Effective faithful tropicalizations associated to linear systems on curves


Shu Kawaguchi
2016-06-24  13:30 - 14:30
Room 103 , Mathematics Research Center Building (ori. New Math. Bldg.)

For a smooth projective curve $X$ of genus $g ¥geq 2$, 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 non-archimedean geometry, in which projective space is replaced by tropical projective space, and an embedding is replaced by a faithful tropicalization. Let $K$ be an algebraically closed field which is complete with respect to a non-trivial non-Archimedean value. We assume that the valuation of $K$ has rational rank $1$. Suppose that $X$ is defined over $K$, and that $¥Gamma$ is a skeleton (that is allowed to have ends) of the analytification $X^¥an$ of  $X$ in the sense of Berkovich. We show that, if $¥deg(L) ¥geq 5g-4$, then global sections of $L$ give a faithful tropicalization of $¥Gamma$ into tropical projective space. If time permits, we would like to explain a version of this statement for higher dimensional varieties. This is a joint work with Kazuhiko Yamaki.