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54 2. Plane Algebraic Curves

(4.4) Definition. With these notations, the intersection multiplicity of the line L
with the hypersurface H f at P = y 0 : ··· : y n , denoted i(P; L, H f ), is the or-
derofthe zeroof ϕ(t) at t = 0. It is defined only when L ⊂ H f .

Now we can relate intersection multiplicity at P with the order of P on a hyper-
surface H f .

(4.5) Proposition. Let k be an infinite field. A point P = y 0 : · : y n is of order r on
H f (k) if and only if i(P; L, H f ) ≥ r for all lines L through P not contained in H f
and i(P; L, H f ) = r for some line L through P.

Proof. After a projective change of coordinates, we can assume that y 0 : ··· : y n =

1:0:···:0, and, without changing the line L, we can assume that the point y :···: y
0 n

has the property that y = 0, that is, the point lies on the hyperplane at infinity. By
0
definition of the order of 1:0:···:0 on H f , the polynomial f (1, x 1 ,... , x n ) has the
form
f (1, x 1 ,... , x n ) = f r (x 1 ,... , x n ) + ··· + f d (x 1 ,... , x n ),
where f 1 (x 1 ,... , x n ) is homogeneous of degree i and f r = 0. The polynomial ∂(t)
associated with f and the two points has the form

ϕ(t) = f r (ty ,... , ty ) + ··· + f d (ty ,... , ty )

1 n 1 n
r
d

= t f r (y ,... , y ) +· · · + t f d (y ,... , y )
1 n 1 n
and ϕ(t) = 0has t = 0 as a root of order ≥ r.For (y ,... , y ) with f r (y ,... , y )

1 n 1 n
= 0, such a point exists since k is infinite, the root t = 0of ϕ(t) has multiplicity
equal to r. This proves the proposition.
For the case n = 2, that of a place curve f (w, x, y) = 0, the above relations for
L at the points 1 : 0 : 0 and 0 : a : b take the form
f (1, x, y) = f r (x, y) +· · · + f d (x, y)
and
ϕ(t) = f r (ta, tb) +· · · + f d (ta, tb)
r
d
= t f r (a, b) + ··· + t f d (a, b).
The condition f r (a, b) = 0 is equivalent to the condition that bx − ay divides
f r (x, y), that is, the line L through 1 : 0 : 0 and 0 : a : b given by the equation bx −
ay = 0 is part of the tangent cone. This leads to the following result extending (4.5).
(4.6) Proposition. Let k be algebraically closed. The point P = w : x : yis of
order r on the plane curve C f if and only if i(P; L, C j ) ≥ r for all lines L through
P with L ⊂ C f , and i(P; L, C f ) = r for some line through P. The line L through
P is part of the tangent cone if and only if L ⊂ C f or i(P; L, C f )> r.