Let $Y$ be an integral projective curve with $g:=p_a\left(Y\right)\ge 2$. For all positive integers $d$, $r$ let $W_d^r\left(Y\right)\left({}^{*A*}\right)$ be the set of all $L\in {\text{Pic}}^d\left(Y\right)$ with $h^0\left(Y,L\right)\ge r+1$ and $L$ spanned. Here we prove that if $d\le g-2$, then $dim\left(W_d^r\left(Y\right)\left({}^{*A*}\right)\right)\le d-3r$ except in a few cases (essentially if $Y$ is a double covering).

Let k be an algebraically closed field of characteristic 0. Let C be an irreducible nonsingular curve in ℙⁿ such that 3C = S ∩ F, where S is a hypersurface and F is a surface in ℙⁿ and F has rational triple points. We classify the rational triple points through which such a curve C can pass (Theorem 1.8), and give an example (1.12). We only consider reduced and irreducible surfaces.

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...