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