Effective faithful tropicalizations associated to linear systems on curves

Abstract

For a connected smooth projective curve $X$ of genus $g$, global sections of any line bundle $L$ with $\deg(L) \geq 2g+ 1$ give an embedding of the curve into projective space. We consider an analogous statement for a Berkovich skeleton in nonarchimedean geometry, in which projective space is replaced by tropical projective space, and an embedding is replaced by a homeomorphism onto its image preserving integral structures (called a faithful tropicalization). Let $K$ be an algebraically closed field which is complete with respect to a non-trivial nonarchimedean value. Suppose that $X$ is defined over $K$ and has genus $g \geq 2$ and that $\Gamma$ is a skeleton (that is allowed to have ends) of the analytification $X^{\mathrm{an}}$ of $X$ in the sense of Berkovich. We show that if $\deg(L) \geq 3g-1$, then global sections of $L$ give a faithful tropicalization of $\Gamma$ into tropical projective space. As an application, when $Y$ is a suitable affine curve, we describe the analytification $Y^{\mathrm{an}}$ as the limit of tropicalizations of an effectively bounded degree.