Graphics Reference
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its algebraic closure, the complex numbers
C
, respectively. On a practical level we are
of course most interested in the field
R
and subsets of
R
n
.
Definition.
Let k be any field and let f(X
1
,X
2
,...,X
n
), f
i
(X
1
,X
2
,...,X
n
) Πk[X
1
,X
2
,
...,X
n
]. A set of points in k
n
of the form
{
}
()
=Œ
n
()
=
Vf
c
k
f
c
0
is called a
hypersurface
in k
n
. If n = 2, then we call V(f) an
algebraic plane curve
(often
abbreviated to
plane curve
in this chapter). A set of points in k
n
of the form
{
}
(
{}
)
=Œ
n
()
=
V
f
c
k
f
c
0
for all i
i
i
is called an
affine algebraic variety
, or simply
variety
, in k
n
defined by the set of
polynomials {f
i
}. If the field k has to be made explicit, we shall use the terms “k-
hypersurface” and “k-variety” and use the notation V
k
(f ) and V
k
({f
i
}). A
real
or
complex
variety
will mean a k-variety where k is
R
or
C
, respectively.
The justification for the term “hypersurface” will be found later in Theorem 10.16.8,
which shows that these sets are typically (n - 1)-dimensional subsets of k
n
.
Algebraic geometry is the study of varieties, but we shall see that thorough
answers to many of the questions we want to ask all involve projective space. In fact,
the natural setting for most of the topics in this chapter is projective space. Even
though we may be motivated by problems about algebraic curves in affine space, the
general approach is to study the curves in projective space first and then translate the
appropriate answers back to the affine context.
Affine space
, namely, k
n
, should be
thought of as dealing with local properties of curves, whereas projective space deals
with their intrinsic properties that are both local
and
global. We need to extend the
definitions above to that setting. In fact, we need to generalize our earlier definition
of projective space (as given in Section 3.4).
Definition.
Define
n-dimensional projective space
P
n
(k)
over a field k
to be the
quotient space
(
)
n
n
kk
+1
()
=
P
-
0
~,
where ~ is the equivalence relation on k
n+1
-
0
defined by
(
)
(
)
cc
,
,...,
c
~
dcdc
,
,...,
dc
12
n
+
1
1
2
n
+
1
for all c
i
, d Œ k, and d π 0. The equivalence class of an n-tuple
c
= (c
1
,c
2
,...,c
n+1
) will
be denoted by [
c
] or [c
1
,c
2
,...,c
n+1
].
Note that the space
P
n
, as defined in Section 3.4, is just
P
n
(
R
), the important
special case of n-dimensional projective space over the reals.
To define a concept of an algebraic variety in
P
n
(k), meaning the zeros of poly-
nomials, one has to be careful. The problem is that if f(X
1
,X
2
,...,X
n+1
) is a polyno-
mial in k[X
1
,X
2
,...,X
n+1
], then the obvious definition of the value of f at a point
p
in
P
n
(k), namely,
