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Mixed volume and an analogue of intersection theory of divisors for non-complete varieties | | Kiumars Kaveh
; A. G. Khovanskii
; | | Date: |
2 Dec 2008 | | Abstract: | Let K(X) be the collection of all finite dimensional subspaces of rational
functions on a complex n-dimensional 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 f_i is a
generic function from the space L_i. In counting the solutions, we neglect
solutions x at which all the functions in some space L_i vanish as well as
solutions at which at least one function from some space L_i has a pole. The
set K(X) is a commutative semi-group with respect to a natural multiplication.
The intersection index {L_1,..., L_n} is multi-linear with respect to this
multiplication and can be extended to the Grothendieck group of K(X). We hence
obtain an analogue of the intersection theory of divisors for a (possibly)
non-complete variety X. We show that the intersection index enjoys all the main
properties of the mixed volume of convex bodies. This paper was inspired by
Bernstein-Kushnirenko theorem from the theory of Newton polyhedra. | | Source: | arXiv, 0812.0433 | | Services: | Forum | Review | PDF | Favorites | |
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