On the Hilbert scheme of smooth curves of degree 15 and genus 14 in \(\mathbb{P}^5\) (Q6612009)

The study of Hilbert schemes of smooth irreducible curves of given degree \(d\) and genus \(g\) in a projective spaces \({\mathbb P}^r\) is a classical one, of permanent interest. One of the questions is the conjectured irreducibility of the Hilbert schemes for given \(r\) and certain range of \(g\) and \(d\), (cf. [\textit{F. Severi}, Vorlesungen über algebraische Geometrie. (Übersetzung von E. Löffler.). Leipzig-Berlin: B. G. Teubner (1921; JFM 48.0687.01)], quoted by the authors). After reviewing the results obtained so far, this paper treats the ``next case'', ie \(r=5, d=15, g=14\). The main result is the fact that this Hilbert scheme has two irreducible components, both of expected dimension \(64\). Moreover, the gonality is also studied. One shows that ``one component is formed by \(5\)-gonal curves'' and ``the general element of the other component is \(7\)-gonal''; also no curve in this Hilbert scheme has gonality \(\le 4\). The paper contains many interesting comments and also connections to the known results.