A Note on Planarity Stratification of Hurwitz Spaces. Issue 3 (1st September 2015)

• Record Type: Journal Article
• Authors: Ongaro, Jared; Shapiro, Boris
• Is Part Of: Canadian mathematical bulletin =. Volume 58:Issue 3(2015)
• Page Start: 596
• Page End: 609
• Publication Date: 2015-09-01
• Subjects: 14H50 -- 14H05; Hurwitz spaces -- meromorphic functions -- Severi varieties; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/CMB-2015-015-x
• Languages: English
• ISSNs: 0008-4395

Abstract

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 $\mathbb{C}{{\mathbb{P}}^{2}}$ 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.