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≤g≤10 such that the variety of nets Wg-12(C) has dimension g-7. We show that dimWg-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 P3 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.