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In this paper some questions about the variety $\mathrm{\Phi }$ of ${𝐏}^{g-1}(k)$ on which lies the canonical curve of a k-gonal curve, are studied.

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 $m\mathrm{—}g\equiv 0$ 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 ${ℳ}_{g,3;m}^1$ of moduli of trigonal curves of genus $g$ and species $m$. We note that ${ℳ}_{g,3;m}^1$ is irreducible...

We consider the representation of a Siegel half-plane associated with the polarized torus of dim 2m obtained by the periods of the holomorphic 2m-forms. In particular, for m=1 we obtain the geometric significance of the isomorphism between a Siegel half-plane of dimension three and the domain of type IV of the same dimension.

In this note we show that every smooth “general enough” hypersurface of bidegree $(p,2)$ with $p>1$ in ${𝐏}^2\times {𝐏}^2$ is an ordinary non rational conic-bundle. Moreover we construct an example, for $p=3$, which is unirational.

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 $m\mathrm{—}g\equiv 0$ 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 ${ℳ}_{g,3;m}^1$ of moduli of trigonal curves of genus $g$ and species $m$. We note that ${ℳ}_{g,3;m}^1$ is irreducible...