Osservazioni sullo spazio dei moduli delle curve trigonali

Bardelli, Fabio ; Del Centina, Andrea

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 70 (1981) no. 2, pp. 96-100

Résumé

Let $C$ be an algebraic projective smooth and trigonal curve of genus $g \ge 5$. In this paper we define, in a way equivalent to that followed by A. Maroni in [1], an integer $m$, called the species of $C$, which is a birational invariant having the property that $0 \le m \le \frac{g+2}{3}$ and $m—g \equiv 0$ mod(2). In section 1 we prove that for every $g (\ge 5)$ and every $m$, as before, there are trigonal curves of genus $g$ and species $m$. In section 2 we define the space $\mathcal{M}_{g,3;m}^{1}$ of moduli of trigonal curves of genus $g$ and species $m$. We note that $\mathcal{M}_{g,3;m}^{1}$ is irreducible and unirational and we prove that $dim \, \mathcal{M}_{g,3;m}^{1} = 2g+2—m$ if $m \ne 0$ and $dim \, \mathcal{M}_{g,3;0}^{1} = 2g+1$. As Corollaries we obtain the following facts: the general trigonal curve of even genus is of species $0$, the general trigonal curve of odd genus is of species 1 and the space $\mathcal{M}_{g,3}^{1}$ of moduli of trigonal curves of genus $g$ is unirational. The results of this note are valid over any algebraically closed field of any characteristic.