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Rationality of the moduli variety of curves of genus 3. (English) Zbl 0885.14013
Let \({\mathcal M}_3\) be the moduli space of smooth curves of genus 3 defined over the complex field \(\mathbb{C}\). The aim of the paper under review is to prove that \({\mathcal M}_3\) is a rational variety. The proof goes as follows:
For \(n\geq 0\) denote by \(V(n)\) the space of forms of degree \(n\) in the variables \(z_1, z_2\) and consider the canonical action of \(\text{SL}_2\) on \(V(n)\) and of \(\text{PSL}_2\) on \(V(2d)\). For \(\lambda= (\lambda_0,\lambda_1, \lambda_4, \lambda_6)\in \mathbb{C}^4\), let \(\delta_\lambda: V(8)\oplus V(0)\oplus V(4)\to V(4)\) be the degree 2 homogeneous \(\text{PSL}_2\)-morphism defined by \[ f_8+ f_0+ f_4\mapsto \lambda_6\psi_6 (f_8,f_8)+ 2\lambda_4\psi_4 (f_8,f_4)+ \lambda_2\psi_2 (f_4,f_4)+ 2\lambda_0f_4f_0 \] where \(\psi_i\) is the bilinear \(\text{SL}_2\)-mapping \(\psi_i: V(d_1)\times V(d_2)\to V(d_1+d_2-2i)\) \[ \psi_i(h_1,h_2)= \frac{(d_1-i)(d_2-i)} {d_1!d_2!} \sum_{0\leq j\leq i} (-1)^j \binom{j}{i} \frac{\partial^ih_1} {\partial z_1^{i-j}\partial z_2^j} \frac{\partial^ih_2} {\partial z_1^j\partial z_2^{i-j}}. \] There is a unique 10-dimensional irreducible component \(U_\lambda\) of \(\delta_\lambda^{-1}(0)\) containing 1. In a previous paper [in: Lie groups, their discrete subgroups, and invariant theory, Adv. Sov. Math. 8, 95-103 (1992; Zbl 0778.14006)], P. T. Katsylo proved that \(\mathbb{C} (\mathbb{P} (S^4\mathbb{C}^{3*} ))^{\text{SL}_3}\simeq \mathbb{C}(U_{\lambda_0})^{\text{PSL}_2\times \mathbb{C}^*}\) where \(\lambda_0= (- \frac{7}{72}, \frac{11}{54}, \frac{1}{1680}, -\frac{6}{1225})\). Since \(\mathbb{C}({\mathcal M}_3)\simeq \mathbb{C} (\mathbb{P}(S^4 \mathbb{C}^{3*}))^{\text{SL}_3}\) the rationality of \({\mathcal M}_3\) is a corollary of the following theorem: For all \(\lambda\neq 0\) the field \(\mathbb{C}(U_\lambda )^{\text{PSL}_2\times \mathbb{C}^*}\simeq \mathbb{C} (\mathbb{P} U_\lambda)^{\text{PSL}_2}\) is rational.
For the proof of this theorem the author proceeds as follows. He finds a \((\text{PSL}_2, N(H))\)-section \(\mathbb{P} X_\lambda^0\) of \(\mathbb{P} U_\lambda\), where \(N(H)\) denotes the normalizer of a suitable subgroup \(H\) of \(\text{PSL}_2\), so that \(\mathbb{C} (\mathbb{P} U_\lambda)^{\text{PSL}_2}\simeq \mathbb{C} (\mathbb{P} X_\lambda^0)^{N(H)}\) and then he proves the rationality of the last field by using the “no name method” and Castelnuovo’s theorem.

14H10 Families, moduli of curves (algebraic)
14M20 Rational and unirational varieties
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