4
SEAN KEEL AND JAMES
MCKERNAN
those with a tiger (defined below), we give a short and elegant deformation theoretic proof of
(1.3), see (6.1). Boundedness results of Alexeev and Kollar imply that rank one log del Pezzos
without a tiger are bounded, see (23.1). We complete the proof by constructing an explicit finite
list of families of rank one log del Pezzos which includes any with no tiger and whose smooth
locus has trivial algebraic fundamental group (it is easy to reduce (1.1) to this case, see §7). For
each of the surfaces in this list, we directly construct a dominating family of rational curves.
We will explain all this in much greater detail below, but first we present some additional
results which are of independent interest. Most are corollaries of (1.1). Proofs are given in §20.
We will use the following notation.
A\ = A1 \{0}. In view of (5.11), it is natural to think of A1 and Aj as the open analogues of P1
and elliptic curves, since the existence of dominating rational, or elliptic families has analogous
implications on ordinary Kodaira dimension.
1.4 Proposition. Let B C S be a reduced curve on a normal projective surface and set U =
S \ (B U Sing(S)). Consider the following conditions:
(1) The Kodaira dimension of K$ 4- B is negative.
(2) Ks + B is numerically trivial, but not log canonical.
(3) Ks + B is numerically trivial, and B ^ 0.
(4) Ks is numerically trivial, B = 0. and S has a singularity which is not a quotient
singularity.
If any of the above hold, then U is dominated by images of A\, and by images of A1 if (1) or
(2) holds.
We note one interesting implication of (1.4.2) and (5.11): The complement of a integral plane
cubic, B, is dominated by images of A1 iff B has a cusp, or B is the union of a smooth conic
and a tangent line, or B is the union of three lines meeting at a point (indeed this is just a list
of the non log canonical possibilities).
Next we have a version of (1.3) for any boundary:
1.5 Corollary. Suppose the pair (5, A) consists of a projective surface S and boundary A, such
that Ks -f- A is log canonical. Then either
(1) \m(Ks + A)| is non-empty for some m 0, or
(2) There is a covering family of rational curves C, such that (Ks -f A) C 0.
(1.5) thus says that either some multiple of Ks + A has a section, or there is a good geometric
reason why it does not.
It may be tempting to believe that (1.5) follows automatically from the MMP: One can assume
the (Ks -f A)-MMP gives a composition of contractions / : S S' such that —(K's + A') is
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