Vol. 5, No. 8, 2011

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Linear determinantal equations for all projective schemes

Jessica Sidman and Gregory G. Smith

Vol. 5 (2011), No. 8, 1041–1061
Abstract

We prove that every projective embedding of a connected scheme determined by the complete linear series of a sufficiently ample line bundle is defined by the $2×2$ minors of a $1$-generic matrix of linear forms. Extending the work of Eisenbud, Koh and Stillman for integral curves, we also provide effective descriptions for such determinantally presented ample line bundles on products of projective spaces, Gorenstein toric varieties, and smooth varieties.

Keywords
determinantally presented, linear free resolution, Castelnuovo–Mumford regularity
Mathematical Subject Classification 2000
Primary: 14A25
Secondary: 14F05, 13D02
Milestones
Revised: 31 May 2011
Accepted: 30 June 2011
Published: 5 June 2012
Authors
 Jessica Sidman Department of Mathematics and Statistics Mount Holyoke College 415A Clapp Lab South Hadley, MA 01075 United States http://www.mtholyoke.edu/~jsidman/ Gregory G. Smith Department of Mathematics and Statistics Queen’s University 512 Jeffery Hall, University Avenue Kingston, ON K7L 3N6 Canada http://www.mast.queensu.ca/~ggsmith/