  The Gale transform, an involution on sets of points in projective space,
appears in a multitude of guises, 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. 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 Goppa's theorem in coding
theory and we give new applications to Castelnuovo theory. We give references
to classical and modern sources.
