Effective faithful tropicalizations associated to adjoint linear systems

TL;DR

This paper investigates conditions under which the skeleton of a smooth projective variety over a discretely valued field can be faithfully tropicalized using adjoint linear systems, linking algebraic geometry with tropical geometry.

Contribution

It establishes that for sufficiently large and basepoint free adjoint line bundles, the associated linear system provides a faithful tropicalization of the skeleton.

Findings

Faithful tropicalization is achievable with adjoint linear systems.

Basepoint freeness of the adjoint bundle is key.

Results connect algebraic and tropical geometry through skeletons.

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 $S(\mathscr{X})$.