[Paper Review] Convex Bodies Associated to Linear Series

Robert Lazarsfeld, Mircea Mustaţă|ArXiv.org|May 29, 2008

Algebraic Geometry and Number Theory23 references20 citations

TL;DR

This paper introduces Okounkov bodies as convex geometric objects associated to linear series on projective varieties using flag-based valuation constructions. It establishes that the Euclidean volume of the Okounkov body equals the volume of the divisor up to a factorial scaling, generalizing classical results in algebraic geometry and linking asymptotic linear series theory to convex geometry.

ABSTRACT

In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens the door to a number of extensions. The purpose of this paper is to initiate a systematic development of the theory, and to give a number of applications and examples.

Motivation & Objective

• To develop a systematic theory of convex bodies associated to linear series on projective varieties using valuation-based constructions.
• To generalize classical results in asymptotic algebraic geometry—such as volume computations—using convex geometric methods.
• To study the variation of these convex bodies as functions of the divisor class, particularly in relation to numerical equivalence and rational classes.
• To investigate the conditions under which the intersection of a semigroup with a rational subspace yields a cone that matches the intersection of the associated cone with that subspace.
• To explore the geometric and algebraic properties of these convex bodies, including non-polyhedral and non-rational examples.

Proposed method

• Construct a valuation-like function ν on the space of global sections of a big divisor D using a flag of subvarieties Y• of codimension i in X.
• Define the image v(D) as the set of valuation vectors of non-zero sections of O_X(D), which captures the asymptotic behavior of sections.
• Form the Okounkov body Δ(D) as the closed convex hull of the union of (1/m)·v(mD) over m ≥ 1, embedded in R^d.
• Prove that the volume of Δ(D) equals (1/d!) times the volume of the divisor D, using results from Khovanskii and asymptotic analysis.
• Establish that Δ(D) depends only on the numerical equivalence class of D, and extend the construction to rational and real numerical classes.
• Use semigroup theory and cone geometry to analyze the behavior of Okounkov bodies under restriction to rational subspaces, proving that the semigroup intersection cone coincides with the cone intersection under full-dimensionality and interior intersection conditions.

Experimental results

Research questions

• RQ1 How can convex geometry be systematically used to study linear series on projective varieties?
• RQ2 What is the precise relationship between the volume of a divisor and the volume of its associated Okounkov body?
• RQ3 Under what conditions does the intersection of a semigroup with a rational subspace yield a cone that matches the intersection of the associated cone with that subspace?
• RQ4 How does the Okounkov body vary as a function of the divisor class, particularly under numerical equivalence?
• RQ5 Can Okounkov bodies be non-polyhedral or non-rational, even for ample divisors?

Key findings

• The Euclidean volume of the Okounkov body Δ(D) is equal to (1/d!) times the volume of the divisor D, i.e., vol_{R^d}(Δ(D)) = (1/d!)·vol_X(D).
• The construction of Δ(D) depends only on the numerical equivalence class of D, and Δ(pD) = p·Δ(D) for positive integers p.
• For any rational numerical class ξ ∈ N^1(X)_Q, there exists a well-defined Okounkov body Δ(ξ) ⊆ R^d with vol_{R^d}(Δ(ξ)) = (1/d!)·vol_X(ξ).
• The paper constructs an example of a divisor on an abelian surface with an irrational Okounkov body that is trapezoidal but not rational, demonstrating non-rationality even in the ample case.
• A four-dimensional example is given where the Okounkov body is not polyhedral, showing that the convex bodies can have complex geometric structure.
• Under the assumption that the semigroup Γ generates a subgroup of finite index in Z^n and the subspace L intersects the interior of the cone Σ, it is proven that Σ(Γ) ∩ L = Σ(Γ ∩ L), establishing a key compatibility between semigroup and cone intersections.