Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/16086
Title: Noncommutative quadric surfaces
Authors: Smith, S. Paul
VAN DEN BERGH, Michel 
Issue Date: 2013
Source: JOURNAL OF NONCOMMUTATIVE GEOMETRY, 7 (3), p. 817-856
Abstract: The 4-dimensional Sklyanin algebra is the homogeneous coordinate ring of a noncommutative analogue of projective 3-space. The degree-two component of the algebra contains a 2-dimensional subspace of central elements. The zero loci of those central elements, except 0, form a pencil of noncommutative quadric surfaces. We show that the behavior of this pencil is similar to that of a generic pencil of quadrics in the commutative projective 3-space. There are exactly four singular quadrics in the pencil. The singular and non-singular quadrics are characterized by whether they have one or two rulings by noncommutative lines. The Picard groups of the smooth quadrics are free abelian of rank two. The alternating sum of dimensions of Ext allows us to define an intersection pairing on the Picard group of the smooth ntive quadrics. A surprise is that a smooth noncommutative quadric can sometimes contain a "curve" having self-intersection number -2. Many of the methods used in our paper are noncommutative versions of methods developed by Buchweitz, Eisenbud and Herzog: in particular, the correspondence between the geometry of a quadric hypersurface and maximal Cohen-Macaulay modules over its homogeneous coordinate ring plays a key role. An important aspect of our work is to introduce definitions of noncommutative analogues of the familiar commutative terms used in this abstract. We expect the ideas we develop here for 2-dimensional noncommutative quadric hypersurfaces will apply to higher dimensional noncommutative quadric hypersurfaces and we develop them in sufficient generality to make such applications possible.
Notes: Smith, SP (reprint author), Univ Washington, Dept Math, Box 354350, Seattle, WA 98195 USA. smith@math.washington.edu; michel.vandenbergh@luc.ac.be
Keywords: Noncommutative algebraic geometry, noncommutative quadric surfaces, Sklyanin algebra
Document URI: http://hdl.handle.net/1942/16086
ISSN: 1661-6952
e-ISSN: 1661-6960
DOI: 10.4171/JNCG/136
ISI #: 000327472400008
Category: A1
Type: Journal Contribution
Validations: ecoom 2014
Appears in Collections:Research publications

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