Variety of nets of degree \(g-1\) on smooth curves of low genus. (English) Zbl 1033.14015

Cho, Kyung-Hye; Keem, Changho; Ohbuchi, Akira

J. Math. Soc. Japan 55, No. 3, 591-616 (2003).

Summary: We classify smooth complex projective algebraic curves \(C\) of low genus \(7\leq g\leq 10\) such that the variety of nets \(W^2_{g-1}(C)\) has dimension \(g-7\). We show that \(\dim W^2_{g-1} (C)=g-7\) 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 \(\mathbb{P}^3\) 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 \(g^2_6\) for \(g=8\) or \(g=7\).

MSC:

14H10	Families, moduli of curves (algebraic)
14H45	Special algebraic curves and curves of low genus
14C20	Divisors, linear systems, invertible sheaves
14C21	Pencils, nets, webs in algebraic geometry

Keywords:

low genus; nets; double covering; trigonal; tetragonal curve