Cayley-Bacharach theorem

Cayley-Bacharach theorem

In mathematics, the Cayley-Bacharach theorem is a statement in projective geometry which contains as a special case Pascal's theorem. The Cayley-Bacharach theorem pertains to the family of cubic curves (plane curves of degree three) passing through eight given points

:"P"1, ..., "P"8

that lie "in general position" in the projective plane mathbb{P}^2 (k) over a field "k". It was first proved by the French geometer Michel Chasles and later generalized (to curves of higher degree) by Arthur Cayley and Bacharach.

Assume that the set of eight points satisfies the following condition (of 'general position'):: Cond4,7: no four of the points are collinear and no seven of the points lie on a conic.

Then one version of the Cayley-Bacharach theorem reads as follows:

: "every cubic curve "C" which passes through the given eight points also passes through a certain (fixed) ninth point "P"9".

According to Bézout's theorem two different cubic curves which have no common irreducible component meet in nine points (counted with multiplicity), the Cayley-Bacharach theorem thus asserts that the last point of intersection of any two members in the family of curves does not move if eight intersection points (satisfying the condition Cond4,7 mentioned above) are already prescribed. A more compact form of the Cayley-Bacharach theorem is:

: "Assume that two cubics "C"1 and "C"2 in the projective plane meet in nine (different) points. Then every cubic which passes through any eight of the points also passes through the ninth point."

Note: Because the vector space of homogeneous polynomials "P"("x","y","z") of degree three in three variables "x","y","z" has dimension 10, the system of cubic curves passing through eight (different) points is parametrized by a vector space of dimension ≥ 2 (the vanishing of the polynomial at one point imposes a single linear condition). It can be shown that the dimension is "exactly" two if no four of the points are collinear and no seven points lie on a conic. The Cayley-Bacharach theorem can be e.g. deduced from this fact (see for example the book by Hartshorne).

The general phenomenon is called "superabundance".


# M. Chasles, "Traité des sections coniques", Gauthier-Villars, Paris, 1885.
# I. Bacharach, "Über den Cayley’schen Schnittpunktsatz", Math. Ann. 26 (1886) 275-299.
# A. Cayley, "On the Intersection of Curves" (published by Cambridge University Press, Cambridge, 1889).
# E. D. Davis, A.V. Geramita, and F. Orecchia, "Gorenstein algebras and Cayley-Bacharach theorem", Proceedings Amer. Math. Soc. 93 (1985) 593 - 597.
# D. Eisenbud, M. Green, and J. Harris, "Cayley-Bacharach theorems and conjectures", Bull. Amer. Math. Soc. 33 (1996) 295—324.
# Robin Hartshorne, "Algebraic geometry", chapter 5, section 4 (The cubic surface in "P"3), Corollary 4.5.

External links

Survey articles on the Cayley-Bacharach theorem and related topics (the second article is an online-version of reference [5] above):

*Gabriel Katz: [ Curves in cages: an algebro-geometric zoo]
*D. Eisenbud, M. Green and J. Harris: [ Cayley-Bacharach theorems and conjectures ]

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Bacharach (disambiguation) — Bacharach (or Bachrach) is a German surname and may refer to: Places * Bacharach, a town in Western Germany People * Arthur J. Bachrach (born 1926), American psychologist * Burt Bacharach, 20th century composer * Harry Bacharach, mayor of… …   Wikipedia

  • De Franchis theorem — In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1. The simplest is that the automorphism… …   Wikipedia

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

  • Cubic plane curve — A selection of cubic curves. See information page for details. Cubic curve redirects here. For information on polynomial functions of degree 3, see Cubic function. In mathematics, a cubic plane curve is a plane algebraic curve C defined by a… …   Wikipedia

  • Riemann surface — For the Riemann surface of a subring of a field, see Zariski–Riemann space. Riemann surface for the function ƒ(z) = √z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real… …   Wikipedia

  • Plane curve — In mathematics, a plane curve is a curve in a Euclidean plane (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. A smooth plane curve is a curve in a …   Wikipedia

  • Moduli of algebraic curves — In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on… …   Wikipedia

  • Genus–degree formula — In classical algebraic geometry, the genus–degree formula relates the degree d of a non singular plane curve with its arithmetic genus g via the formula: A singularity of order r decreases the genus by .[1] Proofs The proof follows immediately… …   Wikipedia

  • Linear system of divisors — A linear system of divisors algebraicizes the classic geometric notion of a family of curves, as in the Apollonian circles. In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of… …   Wikipedia

  • Теорема о 9 точках на кубике — Иллюстрация к теореме о 9 точках Теорема о 9 точках на кубике  теорема аналитической геометрии, которая гласит, что[1] …   Википедия