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§4. Multiple or Singular Points 53

The leading form is well defined up to a projective change of variables coming
from the subgroup of the projective linear leaving one point fixed.
(4.2) Definition. A point on a hypersurface is called simple or nonsingular provided
its order is 1 and is called multiple or singular provided the order is strictly greater
than 1.

For example all points on a multiple component, defined by a power of an ir-
reducible polynomial, are singular with order equal to an integral multiple of the
multiplicity of the multiple component. If H is a hypersurface of degree d consisting
of the union of d hyperplanes H 1 ∪· · · ∪ H d , then P has order r on H if and only
if P is on exactly r of the hyperplanes H i for i = 1,... , d. Thus every point on
H i ∩ H j for i = j is singular.

(4.3) Definition. Let P be a point of order r on a plane algebraic curve C f (k) of
degree m where k is algebraically closed. The tangent cone to C f at P is the union
of the lines whose equations are the linear factors of the leading form of the point P.
If P is a nonsingular point, then the tangent cone reduces to a single line called the
tangent line.

If P is transformed by a projective tranformation to 1 : 0 : 0, then the leading

form is f r (x, y) = (a i x + b i y) where the equation of the curve is
1≤i≤r
m−r
f (w, x, y) = w f r (x, y) +· · · + f m (x, y).
For f (w, x, y) a homogeneous polynomial of degree m, the following formula,
called Euler’s formula, holds:

∂ f ∂ f ∂ f
mf (w, x, y) = w + x + y .
∂w ∂x ∂y
The reader can easily check that for a simple point (w 0 , x 0 , y 0 ) on C f (k) the tangent
line is of the form
∂ f ∂ f ∂ f
(w 0 , x 0 , y 0 )w + (w 0 , x 0 , y 0 )x + (w 0 , x 0 , y 0 )y = 0.
∂w ∂x ∂y
Now we relate the order of a point to a special case of intersection multiplicities.

The line L determined by two distinct points y 0 : ··· : y n and y : ··· : y in
0 n
P n (k), can be parametrized by the function which assigns to each s : t ∈ P 1 (k)

the point (sy 0 + ty ) : ··· : (sy n + ty ) in P n (k). The function which assigns to
0 n

each t ∈ k the point (y 0 + ty ) : ··· : (y n + ty ) parametrizes the affine line

0 n
aff
L −{y : ··· : y }= L .

0 n
If H f is a hypersurface of degree d, then the intersection set of L aff ∩ H f consists
of points P t , where t is a root of the polynomial equation 0 = ϕ(t) = f (y 0 +

ty ,... , y n + ty ). Observe that y 0 : ··· : y n ∈ H f (k) if and only if ϕ(0) = 0, and
0 n
L ⊂ H f if and only if ϕ is not identically zero.