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

Dedicated to Professor Makoto Namba on the occasion of his sixtieth birthday

Abstract

We classify smooth complex projective algebraic curves C of low genus 7≤ g≤ 10 such that the variety of nets W_g-1²(C) has dimension g-7. We show that \dim W_g-1²(C)=g-7 is equivalent to the following conditions according to the values of the genus g. (i) C is either trigonal, a double covering of a curve of genus 2 or a smooth plane curve degree 6 for g=10. (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 \bm{P}³ or a tetragonal curve with a plane model of degree 6 for g=9. (iii) C is either trigonal or has a birationally very ample g₆² for g=8 or g=7.