The rationality of the moduli spaces of trigonal curves of odd genus (Q2861478)

Let \(\mathcal T_g\) be the moduli space of trigonal curves of genus \(g\), i.e., the moduli space parametrizing irreducible smooth projective curves which admit a degree \(3\) morphism to \(\mathbb P^1\). For \(g\geq 5\), \(\mathcal T_g\) can be regarded as a sub locus of the moduli space \(\mathcal M_g\) of smooth curves of genus \(g\) as all such trigonal curves have a unique \(g^1_3\).NEWLINENEWLINEIn the paper under review, the author studies the problem of giving a birational classification for \(\mathcal T_g\). More precisely, the author shows that if \(g=2b+1\) and \(b\geq 2\), then \(\mathcal T_g\) is rational.NEWLINENEWLINEThe proof consists of a geometrical classical argument relating \(\mathcal T_g\) with Hirzebruch surfaces \(\mathbb F_N=\mathbb P(\mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(N))\). In detail, for odd genus, \(\mathcal T_g\) is birationally equivalent to the rational quotient \(|L_{3,b}|/\text{ Aut}(\mathbb F_1)\), where \(L_{3,b}=\mathcal O_{F_1}(3(\Sigma+F)+bF)\) is a line bundle determined by the fiber \(F\) of the natural projection \(\pi:\mathbb F_1\to\mathbb P^1\) and by a section \(\Sigma\) of \(\pi\) with self-intersection \(-1\). Therefore, the result follows by showing that \(|L_{3,b}|/\text{ Aut}(\mathbb F_1)\) is rational for \(b\geq 2\), which the author does.NEWLINENEWLINEThe fact that \(\mathcal T_g\) is rational for \(g=4n+2\) with \(n\geq 2\) was already known due to work of \textit{N. I. Shepherd-Barron} [Proc. Symp. Pure Math. 46, 165--171 (1987; Zbl 0669.14015)]. The remaining case of curves of genus divisible by four has been successively studied by the author in [``The Rationality of the Moduli Spaces of Trigonal Curves'', Int. Math. Res. Not. IMRN No. 14, 5456--5472 (2015)].