Cayley-Bacharach theorem

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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

P₁, ..., P₈

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'):

Cond_4,7: 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 P₉.

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 Cond_4,7 mentioned above) are already prescribed. A more compact form of the Cayley-Bacharach theorem is:

Assume that two cubics C₁ and C₂ 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.

[edit] References

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 P³), Corollary 4.5.