In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane P2. The original form states:
Assume that two cubics C1 and C2 in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field. Then every cubic that passes through any eight of the points also passes through the ninth point.
A more intrinsic form of the Cayley–Bacharach theorem reads as follows:
Every cubic curve C over an algebraically closed field that passes through a given set of eight points P1, ..., P8 also passes through (counting multiplicities) a ninth point P9 which depends only on P1, ..., P8.
A related result on conics was first proved by the French geometer Michel Chasles and later generalized to cubics by Arthur Cayley and Isaak Bacharach.
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Details
If seven of the points P1, ..., P8 lie on a conic, then the ninth point can be chosen on that conic, since C will always contain the whole conic on account of Bézout's theorem. In other cases, we have the following.
If no seven points out of P1, ..., P8 are co-conic, then the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) P1, ..., P8 (with multiplicity for double points) has dimension two.
In that case, every cubic through P1, ..., P8 also passes through the intersection of any two different cubics through P1, ..., P8, which has at least nine points (over the algebraic closure) on account of Bézout's theorem. These points cannot be covered by P1, ..., P8 only, which gives us P9.
Since degenerate conics are a union of at most two lines, there are always four out of seven points on a degenerate conic that are collinear. Consequently:
If no seven points out of P1, ..., P8 lie on a non-degenerate conic, and no four points out of P1, ..., P8 lie on a line, then the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) P1, ..., P8 has dimension two.
On the other hand, assume P1, P2, P3, P4 are collinear and no seven points out of P1, ..., P8 are co-conic. Then no five points of P1, ..., P8 and no three points of P5, P6, P7, P8 are collinear. Since C will always contain the whole line through P1, P2, P3, P4 on account of Bézout's theorem, the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) P1, ..., P8 is isomorphic to the vector space of quadratic homogeneous polynomials that vanish (the affine cones of) P5, P6, P7, P8, which has dimension two.
Although the sets of conditions for both dimension two results are different, they are both strictly weaker than full general positions: three points are allowed to be collinear, and six points are allowed to lie on a conic (in general two points determine a line and five points determine a conic). For the Cayley–Bacharach theorem, it is necessary to have a family of cubics passing through the nine points, rather than a single one.
According to Bézout's theorem, two different cubic curves over an algebraically closed field which have no common irreducible component meet in exactly 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 (without seven co-conic ones) are already prescribed.
Applications
A special case is Pascal's theorem, in which case the two cubics in question are all degenerate: given six points on a conic (a hexagon), consider the lines obtained by extending opposite sides – this yields two cubics of three lines each, which intersect in 9 points – the 6 points on the conic, and 3 others. These 3 additional points lie on a line, as the conic plus the line through any two of the points is a cubic passing through 8 of the points.
A second application is Pappus's hexagon theorem, similar to the above, but the six points are on two lines instead of on a conic.
Finally, a third case is found for proving the associativity of the group law for elliptic curves. Consider two cubics containing the following lines:
cubic 1:
b
c
,
o
(
a
+
b
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,
a
(
b
+
c
)
cubic 2:
a
b
,
o
(
Dimension counting
One can understand the Cayley–Bacharach theorem, and why it arises for degree 3, by dimension counting. Simply stated, nine points determine a cubic, but in general define a unique cubic. Thus if the nine points lie on more than one cubic, equivalently on the intersection of two cubics (as 3 × 3 = 9), they are not in general position – they are overdetermined by one dimension – and thus cubics passing through them satisfying one additional constraint, as reflected in the "eight implies nine" property. The general phenomenon is called superabundance; see Riemann–Roch theorem for surfaces.
Details
Formally, first recall that given two curves of degree d, they define a pencil (one-parameter linear system) of degree d curves by taking projective linear combinations of the defining equations; this corresponds to two points determining a projective line in the parameter space of curves, which is simply projective space.
The Cayley–Bacharach theorem arises for high degree because the number of intersection points of two curves of degree d, namely d 2 (by Bézout's theorem), grows faster than the number of points needed to define a curve of degree d, which is given by
(
d
+
1
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(
d
+
2
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2
−
1
=
d
2
+
3
d
2
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{\displaystyle {\frac {(d+1)(d+2)}{2}}-1={\frac {d^{2}+3d}{2}}.}
These first agree for d = 3, which is why the Cayley–Bacharach theorem occurs for cubics, and for higher degree d 2 is greater, hence the higher degree generalizations.
In detail, the number of points required to determine a curve of degree d is the number of monomials of degree d, minus 1 from projectivization. For the first few d these yield: