Algebra & Number Theory

Linear determinantal equations for all projective schemes

Jessica Sidman and Gregory Smith

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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.

Article information

Algebra Number Theory, Volume 5, Number 8 (2011), 1041-1061.

Received: 20 May 2010
Revised: 31 May 2011
Accepted: 30 June 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14A25: Elementary questions
Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 13D02: Syzygies, resolutions, complexes

determinantally presented linear free resolution Castelnuovo–Mumford regularity


Sidman, Jessica; Smith, Gregory. Linear determinantal equations for all projective schemes. Algebra Number Theory 5 (2011), no. 8, 1041--1061. doi:10.2140/ant.2011.5.1041.

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