A Note on Planarity Stratification of Hurwitz Spaces. Issue 3 (1st September 2015) - Ongaro, Jared; Shapiro, Boris
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 froman appropriate point pCP^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. We observe that they are closely related to a family of Severi varieties studied earlier by J. Harris, Z. Ran, and I. Tyomkin.