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Resultants and Chow forms via exterior syzygies

Authors: David Eisenbud, Frank-Olaf Schreyer and Jerzy Weyman
Journal: J. Amer. Math. Soc. 16 (2003), 537-579
MSC (2000): Primary 13P05, 14Q99; Secondary 13D25, 14F05.
Published electronically: February 27, 2003
MathSciNet review: 1969204
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Abstract: Given a sheaf on a projective space ${\mathbf P}^n$, we define a sequence of canonical and effectively computable Chow complexes on the Grassmannians of planes in ${\mathbf P}^n$, generalizing the well-known Beilinson monad on ${\mathbf P}^n$. If the sheaf has dimension $k$, then the Chow form of the associated $k$-cycle is the determinant of the Chow complex on the Grassmannian of planes of codimension $k+1$. Using the theory of vector bundles and the canonical nature of the complexes, we are able to give explicit determinantal and Pfaffian formulas for resultants in some cases where no polynomial formulas were known. For example, the Horrocks–Mumford bundle gives rise to a polynomial formula for the resultant of five homogeneous forms of degree eight in five variables.

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Additional Information

David Eisenbud
Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
MR Author ID: 62330
ORCID: 0000-0002-5418-5579

Frank-Olaf Schreyer
Affiliation: Mathematik und Informatik, Geb. 27, Universität des Saarlandes, D-66123 Saarbrücken, Germany
MR Author ID: 156975

Jerzy Weyman
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
MR Author ID: 182230
ORCID: 0000-0003-1923-0060

Keywords: Chow form, resultants, Beilinson monad, Ulrich modules.
Received by editor(s): November 16, 2001
Published electronically: February 27, 2003
Article copyright: © Copyright 2003 American Mathematical Society