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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.
DOI: https://doi.org/10.1090/S0894-0347-03-00423-5
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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    [Angéniol and Lejeune-Jalabert 1989] AngeniolandLejeune-Jalabert1989 B. Angéniol and M. Lejeune-Jalabert: Calcul différentiel et classes caractéristiques en géométrie algébrique. Travaux en Cours 38, Hermann, Paris, 1989. J. Backelin and J. Herzog: On Ulrich-modules over hypersurface rings. Commutative algebra (Berkeley, CA, 1987), 63–68, Math. Sci. Res. Inst. Publ., 15, Springer, New York, 1989. A. Beauville: Determinantal hypersurfaces. Dedicated to William Fulton on the occasion of his 60th birthday. Mich. Math. J. 48 (2000) 39–64. A. Beilinson: Coherent sheaves on ${\mathbf P}^n$ and problems of linear algebra. Funct. Anal. and its Appl. 12 (1978) 214–216. (Trans. from Funkz. Anal. i. Ego Priloz 12 (1978) 68–69.)
  • I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Algebraic vector bundles on ${\bf P}^{n}$ and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66–67 (Russian). MR 509387
  • [Bézout 1779] Bezout1779 E. Bézout: Théorie générale des équation algébriques, Pierres, Paris 1779. [Brennan et al. 1987] Brennanetal.1987 J. Brennan, J. Herzog, and B. Ulrich: Maximally generated Cohen–Macaulay modules. Math. Scand. 61 (1987) 181–203. R.-O. Buchweitz and F.-O. Schreyer: Complete intersections of two quadrics, hyperelliptic curves and their Clifford Algebras, manuscript, 2002. R.-O. Buchweitz and F.-O. Schreyer: Complete intersections of two quadrics, hyperelliptic curves and their Clifford Algebras, manuscript, 2002.
  • Ragnar-Olaf Buchweitz, David Eisenbud, and Jürgen Herzog, Cohen-Macaulay modules on quadrics, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 58–116. MR 915169, DOI https://doi.org/10.1007/BFb0078838
  • [Busé et al. 2001] Buseetal.2001 L. Busé, M. Elkadi and B. Mourrain: Resultant over the residual of a complete intersection. J. Pure Appl. Algebra 164, No.1-2, 35-57 (2001). C. D’Andrea: Macaulay style formulas for sparse resultants. Trans. Am. Math. Soc. 354, No.7, 2595-2629 (2002). C. D’Andrea and A. Dickenstein: Explicit formulas for the multivariate resultant. J. Pure Appl. Algebra 164, No.1-2, 59-86 (2001). A. Cayley: On the theory of elimination. Cambridge and Dublin Mathematical J. 3 (1848) 116–120. S. Donkin: Rational representations of algebraic groups. Tensor products and filtration. Lecture Notes in Math. 1140. Springer, Berlin, 1985. D. Eisenbud: Linear sections of determinantal varieties. American J. Math. 110 (1988) 541–575. D. Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry. Springer Verlag, 1995. D. Eisenbud and S. Goto: Linear free resolutions and minimal multiplicity. J. Alg. 88 (1984) 89–133. [Eisenbud et al. 2001]Eisenbudetal.2001 D. Eisenbud, G. Fløystad and F.-O. Schreyer: Sheaf cohomology and free resolutions over the exterior algebra. Preprint (2001), arXiv.org/abs/math.AG/0104203. W. Fulton: Intersection Theory. Springer, New York, 1984. W. Fulton: Young tableaux : with applications to representation theory and geometry. London Mathematical Society student texts 35, Cambridge University Press 1997. [Gel’fand et al. 1994]Gelfandetal.1994 I. M. Gelfand, M. Kapranov, and A. Zelevinsky: Discriminants, resultants, and multidimensional determinants. Birkhäuser, Boston, 1994. D. Grayson and M. Stillman: Macaulay2. Available online at http://www.math.uiuc.edu/~Macaulay2/. W. Haboush: A short proof of the Kempf vanishing theorem. Invent. Math. 56, 109-112 (1980). D. Hanes: Special condition on maximal Cohen–Macaulay modules, and applications to the theory of multiplicities. Thesis, Michigan, 2000. R. Hartshorne: Algebraic Geometry. Springer Verlag, 1977. R. Hartshorne and A. Hirschowitz: Cohomology of the general instanton bundle. Ann. scient. Éc. Norm. Sup. a série 15 (1982) 365–390. G. Horrocks: Vector bundles on the punctured spectrum of a local ring. Proc. Lond. Math. Soc., III. Ser. 14, 689–713 (1964). G. Horrocks, D. Mumford: A rank $2$ vector bundle on ${\mathbf P}^4$ with 15,000 symmetries. Topology 12 (1973) 63–81. B. Iverson: Cohomology of sheaves. Springer-Verlag, New York, 1986. J. C. Jantzen: Representations of algebraic groups. Pure and Applied Mathematics, 131, Academic Press, Boston, MA, 1987. J.-P. Jouanolou: Aspects invariants de l’élimination. Adv. Math. 114 (1995) 1–174. G. R. Kempf: Linear systems on homogeneous spaces. Ann. of Math., II. Ser. 103, 557-591 (1976). A. Khetan: Determinantal Formula for the Chow Form of a Toric Surface, In ISSAC Proceedings, 145–150, ACM 2002. M. Kline: Mathematical thought from ancient to modern times. Oxford University Press, New York 1972. F. Knudsen and D. Mumford: The projectivity of the moduli space of stable curves I: Preliminaries on “det” and “div”. Math. Scand. 39 (1976) 19–55. G. W. Leibniz: Mathematische Schriften, herausgegeben von C. I. Gerhardt, Letter to l’Hospital 28 April 1693, volume II, p. 239; Vorwort volume VII, pp. 5. Georg Olms Verlag Hildesheim, New York 1971. , Ch. Okonek, M. Schneider, and H. Spindler: Vector Bundles on Complex Projective Spaces. Birkhäuser, Boston 1980. C. Peskine and L. Szpiro: Liaison des variétés algébriques I. Invent. Math. 26 (1974) 271–302. R. Stanley: Theory and application of plane partitions. Studies in Appl. Math. 1 (1971) 167–187 and 259–279. R. G. Swan: $K$-theory of quadric hypersurfaces. Ann. of Math. 122 (1985) 113–153. J. J. Sylvester: A method of determining by mere inspection the derivatives from two equations of any degree. Philosophical Magazine XVI (1840) 132–135. Memoir on the dialytic method of elimination. Part I. Philsophical Magazine XXI (1842) 534–539. Reprinted in: The collected Mathematical Papers of James Joseph Sylvester, Vol. I. Chelsea New York 1973, reprint of the Cambridge 1904 edition. A. A. Tikhomirov: The main component of the moduli space of mathematical instanton vector bundles on ${\mathbf P}^3$. Journal of the Mathematical Sciences 86 (1997) 3004–3087. (Translation from the Russian of Itogi Nauki i Tkhniki. Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematischeskie Obzory, Vol 25, Algebraic Geometry-2, 1995.) B. Ulrich: Gorenstein rings and modules with high numbers of generators. Math. Z. 188 (1984) 23–32. V. Vinnikov: Complete description of determinantal representations of smooth irreducible curves. Linear Algebra Appl. 125 (1989), 103–140. J. Weyman and A. Zelevinsky: Determinantal formulas for multigraded resultants. J. Alg. Geom. 3 (1994) 569–598.

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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
Email: eisenbud@math.berkeley.edu

Frank-Olaf Schreyer
Affiliation: Mathematik und Informatik, Geb. 27, Universität des Saarlandes, D-66123 Saarbrücken, Germany
MR Author ID: 156975
Email: schreyer@math.uni-sb.de

Jerzy Weyman
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
MR Author ID: 182230
ORCID: 0000-0003-1923-0060
Email: weyman@neu.edu

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