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46 2. Plane Algebraic Curves

(1.1) Definition. The r-dimensional projective space P r (k) over a field k consists of
equivalence classes of (r + 1)-tuples y 0 : ··· : y r , where the y i are not all zero. The

equivalence relation is defined by y 0 : ··· : y r = y : ··· : y provided there is a
0 r

nonzero constant a with y = ay i for i = 0,... ,r.
i
If k is a subfield of K, then there is an obvious inclusion P r (k) ⊂ P r (K).
(1.2) Definition. A hyperplane in P r (k) is the set of all y 0 : ··· : y r in P r (k) satis-
fying an equation
a 0 y 0 + ··· + a r y r = 0,

where not all a i are zero.

Two equations a 0 y 0 +· · · + a r y r = 0and a y 0 +· · · + a y r = 0 determine the

0 r

same hyperplane if and only if there is a nonzero b with a = ba i for all i = 0,... ,r.
i
Hence the set of hyperplanes form a projective space where the point a 0 : ··· : a r
corresponds to the hyperplane given by the equation a 0 y 0 +· · · + a r y r = 0.
(1.3) Remark. Let H i denote the hyperplane y i = 0 for i = 0,... ,r. The subset
r
r
P r (k) − H i can be parametrized as k where (x 1 ,... , x r ) in k corresponds to the
point x 1 : ··· : x i :1: x i+1 : ··· : x r in P r (k) − H i .If y i = 0in y 0 : ··· : y r or
y 0 : ··· : y r ∈ P r (k) − H i , then it corresponds to (y 0 /y i ,... , y i−1 /y i , y i+1 /y i ... ,
n
y r /y i ) in k . Observe that
P r (k) = (P r (k) − H 0 ) ∪ ··· ∪ (P r (i) − H r ) .
We speak of H 0 as the hyperplane at infinity.
There is a coordinate-free version of projective space where we assign to any
finite-dimensional vector space V of dimensional r + 1the r-dimensional projective
space P(V ) of all one-dimensional subspaces P ⊂ V .An s-dimensional linear sub-
space M ⊂ P(V ) is determined by M ,an (s +1)-dimensional subspace of V where
+
+
for P ∈ P(V ) we have P ∈ M provided P ⊂ M .
A point P ∈ P(V ) is represented P = kv where v ∈ P is nonzero vector in V
defined over the field k. Properties of sets of nonzero vectors in V can be transferred
to those of sets of points in P(V ).
(1.4) Definition. Let be a set of points in P(V ). Let be the set of representa-

tives of nonzero v ∈ V with kv ∈ . The set is a general position provided any

subset ⊂ with # ≤ dim(V ) is a linearly independent set.
Observe that for points P 0 ... , P m in P(V ) with m ≤ r, the following conditions
are equivalent. We say they define the property of being in general position:
(1) No s + 1 of the points P 0 ,... , P m lie on an (s − 1)-dimensional plane M in
P(V ) for each s ≤ m.
(2) For nonzero vectors v i ∈ P i the set of vectors v 0 ... ,v m in V is linearly inde-
pendent.