SIMULTANEOUS RESOLUTION

5

To prove (1.2) it suffices to note that L is a splitting ofF{ over K.

To prove (1.3), since A1 E

M(R)

with B1 =

Y

E

M(R) \

M(R)

2,

we see

that S1 is a two dimensional regular local domain with M(S1)

=

(X, H)S1

•

Also

y

=

H 2

+

xya H

and substituting this in

we get

where

and hence

F~(Z)

=

F2(Z +H)= Z

2

+ XYa EZ + (Y + H

2

+ xya EH)

F~(Z)

=

Z

2

+

XYa EZ

+

XYaH(1 +E)

{

ya

= (H2

+

XYaH)a

=

(H2 +X[H2 +XYaH]aH)a

=

(H2 + X[H2 +

X(H2

+ XYaH)aH]H)a

=

H

2aD*

with

D*

E

R \ M(R)

F~(Z)

= Z

2

+

XH2aED*Z

+

XH2a+1D**

with

D**

E

R \ M(R)

and therefore

F~'(Z)

=

H- 2 aF~(zHa)

=

Z

2

+

XHaED*Z

+

XHD**.

Consequently "the irreducible surface

F~'

( Z)

= 0 is devoid of singular curves" and

hence, by the following Normality Theorem 3, we see that

S

=

S1[J]

where

I

is a

root

ofF~'

in L. Since the coefficients of Z

1

and

Z

0

in

F~'(Z)

belong to

M(R)

and

M(R)

2

respectively, it follows that Sis a two dimensional nonregular local domain.

Since

F1(Z)

and

F~'

( Z)

are irreducible, we also see that

L / K

is Galois with Galois

group Z2 EB Z2.

THEOREM 2. Let (Ri)i=0,1,2, ... be a two dimensional quadratic sequence with

R

0

=

R,

and let

Si

be the integral closure of

Ri

in

L.

Then we have the following.

(2.1) If char(K)

-I

2 and

R

is pseudogeometric, then

Si

is a two dimensional

semilocal regular domain for infinitely many i.

(2.2) If

n

=

1 with

R

pseudogeometric and

R/M(R)

algebraically closed, then

si

is a two dimensional semilocal regular domain for infinitely many

i.

(2.3) If char(K)

=

2

=

n

with F1 and F2 as in Lemma (1.3) and for i

=

0, 1, 2, ...

we have

Ri

=

R[X/Yi]P;

where

Pi

is the prime ideal in

R[X/Yi]

gen-

erated by

XjYi

andY, then

Si

is a two dimensional nonregular local domain for

every i, and Lj K is Galois with galois group Z2 EB Z2.

PROOF. To prove (2.1), by applying the following Total Embedded Curve

Resolution Theorem 4 to "the plane curve

C

=

0," for all sufficiently large i we

can write

C

=

DX[Y/,

with

DE Ri \ M(Ri)

and nonnegative integers

r, s,

where

M(Ri)

=

(Xi, Yi)Ri.

This amounts to writing

Cj

=

DjX?Y;si

for all j

E

J with

Dj

E

Ri \ M(Ri)

and nonnegative integers

Tj, Sj.

Now we are done by Lemma

(1.1).

To prove (2.2), in view of Lemma (1.2) and Theorem (2.1), it suffices to show

that, assuming char(K)

=

2 with

R

pseudogeometric and

R/M(R)

algebraically

closed, given any irreducible

F(Z)

=

Z

2

+

AZ

+

B

E

R[Z], for infinitely many i

there exists 0

"I

o:i

E

K and f3i

E

K such that for Ff ( Z)

=

o:;

2

F (

O:i

Z

+

f3i) we have