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
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