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## 3.iv. Real rational cubics through 8 points in P2R

There are 12 singular (rational) cubic curves containing 8 general points in the plane.

Theorem 3.6 (Degtyarev and Kharlamov [DK, Proposition 4.7.3]) Given 8 general points in P2R, there are at least 8 singular real cubics containing them, and there are choices of the 8 points for which all 12 singular cubics are real.

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

f   :   Z   --->   P1 ,

where f-1([a, b]) is the cubic curve defined by the polynomial aP + bQ .

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

The real nodal curve is homeomorphic to a figure 8 and has Euler characteristic -1, while the complex nodal curve is the union of a S1 with a solitary point and so has Euler characteristic 1.

Among the singular fibres, we have

 -8 = #{complex nodal} - #{real nodal},     with 12 = #{complex nodal} + #{real nodal},

Thus there are at least 8 real nodal curves containing 8 general points in P2R. The pencil of cubics containing the 2 complex nodal cubics of Figure 8 has 10 real nodal cubics. Thus there are 12 real rational cubics containing any 8 of the 12 points in Figure 8.

 (y-28)2 = 4x3-85x2 +504x-18yx (x-28)2 = 4y3-85y2 +504y-18xy Figure 8: Complex nodal curves meeting in 9 points

Remark 3.7   This classical problem of 12 plane cubics containing 8 points generalizes to the problem of enumerating rational plane curves of degree d containing 3d - 1 points. Let Nd be the number of such curves, which satisfies the recursion [FP]

The values N1 = N2 = 1 are trivially fully real, and we have just seen that N3 = 12 is fully real. The next case of N4 = 620 (computed by Zeuthen [Ze1]) seems quite challenging.

Remark 3.8   The most interesting feature of Theorem 3.6 is the existence of a lower bound on the number of real solutions, which is a new phenomenon. In Section 6 we shall see evidence that this may be a pervasive feature of this field.

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