Graphics Reference
In-Depth Information
r
'
1
(
)
=
(
)
+
(
)
fXY
,
Y m
-
X
... ,
n
i
i
=
which shows that the sum of the intersection multiplicities of the line X = 0 with the
curve
C
n
is r.
Now if two of the m
i
are distinct, then, since the intersection multiplicities at those
points would be each less that r, we could prove this case of Theorem 10.12.6 using
induction on r. But what if all the m
i
are equal? We would have
r
(
)
=-
(
)
+
(
)
fXY
,
Y m
X
... .
n
1
We can now translate
C
n
so as to move the point (0,m
1
) to the origin. The new curve
does not have a vertical tangent at the origin, so that we can apply a quadratic trans-
formation to it. Unfortunately, the new curve could also have an r-fold singularity with
multiplicity r. The question arises, if we keep repeating this process of translating to
the origin and applying a quadratic transformation, will we continue to get r-fold sin-
gularities at the origin with multiplicity r? The answer is no. See [Seid68] for the
details. Therefore, after a finite number of steps our procedure will lead us to a situ-
ation where we have a simple point. Induction works.
Places provide an alternate way to prove various theorems of algebraic geometry
such as Bèzout's theorem. See [Walk50]. Some definitions can also be phrased in terms
of places. For example,
10.12.16. Theorem.
A point of a plane curve is nonsingular if it is the center of just
one linear place.
Proof.
See [Walk50]. The only if part of this theorem is Theorem 10.12.8.
The definition of a place given in this section was chosen because it is less abstract
than other definitions. However, it is worth being aware of another common defini-
tion that is used in “valuation theory.” Let
n
c
=
(
)
Œ
cc
,
,...,
c
k
.
Then
c
defines a
12
n
homomorphism
[
]
Æ
vkXX
:
,
,...,
X
k
12
n
..,
X
(
)
Æ
()
=
(
)
fX X
,
,.
f
c
fc c
,
,...,
c
.
12
n
12
n
The element f(
c
) is called a
value
of f. Now k[X
1
,X
2
,...,X
n
] is a subring of the quo-
tient field K = k(X
1
,X
2
,...,X
n
) and v extends to a homomorphism on K except at those
elements whose denominators vanish on
c
. This motivates the following:
Let K be a field. A subring R of K is called a
valuation ring
if for all x ΠK, either
x or 1/x belongs to R. A
place
for K is a nonzero homomorphism p from a valua-
tion ring K
p
in K to a field F, so that if x Œ K and x œ K
p
, then p(1/x) = 0.
