Full-rank Valuations and Toric Initial Ideals

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.