Linear determinantal equations for all projective schemes

Sidman, Jessica; Smith, Gregory

doi:10.2140/ant.2011.5.1041

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Linear determinantal equations for all projective schemes

Jessica Sidman and Gregory G. Smith

Vol. 5 (2011), No. 8, 1041–1061

DOI: 10.2140/ant.2011.5.1041

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 $2 \times 2$ 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

Received: 20 May 2010

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/

Vol. 5, No. 8, 2011