Bardelli, Fabio and Del Centina, Andrea:
Osservazioni sullo spazio dei moduli delle curve trigonali
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Serie 8 70 (1981), fasc. n.2, p. 96-100, (Italian)
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Sunto
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.
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