**Proof. **
Since a cubic equation in the plane has 10 coefficients,
the space of cubics is identified with **P**^{9}.
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 (**P**^{1}) of cubics containing 8 general points in
**P**^{2}.

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 P^{1},
any two curves in the pencil meet transversally in these 9 points.
Let Z be P^{2} blown up at these same 9 points.
We have a map
*

where

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
**P**^{1}_{C} and thus increases the Euler
characteristic by 1.
Since **P**^{1}_{C} 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 (*S*^{1} x *S*^{1}), 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 **P**^{2}_{C}.

Consider now the Euler characteristic of *Z*_{R}.
Blowing up a smooth point on a real surface replaces the point by
**P**^{1}_{R} = *S*^{1}, and hence
decreases the Euler characteristic by 1.
Since **P**^{2}_{R} has Euler characteristic
1, the Euler characteristic of *Z*_{R} is 1 - 9 = -8.
A nonsingular real cubic is homeomorphic to either one or two disjoint copies
of *S*^{1}, and hence has Euler characteristic 0.
Again the Euler characteristic of *Z*_{R} 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} |
= | 4x^{3}-85x^{2}
+504x-18yx |
||

(x-28)^{2} |
= | 4y^{3}-85y^{2}
+504y-18xy | ||

Figure 8:
Complex nodal curves meeting in 9 points |

The values