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 elliptic curve point addition. Let a first cubic contain the three lines BC, O(A+B) and A(B+C); and a second cubic containing the three lines AB, O(B+C) and C(A+B). The following eight points are common to both cubics: A, B, C, A+B, -A-B, B+C, -B-C, O. Hence their ninth points must be the same -A-(B+C)=-(A+B)-C, giving the associativity. 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 ), 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