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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