A note on planarity stratification of hurwitz spaces

Abstract

Abstract. One can easily show that any meromorphic function on a comple x closed Riemann surface can be represented as a composition o f a birational map of this surface to CP 2 and a projection of the image curve from an ap- propriate point p ∈ CP 2 to the pencil of lines through p . We introduce a natural stratification of Hurwitz spaces according to the mi nimal degree of a plane curve such that a given meromorphic function can be rep resented in the above way and calculate the dimensions of these strata. We ob serve that they are closely related to a family of Severi varieties studied e arlier by J. Harris, Z. Ran and I. Tyomkin