {P}^4\) of degree \(g+2\) and genus \(g\)
Let \(\mathcal H_{d,g,r}\) be the union of the Hilbert scheme components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree d and genus g in \(\mathbb P^r\). In this paper the authors study \(\mathcal H_{d,g,r}\) for the case \(r =4\) and \(d = g+2\). The main result shows that any non-empty \(\mathcal H_{g+2,g,4}\) is irreducible without any restriction on \(g\). The same result was previously established by \textit{H. Iliev} [Proc. Am. Math. Soc. 134, No. 10, 2823--2832 (2006; Zbl 1097.14022)] with several low genus cases left untreated.
scientific article; zbMATH DE number 6813951
scientific article