2 6 Ju n 20 07 GALE DUALITY FOR COMPLETE INTERSECTIONS

We show that every complete intersection of Laurent polynomials in an algebraic torus is isomorphic to a complete intersection of master functions in the complement of a hyperplane arrangement, and vice versa. We call this association Gale duality because the exponents of the monomials in the polynomials annihilate the weights of the master functions and linear forms defining the two systems also annihilate each other. We use Gale duality to give a Kouchnirenko theorem for the number of solutions to a system of master functions and to compute some topological invariants of generic master function complete intersections. Introduction A polynomial complete intersection in the torus (C) is a system f1(x1, . . . , xm+n) = f2(x1, . . . , xm+n) = · · · = fn(x1, . . . , xm+n) = 0 of Laurent polynomials whose solutions in (C) form a subscheme of dimension m. Let p1(y), . . . , pl+m+n(y) be linear polynomials defining the hyperplanes in an arrangement A of hyperplanes in C and let β = (b1, . . . , bl+m+n) ∈ Z l+m+n be a vector of integers. A master function of weight β is the rational function p(y) := p1(y) b1 · p2(y) b2 · · · pl+m+n(y) bl+m+n , which is naturally defined on the complement MA := C l+m \ A of the arrangement. A master function complete intersection is a system p(y)1 = p(y)2 = · · · = p(y)l = 1 of master functions defining a subscheme of dimension m in C \ A, where the weights B := {β1, . . . , βl} ⊂ Z l+m+n are linearly independent. We describe a correspondence between polynomial complete intersections and master function complete intersections that we call Gale duality, as the weights of the master functions and the exponent vectors of the monomials in the polynomials annihilate each other. There is also a second duality between the linear polynomials pi and linear forms defining the polynomials fi. Our main result is that the schemes defined by a pair of Gale dual systems are isomorphic. This follows from the simple geometric observation that a complete intersection in a torus is a linear section of the torus in an appropriate projective embedding, and that in turn is a toric section of a linear embedding of a hyperplane complement. We explain this geometry in Section 1. 2000 Mathematics Subject Classification. 14M25, 14P25, 52C35.