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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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Resultants and Chow forms via exterior syzygies
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by David Eisenbud, Frank-Olaf Schreyer and Jerzy Weyman
J. Amer. Math. Soc. 16 (2003), 537-579
Published electronically: February 27, 2003


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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Bibliographic Information
  • David Eisenbud
  • Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
  • MR Author ID: 62330
  • ORCID: 0000-0002-5418-5579
  • Email:
  • Frank-Olaf Schreyer
  • Affiliation: Mathematik und Informatik, Geb. 27, Universität des Saarlandes, D-66123 Saarbrücken, Germany
  • MR Author ID: 156975
  • Email:
  • Jerzy Weyman
  • Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
  • MR Author ID: 182230
  • ORCID: 0000-0003-1923-0060
  • Email:
  • Received by editor(s): November 16, 2001
  • Published electronically: February 27, 2003
  • © Copyright 2003 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 16 (2003), 537-579
  • MSC (2000): Primary 13P05, 14Q99; Secondary 13D25, 14F05
  • DOI:
  • MathSciNet review: 1969204