Okounkov bodies and toric degenerations

By David Anderson. 2010

159 citations recorded by the source corpus.

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 this situation $$X$$ admits a flat degeneration to the corresponding toric variety. This degeneration is functorial in an appropriate sense.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6cf1e7c6d63a4df17289aa66f1918a6180391dd7

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In Algebra and Number Theory

1. Kiumars Kaveh
2. Megumi Harada
3. Askold Khovanskii
4. Chi Li
5. G'abor Sz'ekelyhidi

In Geometry and Topology

1. Kiumars Kaveh
2. Megumi Harada
3. Askold Khovanskii
4. Chi Li
5. Ghislain Fourier
6. P. Littelmann
7. G'abor Sz'ekelyhidi

Citations in the corpus

• Integrable systems, toric degenerations and Okounkov bodies (2012, methodology): “This is explained in more detail in [2].”
• Crystal bases and Newton–Okounkov bodies (2011, background): “is 20 sequence of subvarieties (together with the obvious choices of parameters) coincides with the valuation vw 0 . 6. SAGBI bases,valuationsandtoric degenerations This section is closely related to [And10]. Let C[x1,...,xn] be the polynomial algebra in nvariables. Fix a well-ordering <on Zn ≥0 ”
• Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory (2009, methodology): “Recently the Newton-Okounkov bodies (which LazarsfeldMustata call Okounkov bodies) have been explored and used in the papers of several authors among which we can mention [Yua09], [Nys], [Jow10], [BC11], [And] and [Pet].”