Variety of nets of degree g-1 on smooth curves of low genus

Abstract

We classify smooth complex projective algebraic curves $C$ of low genus $7\le g\le 10$ such that the variety of nets $W_{g-1}^2\left(C\right)$ has dimension $g-$$7$. We show that $\dim W_{g-1}^2\left(C\right)=g-$$7$ is equivalent to the following conditions according to the values of the genus $g$. $\left(i\right)C$ is either trigonal, a double covering of a curve of genus 2 or a smooth plane curve degree 6 for $g=10$. $\left(ii\right)C$ is either trigonal, a double covering of a curve of genus 2, a tetragonal curve with a smooth model of degree 8 in $P^3$ or a tetragonal curve with a plane model of degree 6 for $g=9$. $\left(iii\right)C$ is either trigonal or has a birationally very ample $g_6^2$ for $g=8$ or $g=7$.