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.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2019
- DOI:
- 10.48550/arXiv.1903.11068
- arXiv:
- arXiv:1903.11068
- Bibcode:
- 2019arXiv190311068B
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Commutative Algebra;
- Mathematics - Representation Theory;
- 14D06;
- 14M25;
- 14T05;
- 13A18;
- 14M15
- E-Print:
- 21 pages, 3 pages appendix, 5 figures/tables. arXiv admin note: text overlap with arXiv:1806.02090