Plane models for Riemann surfaces admitting certain half-canonical linear series. II

Abstract

For r ⩾ 2 r \geqslant 2 , closed Riemann surfaces of genus 3 r + 2 3r + 2 admitting two simple half-canonical linear series g 3 r + 1 r , h 3 r + 1 r g_{3r + 1}^r,h_{3r + 1}^r are characterized by the existence of certain plane models of degree 2 r + 3 2r + 3 where the linear series are apparent. The plane curves have r − 2 r - 2 3 3 -fold singularities, one ( 2 r − 1 ) (2r - 1) -fold singularity Q Q , and two other double points (typically tacnodes) whose tangents pass through Q Q . The lines through Q Q cut out a g 4 1 g_4^1 which is unique. The case where the g 4 1 g_4^1 is the set of orbits of a noncyclic group of automorphisms of order four is characterized by the existence of 3 r + 3 3r + 3 pairs of half-canonical linear series of dimension r − 1 r - 1 , where the sum of the two linear series in any pair is linearly equivalent to g 3 r + 1 r + h 3 r + 1 r g_{3r + 1}^r + h_{3r + 1}^r .