[Paper Review] Mixed volume and an extension of intersection theory of divisors

Kiumars Kaveh, Askold Khovanskiĭ|arXiv (Cornell University)|Dec 2, 2008

Polynomial and algebraic computation12 references3 citations

TL;DR

This paper introduces a birationally invariant intersection index for finite-dimensional subspaces of rational functions on an n-dimensional irreducible variety X, extending classical intersection theory beyond complete varieties. By defining an intersection index analogous to mixed volume in convex geometry and extending it to the Grothendieck group of subspaces, the authors establish a framework that generalizes the Bernstein-Kuřnirenko theorem and proves key inequalities like the Hodge-type and Alexandrov-Fenchel-type inequalities.

ABSTRACT

Let K(X) be the collection of all non-zero finite dimensional subspaces of rational functions on an n-dimensional irreducible variety X. For any n-tuple L_1,..., L_n in K(X), we define an intersection index [L_1,..., L_n] as the number of solutions in X of a system of equations f_1 = ... = f_n = 0 where each f_i is a generic function from the space L_i. In counting the solutions, we neglect the solutions x at which all the functions in some space L_i vanish as well as the solutions at which at least one function from some subspace L_i has a pole. The collection K(X) is a commutative semigroup with respect to a natural multiplication. The intersection index [L_1,..., L_n] can be extended to the Grothendieck group of K(X). This gives an extension of the intersection theory of divisors. The extended theory is applicable even to non-complete varieties. We show that this intersection index enjoys all the main properties of the mixed volume of convex bodies. Our paper is inspired by the Bernstein-Kushnirenko theorem from the Newton polytope theory.

Motivation & Objective

• To generalize the Bernstein-Kuřnirenko theorem beyond toric varieties to arbitrary irreducible algebraic varieties.
• To define a birationally invariant intersection index for finite-dimensional subspaces of rational functions on a variety X.
• To extend this intersection index to the Grothendieck group of the semigroup of subspaces of rational functions.
• To establish analogues of mixed volume properties—such as multilinearity, symmetry, and the Alexandrov-Fenchel inequality—within algebraic geometry.
• To connect the theory to Cartier divisors and show that for normal projective varieties, the Grothendieck group of certain subspaces is isomorphic to the group of Cartier divisors.

Proposed method

• Define the intersection index $[L_1, \ldots, L_n]$ as the number of common zeros of generic sections $f_i \in L_i$, excluding points where all $f_i$ vanish or any $f_i$ has a pole.
• Equip the set $\mathbf{K}_{\text{rat}}(X)$ of non-zero finite-dimensional subspaces of rational functions with a commutative semigroup structure under pointwise multiplication of functions.
• Extend the intersection index to the Grothendieck group of $\mathbf{K}_{\text{rat}}(X)$ using its multilinearity with respect to the semigroup operation.
• Use the Kodaira map $\Phi_L: X \dashrightarrow \mathbb{P}(L^*)$ to associate subspaces to rational maps and identify a subsemigroup $\mathbf{K}_{\text{Cart}}(X)$ of subspaces with regular Kodaira maps.
• Apply Hodge theory and integral formulas involving Kähler forms on resolution surfaces to derive inequalities, including a Hodge-type inequality for surfaces.
• Prove the Alexandrov-Fenchel inequality for the intersection index by reducing to the Hodge index theorem on a desingularized model of the variety.

Experimental results

Research questions

• RQ1 Can the Bernstein-Kuřnirenko theorem be generalized beyond toric varieties to arbitrary irreducible algebraic varieties?
• RQ2 How can an intersection index for subspaces of rational functions be defined in a way that is invariant under birational equivalence?
• RQ3 What algebraic structure (e.g., Grothendieck group) underlies the intersection index, and how does it relate to classical divisor theory?
• RQ4 Do the intersection indices satisfy inequalities analogous to those in mixed volume theory, such as the Alexandrov-Fenchel inequality?
• RQ5 Is there a natural correspondence between subspaces of rational functions with regular Kodaira maps and Cartier divisors on projective varieties?

Key findings

• The intersection index $[L_1, \ldots, L_n]$ is well-defined, birationally invariant, and extends linearly to the Grothendieck group of $\mathbf{K}_{\text{rat}}(X)$.
• For normal projective varieties, the Grothendieck group of the subsemigroup $\mathbf{K}_{\text{Cart}}(X)$ of subspaces with regular Kodaira maps is naturally isomorphic to the group of Cartier divisors on $X$.
• The intersection index satisfies the Hodge-type inequality: $[L_1, L_2]^2 \geq [L_1, L_1][L_2, L_2]$ for surfaces.
• The intersection index satisfies the Alexandrov-Fenchel inequality, as shown via Hodge theory on a desingularized model of the variety.
• The theory generalizes the mixed volume of convex bodies to algebraic geometry, with the intersection index mimicking the properties of mixed volume.
• The integral formula for the intersection index involves pullbacks of Kähler forms on projective spaces, linking algebraic geometry to differential geometry.