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