The projective geometry of the Gale transform

The Gale transform, an involution on sets of points in projective space, ap pears in a multitude of guises and in subjects as diverse as optimization, coding theory, theta functions, and recently in our proof that certain gene ral sets of points fail to satisfy the minimal free resolution conjecture ( see Eisenbud and Popescu, 1999, Invent. Math. 136, 419-449). In this paper we reexamine the Gale transform in the light of modern algebraic geometry. We give a more general definition in the context of finite (locally) Gorens tein subschemes. We put in modern form a number of the more remarkable exam ples discovered in the past, and we add new constructions and connections t o other areas of algebraic geometry. We generalize Goppa's theorem in codin g theory and we give new applications to Castelnuovo theory. We also give r eferences to classical and modern sources. (C) 2000 Academic Press.