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A Note on Planarity Stratification of Hurwitz Spaces

Published online by Cambridge University Press:  20 November 2018

Jared Ongaro  and
Boris Shapiro
Jared Ongaro
Affiliation:
Department of Mathematics, Stockholm University, S-10691, Stockholm, Sweden e-mail: ongaro@math.su.seshapiro@math.su.se
Boris Shapiro
Affiliation:
Department of Mathematics, Stockholm University, S-10691, Stockholm, Sweden e-mail: ongaro@math.su.seshapiro@math.su.se
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Abstract

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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 $\mathbb{C}{{\mathbb{P}}^{2}}$ and a projection of the image curve froman appropriate point $p\in \mathbb{C}{{\mathbb{P}}^{2}}$ 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.

Keywords

14H5014H05Hurwitz spacesmeromorphic functionsSeveri varieties
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 3 , 01 September 2015 , pp. 596 - 609
Copyright
Copyright © Canadian Mathematical Society 2015

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