Brill-Noether loci and the gonality stratification of \(\{M_g\}\) (Q2744715)

Let \(C\) be an irreducible smooth projective complex curve of genus \(g\). The Brill-Noether locus is the set \({\mathcal M}^r_{g, d}:= \{[C] \in {\mathcal M}_{g}:C\) carries a \({\mathfrak g}^{r}_{d} \}\), and it is known that when the Brill-Noether number \(\rho(g, r, d) = g - (r + 1)(g - d + r)\) is negative, the general curve of genus \(g\) does not carry a \({\mathfrak g}^r_d\), hence \({\mathcal M}^r_{g, d}\) is a proper subvariety of \({\mathcal M}_{g}\). In this paper, the author studies the gonality of \(C\), when \([C]\) is a general point in a component of \({\mathcal M}^r_{g, d}\) which is generically smooth, of the expected dimension and with a general point corresponding to a curve with a very ample \({\mathfrak g}^r_{d}\). For a wide range of \(d\), \(g\) and \(r\) the author obtains curves of genus \(g\) and degree \(d\) in \(\mathbb{P}^r\) having the expected gonality, in many cases by taking sections of \(K3\) surfaces. When \(r = 3\) the main result of the paper states that for an odd \(g \geq 15\) and an even \(d \geq 14\), if \(\rho(g, 3, d) < 0\) and other technical conditions hold, for any pair \((d', g') \in \{(d, g), (d + 1, g+1), (d + 1, g + 2), (d + 2, g + 3)\}\) there exists a regular component of \(\text{Hilb}_{d', g', 3}\) whose general point \([C']\) is a smooth curve such that \(\text{gonality} (C') = \min \{d' - 4, [(g' + 3)/2]\}\). NEWLINENEWLINENEWLINEThe paper ends with an application of this result, namely, a new proof of a result by the same author, stating that the Kodaira dimension of the moduli space of curves of genus 23 is greater than or equal to 2.