that given two curves of degree , they define a pencil (one-parameter linear system) of degree 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 , namely (by Bézout's theorem), grows faster than the number of points needed to define a curve of degree , which is given by These first agree for , which is why the Cayley–Bacharach theorem occurs for cubics, and for higher degree is greater, hence the higher degree generalizations. In detail, the number of points required to determine a curve of degree is the number of monomials of degree , minus 1 from projectivization. For the first few these yield: 2 and 1: two points determine a line, two lines intersect in a point, 5 and 4: five points determine a conic, two conics intersect in four points, 9 and 9: nine points determine a cubic, two cubics intersect in nine points, 14 and 16. Thus these first agree for 3, and the number of intersections is larger when . The meaning of this is that the 9 points of intersection of two cubics are in special position with respect to cubics, a fortiori for higher degree, but unlike for lower degree: two lines intersect in a point, which is trivially in general linear position, and two quadratics intersect in four points, which (assuming the quadratics are irreducible so no three points are collinear) are in general quadratic position because five points determine a quadratic, and any four points (in general linear position) have a pencil of quadratics through them, since the system is underdetermined. For cubics, nine points determine a cubic, but in general they determine a unique cubic – thus having two different cubics pass through them (and thus a pencil) is special – the solution space is one dimension higher than expected, and thus the solutions satisfy an additional constraint, namely the "8 implies 9" property. More concretely, because the vector space of homogeneous polynomials of degree three in three variables has dimension , the system of cubic curves passing through eight (different) points is parametrized by a vector space of dimension (the vanishing of the polynomial at one point imposes a single