Full-Rank Valuations and Toric Initial Ideals

@article{Bossinger2019FullRankVA,
  title={Full-Rank Valuations and Toric Initial Ideals},
  author={Lara Bossinger},
  journal={arXiv: Algebraic Geometry},
  year={2019}
}
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… Expand

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References

SHOWING 1-10 OF 58 REFERENCES
Khovanskii Bases, Higher Rank Valuations, and Tropical Geometry
TLDR
The notion of a Khovanskii basis for $(A, \mathfrak{v})$ is introduced which provides a framework for far extending Gr\"obner theory on polynomial algebras to general finitely generated algeBRas and construct an associated compactification of $Spec(A)$. Expand
Essential bases and toric degenerations arising from birational sequences
We present a new approach to construct $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, combining ideas coming from the theory of Newton-Okounkov bodies with ideasExpand
Canonical bases for cluster algebras
In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonicalExpand
Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
We use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian $X=Gr_{n-k}(\mathbb C^n)$, as well asExpand
Crystal bases and Newton–Okounkov bodies
Let G be a connected reductive algebraic group. We prove that the string parametrization of a crystal basis for a finite dimensional irreducible representation of G coincides with a natural valuationExpand
Birational sequences and the tropical Grassmannian
We introduce iterated sequences for Grassmannians, a new class of Fang-Fourier-Littelmanns' birational sequences and explain how they give rise to points in $\text{trop}(\text{Gr}(k,\mathbb C^n))$,Expand
Tensor product multiplicities, canonical bases and totally positive varieties
We obtain a family of explicit "polyhedral" combinatorial expressions for multiplicities in the tensor product of two simple finite-dimensional modules over a complex semisimple Lie algebra. HereExpand
Okounkov bodies and toric degenerations
Let $$\varDelta $$ be the Okounkov body of a divisor $$D$$ on a projective variety $$X$$. We describe a geometric criterion for $$\varDelta $$ to be a lattice polytope, and show that in thisExpand
Toric degenerations of Grassmannians from matching fields
We study the algebraic combinatorics of monomial degenerations of Plucker forms which is governed by matching fields in the sense of Sturmfels and Zelevinsky. We provide a necessary condition for aExpand
Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory
Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras, and linear series on varieties. We prove that anyExpand
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