# zbMATH — the first resource for mathematics

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.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14M20 Rational and unirational varieties
##### Keywords:
rationality; moduli space of smooth curves of genus 3
Full Text: