RATIONAL CURVES ON QUASI-PROJECTIVE SURFACES
17
3.6 Lemma. Let S = S{E%). S contains a unique —1-curve D. D C S meets a unique excep-
tional curve, the opposite end of the A4 chain, with normal contact. \ Ks\ is one dimensional,
and has a unique basepoint, a smooth point of S. There are two possibilities for the collection
of rational members of \ Ks\- Either
(1) | Ks\ has exactly three rational members, D and two integral nodal rational curves N\,
N2 C S°, or
(2) | Ks\ has exactly two rational members, D, and a unique integral cuspidal rational
curve, C C .
Proof. Let S = S(Es) be any such surface. Obviously S contains some —1-curve, D. K$ 1,
so | Ks\ is one dimensional by Riemann-Roch, and every member of | Ks\ is reduced and
irreducible. | K$\ and has a smooth elliptic member E C by [17]. OE(E) = OE(Q) for a
unique q G E. Since Hl(Os) = 0, q G is the unique basepoint of | Ks\- Let T S blow
up q. \—Ks\ yields an elliptic fibration g : T P 1 . Let g : T P1 be the induced map. Since
is simply connected, any —1-curve on S is a member of | Ks\- Since any —1-curve must
pass through the singular point, and g has irreducible fibres, D is the unique —1-curve. The
fibre of g containing D has 9 irreducible components, thus by Kodaira's classification of singular
fibres, see page 150 of [4], the fibre containing D is E%, thus D meets the opposite end of the
A4 chain as required. Note e(T) (the topological Euler characteristic) is 12. By formula (11.4)
on page 97 of [4], the additional singular fibres (which we know are reduced and irreducible) are
either exactly two nodal rational curves, or exactly one cuspidal rational curve. D
3.7 Lemma. There are exactly two isomorphism classes of S(E$) corresponding to the two
possibilities in (3.6).
Proof. Let S S{E%). Let D be the unique —1-curve of (3.6). Let B C be any member of
| Ks\. Let q G be the unique basepoint of | Ks\, see (3.6). Let L be the —2-curve of the
Ai branch of the E% singularity. Let T S extract L. T has an A7 singularity, and D C T
contracts 7r : T P 2 , the image of L is a flex line to the cubic curve B C P 2 at (the image of
the) q. The induced map S P 2 is obtained by blowing up 8 times along B over q. B c F 2
is embedded by the full linear system |3g|, thus S depends only on (B,q). The automorphism
group of B acts transitively on # ° , so S depends only on B. By (3.6) we can take for B either
a cuspidal rational curve C, or a nodal rational curve N.
3.8 Lemma. Assume has trivial algebraic fundamental group. If S is not P2 or ¥2 then S
contains a unique —1-curve, D. D has normal crossings with the exceptional locus of S S,
and meets exactly one exceptional curve (of the minimal desingularisation) over each singular
point of S. If S ^ P 2 then there is a —2-curve, E, of S, such that extracting E gives a
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