2
1. INTERSECTIONS OF HYPERSURFACES
1.2. Class of a curve (Plücker)
An important early application of Bezout's theorem was for the calculation of
the class of a plane curve C, i.e., the number of tangents to C through a given
general point Q:
Equivalently, the class of C is the degree of the dual curve C v . If F(x, y, z) is the
homogeneous polynomial defining C and Q = (a:b:c), then the polar curve CQ is
defined by
FQ(X, y, z) = aFx + bFy + cFZJ
where Fx = dF{x,y,z)/dX, Fyi and Fz are partial derivatives. This is defined so
that a nonsingular point Ñ of C is on CQ exactly when the tangent line to C at Ñ
(defined by XFX(P) + YFy(P) -f- ZFZ(P) = 0) passes through Q. One checks that
C meets CQ transversally at Ñ if Ñ is not a flex on C, so
class(C) = # ( C Ð CQ) = degCdegCg = n(n ~ 1),
if ç is the degree of C, and C is nonsingular.
If C has Singular points, however, they are always on C Ð CQ, SO they must
contribute. For example, if Ñ is an ordinary node (resp. cusp) and Q is general,
then
i(P,C*C
Q
) = 2 (resp.
( P , C * C Q ) = 3) .
This gives the first Plücker formula [50]
n(n - 1) = class(C) + 26 + 3«,
if C has degree ç , ä ordinary nodes, ç ordinary cusps, and no other singularities.
1,3. Degree of a dual surface (Salmon)
In 1847 Salmon [53] made a similar study of surfaces. If S C P
3
is a surface,
the degree of the dual (or "reciprocal") surface 5
V
is the number of points Ñ £ S
such that the tangent plane to 5 at Ñ contains a given general line l. (This number
is one of the projective characters of 5, now called the second class of 5.)
For a point Q £
P3,
let SQ be the polar surface of S with respect to Q: if
F(x, y, z, w) defines S and Q = (a:b:c:d), then aFx 4- bFy + cFz 4- dPu, defines SQ.
Taking two points Q\, Q2 on , one sees as before that a nonsingular point Ñ of 5
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