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