Geodesic
c-fans and Newton polyhedra of algebraic varieties
Izvestiya. Mathematics , Tome 67 (2003) no. 3, pp. 439-460

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To every algebraic subvariety of a complex torus there corresponds a Euclidean geometric object called a c-fan. This correspondence determines an intersection theory for algebraic varieties. c-fans form a graded commutative algebra with visually defined operations. The c-fans of algebraic varieties lie in the subring of rational c-fans. It seems that other subrings may be used to construct an intersection theory for other categories of analytic varieties. We discover a relation between an old problem in the theory of convex bodies (the so-called Minkowski problem) and the ring of c-fans. This enables us to define a correspondence that sends any algebraic curve to a convex polyhedron in the space of characters of the torus. 
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     author = {B. Ya. Kazarnovskii},
     title = {c-fans and {Newton} polyhedra of algebraic varieties},
     journal = {Izvestiya. Mathematics },
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B. Ya. Kazarnovskii. c-fans and Newton polyhedra of algebraic varieties. Izvestiya. Mathematics , Tome 67 (2003) no. 3, pp. 439-460. http://geodesic.mathdoc.fr/item/IM2_2003_67_3_a2/