# Full-rank Valuations and Toric Initial Ideals

## TL;DR

This paper establishes a deep connection between valuations, initial ideals, and tropical geometry for projective varieties, providing criteria for when valuations generate the value semi-group and linking to conjectures in flag varieties and Grassmannians.

## Contribution

It proves that the value semi-group is generated by generators if and only if the initial ideal is prime, and shows that associated weight vectors lie in the tropicalization, with applications to flag varieties and Grassmannians.

## Key findings

- Value semi-group generated iff initial ideal is prime.
- Weight vectors always in the tropicalization.
- Criteria for Khovanskii bases in Grassmannians.

## Abstract

Let $V(I)$ be a polarized projective variety or a subvariety of a product of projective spaces and let $A$ be its (multi-)homogeneous coordinate ring. Given a full-rank valuation $\mathfrak v$ on $A$ we associate weights to the coordinates of the projective space, respectively, the product of projective spaces. Let $w_{\mathfrak v}$ be the vector whose entries are these weights. Our main result is that the value semi-group of $\mathfrak v$ is generated by the images of the generators of $A$ if and only if the initial ideal of $I$ with respect to $w_{\mathfrak v}$ is prime. We further show that $w_{\mathfrak v}$ always lies in the tropicalization of $I$. Applying our result to string valuations for flag varieties, we solve a conjecture by \cite{BLMM} connecting the Minkowski property of string cones with the tropical flag variety. For Rietsch-Williams' valuation for Grassmannians our results give a criterion for when the Pl\"ucker coordinates form a Khovanskii basis. Further, as a corollary we obtain that the weight vectors defined in \cite{BFFHL} lie in the tropical Grassmannian.