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 $\mathcal{X}$ of $X$ over $R$ and a relatively ample line bundle $\mathcal{L}$ over $\mathcal{X}$ with $\mathcal{L}{|}_X\cong L$. Let $S(\mathcal{X})$ be the skeleton associated to $\mathcal{X}$ in the Berkovich analytification $X^{\mathrm{an}}$ of $X$. In this article, we study when $S(\mathcal{X})$ 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 $S(\mathcal{X})$.

16 pages. Presentation has been changed. Some minor errors have been corrected

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