4
2. Non-singularity and Resolution of Singularities
Set r0 = mult(/(0, j/)) mult(/). Set
5o = min
:
: j
TQ
and an ^ 0 .
[r0-j J
£0 = oo if and only if / =
uyr°,
where u is a unit in K[[x, y]]. Suppose that
So oo. Then we can write
2+5o:/£on)
with aor0 7^ 0
a n
d the weighted leading form
Ls0(x,y) = 2^
aijx%y^
—
a0r0yr°
+ terms of lower degree in y
i+Soj=Soro
has at least two non-zero terms. We can thus choose 0 ^ c\ G K so that
Ztf0(l,ci)= ^ a y c j = 0 .
Write ^0 — ^5 where qo,Po are relatively prime positive integers. We make
a transformation
x = xf, y = xp1°(y1 + ci).
Then
/ ^ ? P ° / i ( x i , l / i ) ,
where
(2.1) /i(^i,yi) = X ]
a
^'(
C l + yi)3 +xiH(xiiVi)-
i-\-Soj=Soro
By our choice of ci, /i(0,0) = 0. Set ri = mult(/i(0, yi)). We see that
r\ TQ. We have an expansion
h = ^2aij(1)x\yv
Set
Si — min
:
: j r\ and a^-(l) 7^ 0 ,
and write 8\ = ^ with pi,gi relatively prime. We can then choose C 2 Gif
for /1 , in the same way that we chose c\ for / , and iterate this process,
obtaining a sequence of transformations
x = xf, y = x f (yi-hci),
(2.2)
x
i =
xl1*
2/i = ^(jfe + c2),
Either this sequence of transformations terminates after a finite number n
of steps with 5n = 00, or we can construct an infinite sequence of trans-
formations with Sn 00 for all n. This allows us to write y as a series in
ascending fractional powers of x.
