A new look at the moduli space of stable hyperelliptic curves
By results of \textit{D. Mumford} [Enseign. Math., II. Ser. 23, 39--100 (1977; Zbl 0363.14003)] and \textit{D. Gieseker} [Lectures on moduli of curves. Tata Inst. Fund. Res. (1982; Zbl 0534.14012)] the compactified moduli space \(\bar{M}_g\) of curves of genus \(g\) is isomorphic to the GIT quotients \(\text{Hilb}_{g,n}/ /{SL}_{(2n-2)(g-1)}\) and \(\text{Chow}_{g,n}/ /{SL}_{(2n-2)(g-1)}\) of the Hilbert scheme and the Chow variety of \(n\)-canonical images, for any \(n \geq 5\). There are several ways for constructing compact moduli spaces of hyperelliptic curves \(H_g\), two of which are concerned in this paper. The first is to take the closure \(\bar{H}_g\) in \(M_g\); it yields that for any \(n \geq 5\),\ \(\bar{H}_g\) is isomorphic to the GIT quotients \(\text{Hilb}^{he}_{g,n}/ /{SL}_{(2n-2)(g-1)}\) and \(\text{Chow}^{he}_{g,n}/ /{SL}_{(2n-2)(g-1)}\) of the hyperelliptic loci of \(\text{Hilb}_{g,n}\) and \(\text{Chow}_{g,n}\). This paper studies the above GIT quotients for hyperelliptic curves in some of the problematic cases corresponding to the values \(n < 5\). The general hyperelliptic curve \(C\) of genus \(g\) is defined by the \((2g+2)\) branched points of the double covering \(C \rightarrow {\mathbb P}^1\), equivalently -- by the binary forms of degree \(2g+2\). By a result of Avritzer and Lange, there is a natural morphism \(f_g: \bar{H}_g \rightarrow \bar{B}_{2g+2}\) to the GIT compactified moduli space of such forms, contracting \(g-1\) special divisors \(\tilde{B}_3,\dots,\tilde{B}_{g+1}\), [see \textit{D. Avritzer, H. Lange}, Math. Z. 242, No. 4, 615--632 (2002; Zbl 1080.14031)]. This paper studies the maps \(f_g\) from the viewpoint of log Mori theory. For this purpose, for any \(\alpha \in {\mathbb Q}\), the authors introduce \({\mathbb Q}\)-Cartier divisors \(L_{\alpha}\), and define the varieties \(\bar{H}_g(\alpha)\) = \(\text{Proj}\;\bigoplus_{n \geq 0} \Gamma(\bar{H}_g,n(4g+2)L_{\alpha})\). Under some additional conditions (the \(F\)-conjecture -- see section 3) they verify first Theorem 1, which in particular states that for \(g \geq 3\) there is a birational contraction \(\bar{H}_g \rightarrow \bar{H}_g(9/11)\) with exceptional locus \(\tilde{B}_3\); and also Theorem 2, which in particular shows that \(\bar{H}_g(7/10)\) is isomorphic to \(\text{Chow}^{he}_{g,2}//{SL}_{3g-3}\). Especially for genus \(g =3\), Theorem 3 states that \(\text{Chow}^{he}_{3,2}\) is isomorphic to the compactified space of binary forms \(\bar{B}_8\). Together with Theorem 2, this yields an isomoprphism \(\bar{H}_3(7/10) \cong \bar{B}_8\). Theorem 4 shows that this isomorphism extends the natural isomorphism \(H_3 \cong B_8\).
scientific article; zbMATH DE number 5671543
scientific article