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 5° .

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 S° by [17]. OE(E) = OE(Q) for a

unique q G E. Since Hl(Os) = 0, q G S° 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

S° 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 S° be any member of

| — Ks\. Let q G B° 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 S° 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