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The projective geometry of the Gale transform. (English) Zbl 1060.14528
From the paper: Let \(r,s\) be positive integers and let \(\gamma= r+s+2\). The classical Gale transform is an involution that takes a (reasonably general) set \(\Gamma\subset\mathbb{P}^r\) of \(\gamma\) labeled points in a projective space \(\mathbb{P}^r\) to a set \(\Gamma'\) of \(\gamma\) labeled points in \(\mathbb{P}^s\), defined up to a linear transformation of \(\mathbb{P}^s\).
Examples: \(r=1\). The Gale transform of a set of \(s+3\) points in \(\mathbb{P}^1\) is the corresponding set of \(s+3\) points on the rational normal curve that is the \(s\)-uple embedding of \(\mathbb{P}^1\) in \(\mathbb{P}^s\). Conversely, the Gale transform of any \(s+3\) points in linearly general position in \(\mathbb{P}^s\) is the same set in the \(\mathbb{P}^1\) that is the unique rational normal curve through the original points.
\(r=2\), \(s=3\). A set \(\Gamma\) of seven general points in \(\mathbb{P}^3\) lies on three quadrics, which intersect in eight points. The Gale transform of \(\Gamma\) is the projection of \(\Gamma\) from the eighth point. The Gale transform 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 [cf. D. Eisenbud and S. Popescu, Invent. Math. 136, 419–449 (1999; Zbl 0943.13011)].
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 also give references to classical and modern sources.
Reviewer: Reviewer (Berlin)

MSC:
14N05 Projective techniques in algebraic geometry
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14N15 Classical problems, Schubert calculus
14G50 Applications to coding theory and cryptography of arithmetic geometry
Software:
Macaulay2
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