RATIONAL CURVES ON QUASI-PROJECTIVE SURFACES
1.18 LOGICAL STRUCTUR E OF P R O O F OF (1.1)
ASSUME K(KX + B) = - c o . GOAL : (X, B) LOG UNIRULED
13
Run (Kx + £)-negative MMP
{X, B) (5, D), 5 rank one Idp
Goal: (S, D) log
uniruled: two cases:
'if D = 0: Goal ^
uniruled: Two cases:
4
If D ^ 0: (5, D) log
uniruled (6.2)
If Ks has
transform
S®\E do
a tiger. E:
(S,$)-^(Sl,E)
minated by A\s
= uniruled. (6.1)
Bug-Eyed cover, §4, and
Def. Theory §5
Bug-Eyed Cover,
Def. Theory, and
Gorenstein ldps, §3
If Ks has no tiger:
Run the Hunt
4
3 special rat Z C S
dominated by rats ~ mZ
Def. Theory,
Bug-Eyed cover, and
criteria (6.5-6).
ft
Analysis of Hunt, §8-19
See flow chart 8.0.16
§2 GLOSSARY O F NOTATION AND CONVENTIONS
If S is a normal surface we indicate by S its minimal desingularisation. If C C S is an effective
divisor, then C C S will indicate its strict transform.
F
n
= F(C 0 0(n)) denotes the unique minimal rational ruled surface, with a curve a^ C F
n
of self-intersection —n, and F
n
denotes the log del Pezzo surface of rank one, obtained by
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