The Projective Geometry of the Gale Transform

✍ Scribed by David Eisenbud; Sorin Popescu

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 297 KB
• Volume: 230
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.7940

✦ Synopsis

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 Ž satisfy the minimal free resolution conjecture see Eisenbud and Popescu, 1999, . In¨ent. 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 Ž . 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 also give references to classical and modern sources. ᮊ 2000 Academic Press CONTENTS I. The Gale transform 1. History 2. The scheme-theoretic definition 3. A generalized Goppa theorem 1 Both authors are grateful to the NSF for support during the preparation of this work.

It was in joint work with David Buchsbaum that the first author first became familiar with Ž . the notion of a Gorenstein ring. A large part of this paper ''self-associated sets'' is concerned, from an algebraic point of view, with the classification and study of a special type of Gorenstein ring, generalizing some of the examples found in that joint work. It is with special pleasure that we dedicate this paper to David.