The rationality of certain moduli spaces associated to half-canonical extremal curves (Q1304899)

In this article it is proved that some moduli spaces of half-canonical extremal curves are rational. Let \(C \subset \mathbb P^r(\mathbb C)\) be a smooth irreducible algebraic curve of genus \(g\) and degree \(d\) in the complex projective space of dimension \(r\). Then Castelnuovo's inequality holds: \[ g \leq \pi(d, r) :=\binom m 2 (r-1) + m \varepsilon \] where \(d = m(r-1) + \varepsilon + 1\) and \(0 \leq \varepsilon \leq r-2\). It is said that \(C\) is extremal if \(g = \pi(d, r)\). Let \(H = H^0(r, d, g)\) denote the Hilbert scheme of smooth curves of genus \(g\) and degree \(d\) in \( \mathbb P^r(\mathbb C)\). Also, let \( h : H \to \mathcal M_g\) denote the natural regular map into the coarse moduli space of curves of genus \(g\). When \(g = \pi(d, r)\) and under some stated and simple conditions on \((r, d)\), it is known, by work of Ciliberto and Lazarsfeld, that \(H\) is irreducible and \(h\) induces an isomorphism \[ H / PGL(r+1) \cong \text{im}(h) \] . The authors consider the problem of the rationality of the moduli space \(\text{im}(h)\), under some further numerical hypothesis. If \(\varepsilon = 1\) it turns out that \(K_C = \mathcal O_C(m-1)\), that is, \(C\) is sub-canonical of level \(m-1\). The main theorem states that if \(r \geq 7\) is odd, \(\varepsilon = 1\) and \(m=3\) then \(\text{im}(h)\) is rational. There is also an analogous statement for pointed curves. The rough idea of the proof is that such extremal sub-canonical curves may be realized as the smooth members of a certain linear system \(|D|\) on a rational surface \(X\). The moduli space is shown to be birational to \(|D|/ G\) where \(G \subset \text{Aut}(X)\) is a linear group. Finally, the rationality of \(|D|/ G\) is achieved using methods of previous works by Shepherd-Barron, Katsylo and Dolgachev.