Cayley-Bacharach theorems and conjectures
Authors:
David Eisenbud, Mark Green and Joe Harris
Journal:
Bull. Amer. Math. Soc. 33 (1996), 295-324
MSC (1991):
Primary 14N05, 14H05, 14-02; Secondary 13-03, 13H10
DOI:
https://doi.org/10.1090/S0273-0979-96-00666-0
MathSciNet review:
1376653
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: A theorem of Pappus of Alexandria, proved in the fourth century A.D., began a long development in algebraic geometry. In its changing expressions one can see reflected the changing concerns of the field, from synthetic geometry to projective plane curves to Riemann surfaces to the modern development of schemes and duality. We survey this development historically and use it to motivate a brief treatment of a part of duality theory. We then explain one of the modern developments arising from it, a series of conjectures about the linear conditions imposed by a set of points in projective space on the forms that vanish on them. We give a proof of the conjectures in a new special case.
- I. Bacharach, Uber den Cayley’schen Schnittpunktsatz, Math. Ann. 26 (1886), 275–299.
- Carl B. Boyer, A history of mathematics, 2nd ed., John Wiley & Sons, Inc., New York, 1991. With a foreword by Isaac Asimov; Revised and with a preface by Uta C. Merzbach. MR 1094813
- A. Brill and M. Noether, Uber die algebraischen Functionen und ihre Anwendung in der Geometrie, Math. Ann. 7 (1874), 269–310.
- H. S. M. Coxeter, Projective geometry, 2nd ed., University of Toronto Press, Toronto, Ont., 1974. MR 0346652
- A. Cayley, On the intersection of curves, Cambridge Math. J. 3 (1843), 211–213; Collected math papers I, vols. 25–27, Cambridge Univ. Press, Cambridge, 1889.
- M. Chasles, Traité des sections coniques, Gauthier-Villars, Paris, 1885.
- E. D. Davis, A. V. Geramita, and F. Orecchia, Gorenstein algebras and the Cayley-Bacharach theorem, Proc. Amer. Math. Soc. 93 (1985), no. 4, 593–597. MR 776185, DOI https://doi.org/10.1090/S0002-9939-1985-0776185-6
- D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Springer-Verlag, New York, 1994.
- Joe Harris, Curves in projective space, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 85, Presses de l’Université de Montréal, Montreal, Que., 1982. With the collaboration of David Eisenbud. MR 685427
- A. Verschoren, Pour une géometrie algébrique noncommutative, Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 33rd Year (Paris, 1980) Lecture Notes in Math., vol. 867, Springer, Berlin-New York, 1981, pp. 319–350 (French). MR 633525
- David Eisenbud, Mark Green, and Joe Harris, Higher Castelnuovo theory, Astérisque 218 (1993), 187–202. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). MR 1265314
- ---, Hilbert functions and complete intersections (in preparation).
- Ph. Ellia and Ch. Peskine, Groupes de points de ${\mathbf P}^2$: caractère et position uniforme, Algebraic geometry (L’Aquila, 1988) Lecture Notes in Math., vol. 1417, Springer, Berlin, 1990, pp. 111–116 (French). MR 1040554, DOI https://doi.org/10.1007/BFb0083336
- Anthony V. Geramita, Martin Kreuzer, and Lorenzo Robbiano, Cayley-Bacharach schemes and their canonical modules, Trans. Amer. Math. Soc. 339 (1993), no. 1, 163–189. MR 1102886, DOI https://doi.org/10.1090/S0002-9947-1993-1102886-5
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Morris Kline, Mathematical thought from ancient to modern times, Oxford University Press, New York, 1972. MR 0472307
- F. S. Macaulay, Algebraic theory of modular systems, Cambridge Tracts in Math., vol. 19, Cambridge Univ. Press, Cambridge, 1916.
- M. Nagata, The theory of multiplicity in general local rings, Proc. Internat. Sympos. (Tokyo-Nikko, 1955), Sci. Council of Japan, Tokyo, 1956, pp. 191–226.
- M. Noether, Uber ein Satz aus der Theorie der algebraischen Funktionen, Math. Ann. 6 (1873), 351–359.
- D. J. Struik (ed.), A source book in mathematics, 1200–1800, Harvard University Press, Cambridge, Mass., 1969. MR 0238647
Retrieve articles in Bulletin of the American Mathematical Society with MSC (1991): 14N05, 14H05, 14-02, 13-03, 13H10
Retrieve articles in all journals with MSC (1991): 14N05, 14H05, 14-02, 13-03, 13H10
Additional Information
David Eisenbud
Affiliation:
Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254-9110
MR Author ID:
62330
ORCID:
0000-0002-5418-5579
Email:
eisenbud@math.brandeis.edu
Mark Green
Affiliation:
Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1555
MR Author ID:
76530
Email:
mlg@math.ucla.edu
Joe Harris
Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138-2901
Email:
harris@abel.math.harvard.edu
Received by editor(s):
March 24, 1995
Received by editor(s) in revised form:
November 3, 1995
Article copyright:
© Copyright 1996
American Mathematical Society