Gale Duality for Complete Intersections

We show that every complete intersection defined by Laurent polynomials in an algebraic torus is isomorphic to a complete intersection defined by master functions in the complement of a hyperplane arrangement, and vice versa. We call systems defining such isomorphic schemes Gale dual systems because the exponents of the monomials in the polynomials annihilate the weights of the master functions. 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 master function complete intersections. Introduction A complete intersection with support W is a subscheme of the torus (C×)m+n having pure dimension m that may be defined by a system f1(x1, . . . , xm+n) = f2(x1, . . . , xm+n) = · · · = fn(x1, . . . , xm+n) = 0 of Laurent polynomials with support W . Let p1(y), . . . , pl+m+n(y) be degree 1 polynomials defining an arrangement A of hyperplanes in C and let β = (b1, . . . , bl+m+n) ∈ Z be a vector of integers. A master function of weight β is the rational function p(y) := p1(y) b1 · p2(y) · · · pl+m+n(y)l+m+n , which is defined on the complement MA := Cl+m\A of the arrangement. A master function complete intersection is a pure subscheme of MA which may be defined by a system p(y)1 = p(y)2 = · · · = p(y)l = 1 of master functions. We describe a correspondence between systems defining polynomial complete intersections and systems defining master function complete intersections that we call Gale duality, as the exponent vectors of the monomials in the polynomials and the weights of the master functions annihilate each other. There is also a second linear algebraic duality between the degree 1 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 with support W is a linear section of the torus in an appropriate projective embedding, and that in turn is a torus 2000 Mathematics Subject Classification. 14M25, 14P25, 52C35.