Effective faithful tropicalizations associated to linear systems on curves
Shu Kawaguchi, Kazuhiko Yamaki · Memoirs of the American Mathematical Society · 2021
For a connected smooth projective curve X X of genus g g , global sections of any line bundle L L with deg ( L ) ≥ 2 g + 1 \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: We replace projective space by tropical projective space, and an embedding by a homeomorphism onto its image preserving integral structures (or equivalently, since X X is a curve, an isometry), which is called a faithful tropicalization. Let K K be an algebraically closed field which is complete with respect to a non-trivial nonarchimedean value. Suppose that X X is defined over K K and has genus g ≥ 2 g \geq 2 and that Γ \Gamma is a skeleton (that is allowed to have ends) of the analytification X a n X^{\mathrm {an}} of X X in the sense of Berkovich. We show that if deg ( L ) ≥ 3 g − 1 \deg (L) \geq 3g-1 , then global sections of L L </