RATIONAL CURVES ON QUASI-PROJECTIVE SURFACES 19

the second case D C S has the prescribed singularities. In the first either the singularities are

as required, or D meets the opposite end of the A3 chain. But the latter is impossible, for then

D 2 = 0 on S.

If S = S(A4). T has singularities Al + A2. Either D C T°, and Si = S(AX + A2), or D

meets one end of the A2 chain, and Si — S(A\). In the first case, either the singularities are

as prescribed, or D C S meets an end of the A4 chain. But the latter is impossible, for in that

case D2 0 on S.

If S = 5(^4), T has a single singularity, an A2 point. D (£_ T°, from the list, so D C S has

the prescribed singularities.

In the cases, S(EQ), S(EJ), T has a single singularity, a non-cyclic singularity, so by (3.3)

D C T°, and the singularities are as prescribed. D

3.9 Corollary. Let W be a Gorenstein log del Pezzo surface of rank one, which contains two

distinct rational curves D\ and D2 with Kw Di — — 1- The following implications hold:

(1) If W is simply connected then W — S(E$), one of the D% is contained in W°, and is

a rational curve of arithmetic genus one. D\ D D2 is a single smooth point of W, the

unique basepoint of the linear series | — Kw\-

(2) If K\y 4, then K^ — 4, W = S(2A\ 4- ^3) and D\ and D2 each pass through one of

the Ai points and opposite ends of the A3 point.

Proof. (1) is immediate from (3.6), (3.7) and (3.8).

Now suppose Kly 4. By (1) W is not simply connected, and so from the (full) list, S —

S(2A-[-{-A3). Kyy + Di -\-D2 is anti-ample. In particular Dt is smooth. By adjunction, they meet

at only one point, and neither can contain all three singular points. Thus each Di must contain

exactly two singular points, and they cannot both be A\, for otherwise D2 0. Thus each Di

contains one of the A\ points, and they both contain the ^3 point. The described configuration

is now the only possibility, as one checks that otherwise after extracting an appropriate curve

over the A3 point, both have non-positive self-intersection, and at least one is contractible,

violating the Picard number. D

3.10 Corollary. Let S be a rank one Gorenstein log del Pezzo surface, with algebraically simply

connected smooth locus. S is uniquely determined by its singularities (and thus by K\) except

in the case of S(Es) (that is K^ = 1) when there are two possibilities, as described in (3.7).

Proof. We will write S(C) (resp S(N)) for the S(E8) of (3.6-7) with a cuspidal member (resp.

nodal member).

S(Ai) is unique by (3.8). Assume 1 K2S 8.

By the final remarks of (3.8) there are at most two possibilities for S: Start with either S(C)

or S(N) and contract repeatedly the unique — 1-curve. K$ — 1 times. We will show that the two