DEFINITION 08.1
For any ring
, we define its Krull dimension to be
the maximum of ascending chains of primes in
. Namely,

DEFINITION 08.2
A local ring
is called regular if its
Krull dimension is equal to
>

PROPOSITION 08.3Let
be a field of characterictic
.
Then every quadratic curve
(a curve defined by a homogeneous polynomial of degree
)
in
over
is isomorphic to a
curve of the form

In particular, every quadratic curve in
over
is
isomorphic to a curve
.

PROPOSITION 08.4Let
be a field of characterictic
.
Then every cubic curve
(a curve defined by a homogeneous polynomial of degree
)
in
over
is isomorphic to a
curve of the form

It should be meaningful to point out:

PROPOSITION 08.5Let
be a imaginary number in
.
defines a lattice (a discrete subgroup of rank
in
)
.
A complex manifold
may be embedded to the complex projective plane
by the Weierstrass
-function
and its derivative
.
Namely, a rational map defined by

gives a holomorphic map
.
moreover,
satisfy a cubic relation
so that
gives an isomorphism of
and a cubic curve
in
.