Mixed Volume and an Extension of Intersection Theory of Divisors
Kiumars Kaveh, Askold Georgievich Khovanskii · Moscow Mathematical Journal · 2010
Let Krat(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 , . . ., Ln ∈ Krat(X), we define an intersection index [L 1 , . . ., Ln] as the number of solutions in X of a system of equations f 1 = • • • = fn = 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 Krat(X) is a commutative semigroup with respect to a natural multiplication.The intersection index [L 1 , . . ., Ln] can be extended to the Grothendieck group of Krat(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-Kušnirenko theorem from the Newton polytope theory.Contents 6.2.Cartier divisor associated to a subspace of rational functions with a regular Kodaira map 23 7. Topological and algebro-geometric proofs of properties of intersection index 26