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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