Cryptography Reference
In-Depth Information
Proof
Theorem I.5.1 of Hartshorne [
252
].
n
be an affine variety over
Corollary 7.1.12
Let X
=
V
(
f
1
(
x
)
,...,f
m
(
x
))
⊆ A
k
of dimen-
sion d. Let P
∈
X
(
k
)
. Then P
∈
X
(
k
)
is a
singular point
of X if and only if the Jacobian
matrix J
X,P
has rank not equal to n
−
d. The point is
non-singular
if the rank of J
X,P
is
equal to n
−
d.
n
be irreducible and let P
Corollary 7.1.13
Let X
=
V
(
f
(
x
1
,...,x
n
))
⊆ A
∈
X
(
k
)
. Then
P is singular if and only if
∂f
∂x
j
(
P
)
=
0
for all
1
≤
j
≤
n
Exercise 7.1.14
Let
k
be a field such that char(
k
)
=
2 and let
F
(
x
)
∈ k
[
x
] be such that
gcd(
F
(
x
)
,F
(
x
))
=
1. Show that
X
:
y
2
=
F
(
x
)
2
.
is non-singular as an affine algebraic set. Now consider t
he
projective closure
X
⊆ P
Show that if deg(
F
(
x
))
≥
4 then there is a unique point in
X
−
X
and that it is a singular
point.
Finally we can define what we mean by a curve.
Definition 7.1.15
A
curve
is a projective non-singular variety of dimension 1. A
plane
curve
is a curve that is given by an equation
V
(
F
(
x,y,z
))
⊆ P
2
.
Remark 7.1.16
We stress that in this topic a curve is always projective and non-singular.
Note that many authors (including Hartshorne [
252
] and Silverman [
505
]) allow affine
and/or singular dimension 1 varieties
X
to be curves. A fact that we will not prove is
th
at
every finitely gen
er
ated, transcendence degree 1 extension
K
of an algebraic closed field
k
is
the f
un
ction field
k
(
C
) of a curve (see Theorem I.6.9 of Hartshorne [
252
]; note that working
over
k
is essential as there are finitely generated, transcendence degree 1 extensions of
k
that are not
k
(
C
) for a curve
C
defined over
k
). It follows that every irreducible algebraic
set of dimension 1 over
to a non-singular curve (see Theorem 1.1 of
Moreno [
395
] for the details). Hence, in practice one often writes down an affine and/or
singular equation
X
that is birational to the projective, non-singular curve
C
one has in
mind. In our notation, the commonly used phrase “singular curve” is an oxymoron. Instead,
one can say “singular equation for a curve” or “singular model for a curve”.
k
is birational over
k
The following result is needed in a later proof.
Lemma 7.1.17
Let C be a curve over
k
. Let P,Q
∈
C
(
k
)
. Then
O
P,
k
⊆
O
Q,
k
implies
P
=
Q.
Proof
Lemma I.6.4 of Hartshorne [
252
].
