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Newton–Okounkov body

A convex body associated, after choosing valuation or flag data, with a divisor or graded linear series on an algebraic variety, encoding asymptotic section growth and positivity in Euclidean geometry.

Version
v1 · 2026-09-28 · History
Domain-specific #
10986
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Algebraic Geometry, Convex Geometry → Mathematics

Core Idea

A Newton–Okounkov body converts asymptotic algebraic geometry into convex geometry. Starting from a variety, a divisor or graded linear series, and a valuation often defined by an admissible flag, one records valuation vectors of nonzero sections in every positive degree and normalizes by that degree.

The closed convex body generated by those normalized vectors encodes asymptotic information such as section growth and divisor volume. It generalizes the Newton polytope of a projective toric variety and includes several representation-theoretic polytopes under suitable constructions. The auxiliary valuation or flag matters: bodies for the same divisor need not have the same shape, even when some invariants such as volume are preserved.

Structural Signature

Sig role-phrases:

  • projective variety. Supplies the algebraic-geometric space. Constitutive base. If altered: A standalone polytope does not identify a Newton–Okounkov body.
  • divisor or graded linear series. Provides sections across positive multiples. Constitutive geometric data. If altered: Changing the series can change the body.
  • valuation or admissible flag. Maps nonzero sections to lattice vectors in an ordered way. Identity-bearing choice. If altered: The body generally depends on this auxiliary data.
  • value semigroup and normalization. Collects degree-valuation pairs and scales vectors by degree. Constitutive construction. If altered: One degree alone does not capture asymptotic geometry.
  • closed convex body. Takes the appropriate convex closure or cone slice in Euclidean space. Constitutive output. If altered: Volume and shape encode asymptotic invariants under hypotheses.

What It Is Not

  • Newton polytope. Is the general graded valuation construction present?
  • Moment polytope. Which symplectic or representation data define it?
  • Value semigroup. Is the convex normalized body or discrete precursor meant?
  • Effective cone. Is a cone of divisor classes being described?

Scope of Application

Use Newton–Okounkov body with variety, divisor or series, flag or valuation, semigroup, normalization, dimension, and positivity hypotheses stated.

  • Algebraic geometry. Studies divisors and positivity.
  • Convex geometry. Translates asymptotic invariants.
  • Representation theory. Realizes string and related polytopes.
  • Toric geometry. Recovers Newton polytopes.
  • Valuation theory. Builds value semigroups.

Clarity

The body is not intrinsic without auxiliary data, while selected quantities derived from it can be invariant.

Manages Complexity

Finite computation may approximate an object defined through all degrees. Non-finite generation, empty interiors, non-big divisors, and normalization conventions should be addressed before reading geometric claims from a plotted polytope.

Abstract Reasoning

  1. Choose variety and graded linear series.
  2. Specify an admissible flag or valuation.
  3. Form degree-valuation semigroup data.
  4. Normalize vectors and take convex closure.
  5. Interpret volume and faces under applicable theorems.

Knowledge Transfer

Asymptotic-to-convex encoding transfers across graded structures, but algebraic sections, valuations, and divisor geometry delimit Newton–Okounkov bodies. The nearest stopping boundary is explicit: A toric Newton polytope is closest: under suitable toric choices it is recovered as a special case, while the general construction applies beyond toric varieties and depends on flag data. The inclusion test remains: A convex set is a Newton–Okounkov body when it is constructed from normalized valuation data of all graded sections of a divisor or linear series on a variety. The structure no longer applies when the case exits when no graded section series and valuation construction underlie the convex body.

Examples

Canonical

For a projective variety with a big divisor and admissible flag, valuations of sections in H0(X,mD) are divided by m across all m and their convex closure forms the body.

Mapped back: projective variety → X; divisor or graded linear series → multiples mD; valuation or admissible flag → chosen flag; value semigroup and normalization → nu(s)/m; closed convex body → closure of normalized values.

Applied / In Practice

A lattice polytope is drawn from monomial exponents but no variety, graded series, or valuation data is specified. It may be a Newton polytope, not automatically a Newton–Okounkov body.

Mapped back: projective variety → unspecified; divisor or graded linear series → absent; valuation or admissible flag → absent; value semigroup and normalization → monomial set only; closed convex body → polytope.

Structural Tensions

T1: auxiliary choice vs. intrinsic invariant. Shape changes with flag while volume can remain geometric. Diagnostic: Which conclusion is choice-independent?

T2: infinite graded data vs. finite computation. The definition spans all multiples while examples truncate. Diagnostic: What convergence or finite generation is known?

Structural–Framed Character

Description turns on projective variety, divisor or graded linear series, valuation or admissible flag, value semigroup and normalization, closed convex body. Skeletal core. A graded family is embedded into ordered numerical coordinates whose asymptotic convex hull reveals growth. Domain-bound accent. Varieties, divisors, sections, flags, valuations, semigroups, and convex volume define the body. Transfer remains bounded because Why not prime. Convex encoding is portable; this is an algebraic-geometric construction. The negative boundary is concrete: Any Newton polytope, moment polytope, convex hull, valuation cone, effective cone, lattice polytope, Okounkov semigroup, or polytope in representation theory is not automatically a Newton–Okounkov body. The body is structural-representational: asymptotic algebraic section data are mapped into a convex Euclidean surrogate. Its character: divisor geometry compressed into normalized valuation shape.

Structural Core vs. Domain Accent

Skeletal core. A graded family is embedded into ordered numerical coordinates whose asymptotic convex hull reveals growth.

Domain-bound accent. Varieties, divisors, sections, flags, valuations, semigroups, and convex volume define the body.

Why not prime. Convex encoding is portable; this is an algebraic-geometric construction.

  • Newton polytope. Toric cases recover the classical object.
  • Valuation. It produces the coordinate vectors.
  • No strict parent is asserted.

Neighborhood in Abstraction Space

Newton–Okounkov body sits in a crowded region of the domain-specific corpus (40th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Formal Models & Logical Foundations (33 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

  • Newton polytope. Tell: Is the general graded valuation construction present?
  • Moment polytope. Tell: Which symplectic or representation data define it?
  • Value semigroup. Tell: Is the convex normalized body or discrete precursor meant?
  • Effective cone. Tell: Is a cone of divisor classes being described?

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Newton%E2%80%93Okounkov_body (revision 1325516867).
  • Preserved source candidate: https://www.mfo.de/occasion/1422b/www_view
  • Preserved source candidate: http://www.birs.ca/events/2014/5-day-workshops/14w5056
  • Preserved source candidate: http://www.birs.ca/events/2014/5-day-workshops/14w5013
  • Preserved source candidate: https://www.mfo.de/occasion/1134b/www_view

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.