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10.7.9. Corollary.
(1) An irreducible quadratic plane curve has no singular points.
(2) An irreducible cubic plane curve has at most one singular point and that is a
double point.
(3) If an irreducible plane curve of order n has an (n - 1)-fold point then it has
no other singular points.
Proof.
These facts are a trivial consequence of Theorem 10.7.8.
10.8
Some Commutative Algebra
The varieties we have talked about so far were all hypersurfaces, that is, they were
defined by a single polynomial. One polynomial defines a variety of codimension 1
(over
R
or
C
, depending on the context) and so the restriction might not seem unrea-
sonable from a dimensional point of view when one studies curves in the real or
complex planes. On the other hand, even in dimension two, we do not pick up all vari-
eties in this way, because the ring of polynomials in more than one variable is not a
principal ideal domain. If we want to consider varieties in dimensions higher than
two, then the restriction to hypersurfaces is even more inadequate because in n-
dimensional space we want to talk about sets other than (n - 1)-dimensional ones.
We definitely need to allow the possibility that our space is defined by a collection of
polynomials. For example, it takes two equations to define a line in 3-space. In order
to be able to handle higher dimensions, we need some additional machinery. This
section will give a brief overview of some fundamental results relating point sets
(topology) and ideals (algebra). The algebra side of this is falls into the field of com-
mutative algebra. In general, commutative algebra deals with commutative rings with
1. In fact, the rings are usually closely related to polynomial rings over a field or the
integers.
Let
A
be a subset of k
n
. Define the
ideal of
A
, I(
A
), by
Definition.
()
=Œ
[
{
]
}
.,
X
n
()
=
I
A
f
k X
,
X
,..
f
a
0
,
for all
a
Œ
A
.
12
If
A
consists of a single point
a
, then we shall write I(
a
) rather than I(
A
).
10.8.1. Lemma.
I(
A
) is an ideal in k[X
1
,X
2
,...,X
n
].
Proof.
Straightforward.
The map
A
Æ I(
A
) associates ideals to arbitrary sets. We already have a kind of
converse that associates a set of points to a finite set of polynomials, namely their
zeros. It is convenient to extend this notion to arbitrary sets of polynomials.
Definition.
Let S be a set of polynomials in k[X
1
,X
2
,...,X
n
]. Define a subset V(S)
of k
n
, called the
variety of S
, by
