ABSTRACT
Given nine generic points in CP 2, there exists one single cubic passing through them. Given eight generic points in CP 2, there exists a one-parameter family of cubics passing through them. We will call such a family a pencil of cubics. Let F0 and F1 be two cubics of a pencil P. They intersect at a ninth point. As the other cubics of P are linear combinations of F0 and F1, they all pass through this ninth point. We call these nine points the base points of P. If eight of the base points are real, the pencil is real, and hence the ninth base point is also real. A pencil of cubics is a line in the space CP 9 of complex cubics. Let ∆ be the discriminantal hypersurface of CP 9, formed by the singular cubics; this hypersurface is of degree 12. Hence, a generic pencil of cubics intersects ∆ transversally at 12 regular points. Otherwise stated, a generic pencil P has exactly 12 singular (nodal) cubics. A non-generic pencil will be called singular pencil . Let P be a real pencil with nine real base points and denote the real part of P by RP. Recall that a circle embedded in RP 2 is called oval (resp. pseudo-line or odd component) if it realizes the class 0 (resp. 1) of H1(RP 2). The real part of a generic real cubic consists either of one pseudo-line, or of one pseudo-line plus one oval. Let n ≤ 12 be the number of real singular cubics of P. Let C3 be one of these cubics. The node (double point) P of C3 is isolated if the tangents to C3 at P are non-real, otherwise P is non-isolated . If P is non-isolated, C3 \ P = J ∪ O, where [J ∪ P ] 6= 0 and [O ∪ P ] = 0 in H1(RP 2). We say that O is the loop and J is the odd component of C3. Notice that the loop O is convex. The estimation of n presented hereafter is due to V. Kharlamov, see [11].
