## Journal of the Mathematical Society of Japan

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

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

Kyung-Hye CHO, Changho KEEM, and Akira OHBUCHI

#### 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-$$7$. We show that $\mathrm{dim}{W}_{g-1}^{2}\left(C\right)=g-$$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 $g=7$.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 3 October 2007

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/jmsj/1191418991

**Mathematical Reviews number (MathSciNet)**

MR1978211

**Zentralblatt MATH identifier**

1033.14015

**Subjects**

Primary: 14H45: Special curves and curves of low genus 14H10: Families, moduli (algebraic) 14C20: Divisors, linear systems, invertible sheaves

**Keywords**

algebraic curves linear series branched covering

#### 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