The Projective Geometry of the Gale Transform

The Gale transform, an involution on sets of points in projective space, appears in a multitude of guises and in subjects as diverse as optimization, coding theory, theta functions, and recently in our proof that certain general 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) Gorenstein subschemes. We put in modern form a number of the more remarkable examples discovered in the past, and we add new constructions and connections to other areas of algebraic geometry. We generalize Goppaʹs theorem in coding theory and we give new applications to Castelnuovo theory. We also give references to classical and modern sources.