4 1. INTERSECTIONS OF HYPERSURFACES i pinch point If C is the line ÷ = y = 0, and S is the surface Ux2 + Vxy + VFt/2 = 0, then these pinch points are the interseetions of C with the surface V2 = 4UW, so there are 2n 4 pinch points on S. Thus C, together with its pinch points, diminishes the degree of Sv by (5n 8) -l· (2n 4) = 7n 12. For example, a cubic with a double line (e.g. y2 = zx2 + x3) has a dual surface of degree three. Salmon also considers more general curves. If C is a complete intersection of surfaces of degrees á and 6, and C is a component of intersection of three surfaces of degrees m, n, and p, then he finds that the contribution of C to the total number of mnp is ab(m + ç -f ñ (a + b)). Concluding this remarkable paper, he deduces that if such C is an r-fold curve on a surface S, then it diminishes the degree of the dual by ab[(r - l)(3r + l)n - r2(r - l)(a -f ö) - 2r(r - 1)]. 1.4. The problem of five conics Problems of excess intersection arise frequently in enumerative problems. The famous problem of the number of plane conics tangent to five given conics in gen- eral position is a typical example of this. Á plane conic is defined by a quadratic polynomial ax2 + by2 -f cxy + dx + dy + f, unique up to multiplication by a nonzero scalar, so the space of conics can be identified with P5. One checks that the con- dition to be tangent to a fixed nonsingular conic is described by a hypersurface of degree six in P5. The desired conics are then represented by the points in the intersection of five such hypersurfaces H\ Ð · · · Ð H§. There are not 65 = 7776 such conics, however, as originally thought by Steiner and others. Indeed, the Veronese surface V = P2 of conics which are double lines is contained in f)Hi, and one can show (cf. §4 below) that the contribution of V to the intersection is actually 4512, which leaves 3264, the actual number of (nonsingular) conics tangent to five given conics in general position. Note that the conics tangent to a fixed line form a quadric hypersurface in P6. Given five general lines, the Veronese contributes 31 to the predicted intersection
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