Linear series over real and $p$-adic fields

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.