## Journal of the Mathematical Society of Japan

### 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 $7\leq g\leq 10$ such that the variety of nets $W_{g-1}^{2}(C)$ has dimension $g-$$7$. We show that $\dim W_{g-1}^{2}(C)=g-$$7$ is equivalent to the following conditions according to the values of the genus $g$. $(\mathrm{i})C$ is either trigonal, a double covering of a curve of genus 2 or a smooth plane curve degree 6 for $g=10$. $(\mathrm{ii})C$ 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 $g=9$. $(\mathrm{iii})C$ is either trigonal or has a birationally very ample $g_{6}^{2}$ for $g=8$ or $g=7$.

#### Article information

Source
J. Math. Soc. Japan, Volume 55, Number 3 (2003), 591-616.

Dates
First available in Project Euclid: 3 October 2007

https://projecteuclid.org/euclid.jmsj/1191418991

Digital Object Identifier
doi:10.2969/jmsj/1191418991

Mathematical Reviews number (MathSciNet)
MR1978211

Zentralblatt MATH identifier
1033.14015

#### Citation

CHO, Kyung-Hye; KEEM, Changho; OHBUCHI, Akira. Variety of nets of degree $g-1$ on smooth curves of low genus. J. Math. Soc. Japan 55 (2003), no. 3, 591--616. doi:10.2969/jmsj/1191418991. https://projecteuclid.org/euclid.jmsj/1191418991