Some 4-point Hurwitz numbers in positive characteristic

Bouw, Irene; Osserman, Brian

doi:10.1090/S0002-9947-2011-05347-X

Some $4$-point Hurwitz numbers in positive characteristic
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Trans. Amer. Math. Soc. 363 (2011), 6685-6711 Request permission

Abstract:

In this paper, we compute the number of covers of curves with given branch behavior in characteristic $p$ for one class of examples of genus $0$, with four branch points and degree $p$. Our techniques involve related computations in the case of three branch points, and allow us to conclude in many cases that for a particular choice of degeneration, all the covers we consider degenerate to separable (admissible) covers. Starting from a good understanding of the complex case, the proof is centered on the theory of stable reduction of Galois covers.

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