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By Brian Osserman and Communicated Michael Stillman


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 P1 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, ande1,...,en satisfying ∑ i (ei −1) = 2d−2−g, thereexists smooth curves C of genus g together with points P1,...,Pn such that all maps from C to P1 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 = P1, n = 3, and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions. 1. Statements and context Let k be either the real numbers, or a finite extension of Qp for some p. The purpose of this note is to discuss what results may be obtained over k from th

Year: 2005
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