6
SHREERAM S. ABHYANKAR AND MANISH KUMAR
Ff(Z) E Ri[Z]
with
ordR;Ff(Z)
=
1. But this is Abhyankar's Thesis Theorem 5
cited below.
To prove (2.3) note that, for every i, relative to the basis
(XIYi, Y)
of
M(~)
we have
A1
=
(XIYi)Ya+i
and
A2
=
(XIYi)Ya+i E
with 0 /=- E
E
M(Ri)
and positive integer
a+
i. So we are done by Lemma (1.3).
NORMALITY THEOREM 3. This refers to the well-known theorem which
says that if
N
is a nonzero nonunit irreducible element in a regular local domain
Q
such that, for every height one prime ideal
P
in
QI(NQ),
the localization of
QI(NQ)
at
P
is regular, then
QI(NQ)
is normal; for instance see (Q15)(T69),
(Q15)(T70), (Q19)(T86), and (Q19)(T88) of Lecture L5 of [A09]. In our case
Q
=
the localization of
R[Z]
at the maximal ideal generated by
M(R)
and
Z,
and
N
=
F~'(Z).
TOTAL EMBEDDED CURVE RESOLUTION THEOREM 4. In (10.7) on
page 44 of [A07] and again in (5.12) on page 1595 of [A08] it is proved that if
C
is any nonzero element in a two dimensional pseudogeometric local domain
R
and
(Ri)i=0,1,2, ...
is any two dimensional quadratic sequence with
Ro
=
R
then for
all large enough i we have
C
=
DX[Y/,
with
D
E
Ri \ M(Ri)
and nonnegative
integers
r,
s, where
M(Ri)
=
(Xi, Yi)Ri.
ABHYANKAR'S THESIS THEOREM 5. See §8 and §9 of [AOl], Proposition
10 of [A05], and Theorems 1 to 12 of [A04].
3. Global Theory
Let
Klk
be a two dimensional excellent function field, i.e.,
K
is a finitely
generated field extension of the quotient field of an excellent domain k such that
the transcendence degree of the said extension plus the (Krull) dimension of
k
equals two. In [A05] and [A06] it was shown that then there exists a nonsingular
projective model of
Klk
and moreover, after applying a finite number of successive
quadratic transformations to such a model, it can be made to dominate any given
projective model of
K
I
k.
For the case of algebraically closed ground fields, this was
proved in [ZOl] for zero characteristic and in [AOl] for nonzero characteristic.
Note that if
V'
is a model of
Klk
which is obtained by applying a finite number
of successive quadratic transformations to a nonsingular projective model
V
of
K
I
k
then
V'
is again a nonsingular projective model of
Klk.
We call
V'
an iterated
quadratic transform of
V.
Note that in applying a quadratic transformation to
V
we are permitted to simultaneously blow up a "finite number of points of
V."
Also
note that, since we have adopted the model view point, a "point" of
V"
actually
means a two dimensional regular local domain
R
whose residue field is
RIM(R).
THEOREM 6. For any two dimensional excellent function field
Klk
we have
the following.
( 6.1) If char(
K)
1=- 2 then, given any nonsingular projective model
V
of
K
I
k
and
any finite Galois extension
Ll
K
whose Galois group is the direct sum of a finite
number of copies of a cyclic group of order 2, there exists an iterated quadratic
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