RATIONAL CURVES ON QUASI-PROJECTIVE SURFACES

7

used to show that (mod p) some multiple of C moves with a fixed point. If a curve moves with

a point fixed, then rigidity implies there is a rational curve through the fixed point. However

it is not clear (at least to us) how to control how the generated rational curve intersects D. In

(6.9) we give examples indicating some of the difficulties.

In any case, the rational curves we use to prove (1.1) are obtained by an entirely different

procedure, which we discuss below. However, once we have (1.1) we obtain a version of (1.11)

as a:

1.12 Corollary. Assumptions as in (1.10). If X has dimension at most two, or X has dimen-

sion three and the Kodaira dimension of Kx + D is non-negative, then there is a rational curve

through a general point of C, meeting D at most once.

Outlin e of t h e proof of (1.3).

Now we turn to an outline of the proof of (1.3), both to indicate its logical structure, and to

high light many of issues which arise that are of independent interest. The logical order of the

proof is also indicated by the flowchart (1.18) below.

The proof divides into two cases. We will use the notation of (1.3). Thus S is a normal

projective surface with Picard number one with quotient singularities. D is a curve, such that

Ks + D is log terminal and — (Ks 4- D) is ample.

Case I: D ^ 0. The case of non-empty D is proved in (6.2), using deformation theory

and Kollar's Bug-Eyed cover. Here is a sketch: For simplicity, suppose D is irreducible. By

adjunction D itself is a smooth rational curve, and we obtain a covering family of rational curves

by deforming a high multiple of D. The difficulty of course is the presence of singularities. For

this we use the Bug-Eyed cover: Given a normal surface S with quotient singularities, there

exists a unique smooth (but non-separated) algebraic space b : S — S such that b is a

universal homeomorphism, and an isomorphism over S°. See §4. For (1.3) the important point

is that in terms of Hom(£,5 b ), for a proper curve E, 5 b behaves exactly like a smooth variety.

The idea then is to lift the problem to 5^. The condition that Ks + D is log terminal is equivalent

to the condition that b _ 1 (D) C 5 b is smooth, and the condition that Ks + D is anti-ample, is

equivalent to the existence of a endomorphism / : IP1 — D = P1 and a commutative diagram

F 1 —g— S*

'1 4

D 5,

see (4.14). Now deformation theory implies we can deform g away from the singularities of 5,

maintaining a single point of (necessarily high order) contact with D.

The case of D non-empty has been previously proved by Miyanishi and Tsunoda [31] and

[32]. Their proof is based on a classification of pairs (5, D). Deformation theory together with