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 5° ^

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

= 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

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