Advanced Studies in Pure Mathematics

Gorenstein in codimension 4: the general structure theory

Miles Reid

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I describe the projective resolution of a codimension 4 Gorenstein ideal, aiming to extend Buchsbaum and Eisenbud's famous result in codimension 3. The main result is a structure theorem stating that the ideal is determined by its $(k + 1) \times 2k$ matrix of first syzygies, viewed as a morphism from the ambient regular space to the Spin-Hom variety $\mathrm{SpH}_k \subset \mathrm{Mat}(k + 1, 2k)$. This is a general result encapsulating some theoretical aspects of the problem, but, as it stands, is still some way from tractable applications.

Article information

Algebraic Geometry in East Asia — Taipei 2011, J. A. Chen, M. Chen, Y. Kawamata and J. Keum, eds. (Tokyo: Mathematical Society of Japan, 2015), 201-227

Received: 17 February 2012
Revised: 25 February 2013
First available in Project Euclid: 19 October 2018

Permanent link to this document euclid.aspm/1539916454

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 13D25
Secondary: 13D02: Syzygies, resolutions, complexes 14J10: Families, moduli, classification: algebraic theory 14M05: Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) [See also 13F15, 13F45, 13H10]

Gorenstein free resolution spinor coordinates


Reid, Miles. Gorenstein in codimension 4: the general structure theory. Algebraic Geometry in East Asia — Taipei 2011, 201--227, Mathematical Society of Japan, Tokyo, Japan, 2015. doi:10.2969/aspm/06510201.

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