Full-rank Valuations and Toric Initial Ideals

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 Plücker coordinates form a Khovanskii basis. Further, as a corollary we obtain that the weight vectors defined in \cite{BFFHL} lie in the tropical Grassmannian.

21 pages, 3 pages appendix, 5 figures/tables. arXiv admin note: text overlap with arXiv:1806.02090

References in corpus (4)

• Total positivity, Grassmannians, and networks
• Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
• Toric degenerations: a bridge between representation theory, tropical geometry and cluster algebras
• Birational sequences and the tropical Grassmannian

Cited by in corpus (6)

• Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
• Toric degenerations of partial flag varieties and combinatorial mutations of matching field polytopes
• Birational sequences and the tropical Grassmannian
• Newton--Okounkov bodies and minimal models for cluster varieties
• Combinatorial mutations of Newton-Okounkov polytopes arising from plabic graphs
• Generic tropical initial ideals of Cohen-Macaulay algebras