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

Abstract:

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