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§2. Irreducible Plane Algebraic Curves and Hypersurfaces 47

Exercises

1. For the finite field k of q elements determine the cardinality of the projective plane P 2 (k)
and more generally of the projective space P r (k). How many points are there on a line and
how many lines are there in P 2 (k)? Determine the number of s-dimensional subspaces in
P r (k).
2. If M i is an s i -dimensional subspace of P r (k), where k is any field, then show that the
intersection M 1 ∩ M 2 is a subspace and determine its possible dimensions.
3. Show that a projective space of dimensional n−r−1 parametrizes the (r+1)-dimensional
subspaces in P n (k) containing a fixed r-dimensional subspace M 0 .

§2. Irreducible Plane Algebraic Curves and Hypersurfaces

In (1.5) of the Introduction we considered a definition of a plane curve in the context
of the complex numbers. Now we define and study plane curves over any field k using
homogeneous polynomials f (w, x, y) ∈ k[w, x, y]. The polynomial has a degree d
and the fact that it is homogeneous of degree d can be expressed by the relation
d
f (tw, tx, ty) = t f (w, x, y).
Again we use the notation C f (K) for the locus of all w : x : y in P 2 (K) with
f (w, x, y) = 0, where K is any extension field of k.
2
Since f (w, x, y) = 0 if and only if f (w, x, y) = 0, the set of points C f (K)
does not determine the equation for any K. When f factors as f = f 1 ... f r ,the
. Similarly if f
algebraic curve can be represented as a union C f = C f 1 ∪· · · ∪ C f r
divides g, then the inclusion C f ⊂ C g holds. Since the question of whether or not
f factors depends on the field k where the coefficients of f are taken from, we will
have to speak of curves over a given field k.
(2.1) Definition. An irreducible plane algebraic curve C f of degree d defined over a
field k is given by an irreducible homogeneous polynomial f (w, x, y) ∈ k[w, x, y]
of degree d. The points C f (K) of C f in an extension field K of k consists of all
w : x : y in P 2 (K) such that f (w, x, y) = 0.
A curve of degree 1 is called a line, 2 a conic, 3 a cubic, 4 a quartic, 5 a quintic,

and 6 a sextic. For two extensions K ⊂ K of k, the inclusions P 2 (K) ⊂ P 2 (K ) and

C f (K) ⊂ C f (K ) hold. For the reader with a background in categories and functors,
it is now clear that C f is a subfunctor of the functor P 2 defined on the category of
fields over k and k-morphisms to the category of sets.
3
If u : k 3 → k is a nonsingular linear transformation with inverse v,and if
u, v : P 2 (K) → P 2 (K) are the associated projective transformations defined by
direct image on lines, then for each homogeneous f (w, x, y) ∈ k[w, x, y]ofdegree
d the composite f v is homogeneous of degree d and u(C f (K)) = C f v (K). This
follows since ( f v)(u(w, x, y)) = 0ifand onlyif f (w, x, y) = 0. Thus projective
transformations carry algebraic curves to algebraic curves and preserve degree and
the property of irreducibility. In this way we can frequently choose a convenient
coordinate system for the discussion of properties of a curve.