Effective faithful tropicalizations associated to adjoint linear systems

Abstract

Let $R$ be a complete discrete valuation ring of equi-characteristic zero with fractional field $K$. Let $X$ be a connected, smooth projective variety of dimension $d$ over $K$, and let $L$ be an ample line bundle over $X$. We assume that there exist a regular strictly semistable model $\mathscr{X}$ of $X$ over $R$ and a relatively ample line bundle $\mathscr{L}$ over $\mathscr{X}$ with $\mathscr{L}|_{X} \cong L$. Let $S(\mathscr{X})$ be the skeleton associated to $\mathscr{X}$ in the Berkovich analytification $X^{\mathrm{an}}$ of $X$. In this article, we study when $S(\mathscr{X})$ is faithfully tropicalized into tropical projective space by the adjoint linear system $|L^{\otimes m} \otimes \omega_X|$. Roughly speaking, our results show that, if $m$ is an integer such that the adjoint bundle is basepoint free, then the adjoint linear system admits a faithful tropicalization of $S(\mathscr{X})$.