Proof. Since a cubic equation in the plane has 10 coefficients, the space of cubics is identified with P9. The condition for a plane cubic to contain a given point is linear in these coefficients. Given 8 general points, these linear equations are independent and so there is a pencil (P1) of cubics containing 8 general points in P2.
Two cubics P and Q in this pencil meet
transversally in 9 points.
Since curves in the pencil are given by
aP + bQ for [a, b] in P1,
any two curves in the pencil meet transversally in these 9 points.
Let Z be P2 blown up at these same 9 points.
We have a map
Consider the Euler characteristic of Z first over C and then over R. Blowing up a smooth point on a surface replaces it with a P1C and thus increases the Euler characteristic by 1. Since P1C has Euler characteristic 3, we see that Z has Euler characteristic 3 + 9 = 12. The general fibre of f is a smooth plane cubic which is homeomorphic to the 2-torus (S1 x S1), and so has Euler characteristic 0. Thus only the singular fibres of f contribute to the Euler characteristic of Z. Assume that the 8 points are in general position so there are only nodal cubics in the pencil. A nodal cubic has Euler characteristic 1. Thus there are 12 singular fibres of f and hence 12 singular cubics meeting 8 general points in P2C.
Consider now the Euler characteristic of ZR. Blowing up a smooth point on a real surface replaces the point by P1R = S1, and hence decreases the Euler characteristic by 1. Since P2R has Euler characteristic 1, the Euler characteristic of ZR is 1 - 9 = -8. A nonsingular real cubic is homeomorphic to either one or two disjoint copies of S1, and hence has Euler characteristic 0. Again the Euler characteristic of ZR is carried by its singular fibres. There are two nodal real cubics; either the node has two real branches or two complex conjugate branches so that the singular point is isolated. Call these curves real nodal and complex nodal, respectively. They are displayed in Figure 7.
Figure 7: A real nodal and a complex nodal curve |
Among the singular fibres, we have
-8 | = | #{complex nodal} - #{real nodal}, with | 12 | = | #{complex nodal} + #{real nodal}, |
(y-28)2 | = | 4x3-85x2 +504x-18yx | ||
(x-28)2 | = | 4y3-85y2 +504y-18xy | ||
Figure 8: Complex nodal curves meeting in 9 points |