18 SEAN KEEL AND JAMES M^KERNAN

P 1 -fibration, with E a section, and D (in case S ^ ¥2) the only multiple fibre. E, and the

singularities of D are as follows:

(1) If S = S(A\) — F

2

then E is the unique —2-curve and the fibration is sm,ooth.

(2) If S — S{A\ -f A2), then Ks -f D is log terminal, E is the —2-curve over the A2 point

which is disjoint from D.

(3) If S = S(A4) with singularity (2, 2', 2,2), then D meets the pruned curve and E is the

underlined curve.

(4) If S — S(Ds) then D meets one of the (2) branches, and E is the opposite (from the

central curve) end of the A2 chain.

(5) If S — S(Ek) (8 k 6) then D meets the opposite end of an A^_4 chain (this chain

is unique except when k — 6), and E is the opposite end of an A2 chain, different, in

the case k = 6, from, the A2 chain which D meets.

Furthermore, assume 1 Kg 8. Let T — S blow up a point of D not on any —2-curve.

Then T is the minimal desingularisation of a rank one log del Pezzo, Sf with algebraically simply

connected smooth locus, and Ks, = K$ — I. If we repeat this process K$ — 1 times, we obtain

S(Es). The induced map S(Eg) — S is canonical, contracting at each stage the unique — 1-

curve.

Proof. We will prove that there exists a —1-curve, D, meeting the singularities as prescribed.

The final remarks are then immediate from the singularity description, and imply the uniqueness

of the —1-curve by (3.6). One also checks easily that extracting the indicated —2-curve E gives

the required P1-fibration. Hence it is enough to prove the existence of D.

We can assume by (3.6) that K$ 2.

Note if C C S is any —1-curve, then (Ks + C) • C 0 and thus C is smooth, and so meets at

most one exceptional divisor over each singular point, and the contact is normal (see for example

(6.11)).

Let / : T — S extract a —2-curve which according to the statement of the lemma is to have

contact with D. In the case of S(A\ -t- A2) let V be the —2-curve of the A\ point, otherwise

choose any of the possible curves, that is on 5(D

5

) either of the A\ branches, on S(E6) the

opposite end of either A2 chain, and on S(A4) either of the interior — 2-curves (in the other

cases V is unique). Suppose first that T has a P1-fibration. By (3.4) this is only possible if

S = S(EQ). In this case one checks that the fibre through the D

5

point of T is a —1-curve

meeting the singularities as prescribed (for details see (11.5.4)). So we can assume T has a

birational contraction -K : T — S\ of a —1-curve, D. Using (3.3) and the list, one checks in

each case that D meets the singularities as prescribed:

If S = S(D5), T has an A4 singularity. By (3.3) and the simply connected list S\ is either

P 2 , D C T contains the A4 point, and KT + D is log terminal, or 5' = S(A4) and D C T°. In