Fullrank 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 fullrank 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 semigroup 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 RietschWilliams' 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1903.11068
 Bibcode:
 2019arXiv190311068B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Commutative Algebra;
 Mathematics  Representation Theory;
 14D06;
 14M25;
 14T05;
 13A18;
 14M15
 EPrint:
 21 pages, 3 pages appendix, 5 figures/tables. arXiv admin note: text overlap with arXiv:1806.02090