A note on planarity stratification of Hurwitz spaces

TL;DR

This paper introduces a natural stratification of Hurwitz spaces based on the minimal degree of plane curves representing meromorphic functions on Riemann surfaces, and calculates the dimensions of these strata.

Contribution

It defines a new stratification of Hurwitz spaces according to minimal plane curve degree and provides explicit dimension calculations for these strata.

Findings

Stratification based on minimal degree of plane curves

Explicit dimension formulas for each stratum

Connection between meromorphic functions and plane curve representations

Abstract

One can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to CP^2 and a projection of the image curve from an appropriate point p in CP^2 to the pencil of lines through p. We introduce a natural stratification of Hurwitz spaces according to the minimal degree of a plane curve such that a given meromorphic function can be represented in the above way and calculate the dimensions of these strata.