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Linear series over real and $ p$-adic fields

Author: Brian Osserman
Journal: Proc. Amer. Math. Soc. 134 (2006), 989-993
MSC (2000): Primary 14H51, 14P99
Published electronically: September 28, 2005
MathSciNet review: 2196029
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Abstract: We note that the degeneration arguments given by the author in 2003 to derive a formula for the number of maps from a general curve $ C$ of genus $ g$ to $ \mathbb{P}^1$ with prescribed ramification also yields weaker results when working over the real numbers or $ p$-adic fields. Specifically, let $ k$ be such a field: we see that given $ g$, $ d$, $ n$, and $ e_1, \dots, e_n$ satisfying $ \sum_i (e_i -1) = 2d-2 - g$, there exists smooth curves $ C$ of genus $ g$ together with points $ P_1, \dots, P_n$ such that all maps from $ C$ to $ \mathbb{P}^1$ can, up to automorphism of the image, be defined over $ k$. We also note that the analagous result will follow from maps to higher-dimensional projective spaces if it is proven in the case $ C=\mathbb{P}^1$, $ n=3$, and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions.

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Additional Information

Brian Osserman
Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720-3840

Received by editor(s): March 20, 2004
Received by editor(s) in revised form: November 7, 2004
Published electronically: September 28, 2005
Communicated by: Michael Stillman
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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