Okounkov bodies and restricted volumes along very general curves

Jow, Shin-Yao

doi:10.1016/j.aim.2009.09.015

Cited by 38 publications

(35 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recall that the Okounkov body of a big divisor is a numerical invariant. More precisely, by [, Proposition 4.1; , Theorem A], for big divisors

D, D^{'}

X

, we have

{normalΔ}_{Y_{•}} false(D false) = {normalΔ}_{Y_{•}} false(D^{'} false)

for any admissible flag

Y_{•}

if and only if

D \equiv D^{'}

. As Theorem states below, this result can be extended to the limiting Okounkov bodies

{normalΔ}_{Y_{•}}^{trueprefixlim} false(D false)

for pseudoeffective divisors

D

.…”

Section: Introductionmentioning

confidence: 85%

“…If

D

is a big divisor, then by [, Proposition 4.1; , Theorem A], it is known that

{normalΔ}_{Y_{•}} false(D false) = {normalΔ}_{Y_{•}} false(D^{'} false)

for all admissible flags

Y_{•}

X

if and only if

D \equiv D^{'}

. Remark implies that the statement is false in general in the pseudoeffective case for

{normalΔ}_{Y_{•}}^{prefixval} false(D false)

.…”

Section: Okounkov Body Of a Pseudoeffective Divisormentioning

confidence: 99%

“…Proof The equivalence

(1) \Leftrightarrow (2)

is known for big divisors by [, Proposition 4.1; , Theorem A]. Thus, we assume that

D, D^{'}

are not big.…”

Section: Okounkov Body Of a Pseudoeffective Divisormentioning

confidence: 99%

See 2 more Smart Citations

Okounkov bodies associated to pseudoeffective divisors

Choi

Hyun

Park

et al. 2018

J. London Math. Soc.

View full text Add to dashboard Cite

An Okounkov body is a convex subset in Euclidean space associated to a big divisor on a smooth projective variety with respect to an admissible flag. In this paper, we introduce two convex bodies associated to pseudoeffective divisors, called the valuative Okounkov bodies and the limiting Okounkov bodies, and show that these convex bodies reflect the asymptotic properties of pseudoeffective divisors as in the case with big divisors. Our results extend the works of Lazarsfeld–Mustaţă and Kaveh–Khovanskii. For this purpose, we define and study special subvarieties, called the Nakayama subvarieties and the positive volume subvarieties, associated to pseudoeffective divisors.

show abstract

“…Recall that the Okounkov body of a big divisor is a numerical invariant. More precisely, by [, Proposition 4.1; , Theorem A], for big divisors

D, D^{'}

X

, we have

{normalΔ}_{Y_{•}} false(D false) = {normalΔ}_{Y_{•}} false(D^{'} false)

for any admissible flag

Y_{•}

if and only if

D \equiv D^{'}

. As Theorem states below, this result can be extended to the limiting Okounkov bodies

{normalΔ}_{Y_{•}}^{trueprefixlim} false(D false)

for pseudoeffective divisors

D

.…”

Section: Introductionmentioning

confidence: 85%

“…If

D

is a big divisor, then by [, Proposition 4.1; , Theorem A], it is known that

{normalΔ}_{Y_{•}} false(D false) = {normalΔ}_{Y_{•}} false(D^{'} false)

for all admissible flags

Y_{•}

X

if and only if

D \equiv D^{'}

. Remark implies that the statement is false in general in the pseudoeffective case for

{normalΔ}_{Y_{•}}^{prefixval} false(D false)

.…”

Section: Okounkov Body Of a Pseudoeffective Divisormentioning

confidence: 99%

See 1 more Smart Citation

Okounkov bodies associated to pseudoeffective divisors

Choi

Hyun

Park

et al. 2018

J. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…A systematic study of the Newton-Okounkov bodies was introduced about the same time in the papers [LM09] and [KK]. 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]. There is also the nice recent paper [KLM12] which describes these bodies in some interesting cases.…”

Section: Introductionmentioning

confidence: 99%

Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory

2012

View full text Add to dashboard Cite

Generalizing the notion of Newton polytope, we define the NewtonOkounkov body, respectively, for semigroups of integral points, graded algebras and linear series on varieties. We prove that any semigroup in the lattice Z n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone. Applying this we obtain several results. We show that for a large class of graded algebras, the Hilbert functions have polynomial growth and their growth coefficients satisfy a Brunn-Minkowski type inequality. We prove analogues of the Fujita approximation theorem for semigroups of integral points and graded algebras, which imply a generalization of this theorem for arbitrary linear series. Applications to intersection theory include a far-reaching generalization of the Kushnirenko theorem (from Newton polytope theory) and a new version of the Hodge inequality. We also give elementary proofs of the Alexandrov-Fenchel inequality in convex geometry and its analogue in algebraic geometry.

show abstract

“…The following is a generalization of [LM,Theorem 4.26] and [J,Theorem 3.4], and it will play a crucial role in studying the augmented base loci and moving Seshadri constants. …”

Section: By Collecting the Valuesmentioning

confidence: 99%

Asymptotic base loci via Okounkov bodies

Choi

Hyun

Park

et al. 2018

Advances in Mathematics

View full text Add to dashboard Cite

Abstract. An Okounkov body is a convex subset of Euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag. In this paper, we recover the asymptotic base loci from the Okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving Seshadri constants. Consequently, we obtain the nefness and ampleness criteria of divisors in terms of the Okounkov bodies. Furthermore, we compute the divisorial Zariski decomposition by the Okounkov bodies, and find upper and lower bounds for moving Seshadri constants given by the size of simplexes contained in the Okounkov bodies.

show abstract

Okounkov bodies and restricted volumes along very general curves

Cited by 38 publications

References 9 publications

Okounkov bodies associated to pseudoeffective divisors

Okounkov bodies associated to pseudoeffective divisors

Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory

Asymptotic base loci via Okounkov bodies

Contact Info

Product

Resources

About