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
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-
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
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
) 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
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
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