Real rational cubics through 8 points in R2 d'Après V. Kharlamov Frank Sottile 26 April 2000 Madison, WI

A cubic in P² is the set of common zeroes of a cubic form

Ax3 + By3 + Cz3 + Dx2y + Ex2z + Fy2x + Gy2z + Hz2x + Iz2y + Jxyz  .

The equation is determined by its zero set only up to multiplication by a non-zero scalar. Since there are 10 coefficients, the space of plane cubics is a 9-dimensional projective space, P⁹. A cubic in P² is rational if and only if it is singular. Thus the space of rational cubics is the hypersurface in P⁹ given by the discriminant of this cubic form. Here, we use geometry and topology to compute the degree of this discriminant, and more interestingly, show a result concerning its real points. Thus while the results are stated in a purely geometric form, we may interpret them in terms of real solutions to an interesting equation.

The condition for a cubic to contain a point is a linear condition on the coefficients A,...,J. In this way, we see that the space of cubics that meet 8 points is a projective line. If p and q are two different cubics that meet 8 given points, then the set of cubics that meet those 8 points is given by all linear combinations sp+tq of p and q. Since p and q meet in a 9th point, it follows that every cubic in this family also meets that same 9th point.

Since every other point in P² contains a unique cubic in this family, we see that there is a map from (P² - 9pts) to the P¹ of cubics sp+tq. If we blow up the 9 points in P², we obtain a variety Z, and the map f from Z to P¹ is our family of cubics. (The fibre over a point sp+tq of P¹ is the plane cubic defined by the polynomial sp+tq.)

The Complex Case.   We first compute the Euler characteristic of Z. Since blowing up a point in a surface amounts to removing that point and replacing it with a P¹, each blow-up adds 1=2-1 to the Euler characteristic of the surface. (The point has Euler characteristic 1, P¹ has Euler characteristic 2, and P² has Euler characteristic 3.) In this way, we see that the Euler characteristic of Z is 12.

Consider the fibres of this map f. A general plane cubic is smooth and is homeomorphic to (S¹)², it has Euler characteristic 0. Thus the only contributions to the Euler characteristic of Z comes from singular fibres, that is rational plane cubics. If the 8 points are in general position, then we may assume that the family Z does not contain any cuspidal cubics, and so its singular fibres must all be nodal cubics. A nodal cubic is obtained from the 2-sphere by identifying 2 points (or contracting a simple S¹ in the torus (S¹)²). Thus the Euler characteristic of a nodal cubic is 1.

From this consideration, we see that there are exactly 12 singular fibres, and each is a nodal rational plane cubic containing the 8 points.

The Real Case.   Suppose the 8 points are real. Then so is the 9th point, and we consider the real points X_R of a variety X. As before, we compute the Euler characteristic of Z_R. When a point on a real surface is blown up, we replace it by P¹_R=S¹. This lowers the Euler characteristic by 1, as S¹ has Euler characteristic zero. Since P²_R has Euler characteristic 1, we see that Z_R has Euler characteristic 1-9=-8.

A non-singular real cubic has either 1 or 2 disjoint components, each homeomorphic to S¹, and so as in the complex case, the nonsingular fibres of Z_R do not contribute to its Euler characteristic. Unlike the complex case, there are 2 different real nodal cubics, distinguished by the tangent lines to the branches at the node---either both tangent directions are real, or else they are complex conjugate. We display both below, the cubic on the left has 2 real tangent directions and the one on the right has complex conjugate directions.
Call these cubics real nodal and complex nodal, respectively. The real nodal cubic is topologically a figure 8, and so has Euler characteristic -1 and the complex nodal cubic is the disjoint union of a point and S¹, and so has Euler characteristic 1.

Thus we see that the number of real nodal cubics minus the number of complex nodal cubics equals 8. We enumerate the different possibilities:

# real nodal	8	9	10
# complex nodal	0	1	2
# real rational cubics	8	10	12

Thus to see that there are 8 points in the plane such that all 12 rational cubics that meet them are real, we need only exhibit 2 complex nodal cubics that meet in 9 points: If we choose any 8 of those points, then the family of cubics meeting those 8 points contains also 10 real nodal cubics. We display 2 such complex nodal cubics below.

We close with the following observation that in this problem of cubics, not all possible numbers of real cubics (0, 2, 4, 6, ....) can occur. In terms of equations, this says that the discriminant of a pencil of cubics (a polynomial in one variable) has degree 12. If the pencil is real and has 9 base points, then it has either 8, 10, or 12 real zeroes. Geometrically, it says that certain (easily described) real lines in P⁹ meet the discriminant hypersurface in at least 8 points.