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

摘要

We classify smooth complex projective algebraic curves C of low genus 7 ≤ g ≤ 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. (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 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 g_6~2 for g = 8 or g = 7.