§1 INTRODUCTION AND STATEMENT O F RESULT S
In classifying smooth projective varieties, one looks for an intrinsic map to projective space.
The natural intrinsic line bundles are the various tensor powers of ux, and so one is led to
consider the pluricanonical linear series \mKx\- The obvious question arises, what if these
series are all empty? The answer is striking, one of the principal accomplishments of Mori's
program:
1.0 Theorem (Miyaoka, Mori, Kawamata, Shokurov, and others). Let X be a smooth
projective variety of dimension at most three.
Then \mKx\ is empty for all m 0 iff X is covered by images off1.
(We say X is covered by images of P1 if for every point p of X , we may find a morphism
/ : P 1 X such that the point p belongs to /(P 1 ), where here, as in every other definition
involving a morphism from a curve to a surface, we require / to be non-constant.)
In this paper we consider the analogous question for quasi-projective (or more generally log)
varieties. If U is a smooth quasi-projective variety, then, following Iitaka, one picks a smooth
compactification U C X such that the complement D X\U is a divisor with normal crossings.
The linear series \m(Kx -f D)\ turn out to depend only on £/, not on X or D, and is thus the
natural analogue of \mKx\- \m(Kx + D)\ is called the log pluricanonical series, and the problem
is to characterise those smooth quasi-projective varieties for which the log pluricanonical series
are all empty.
When U is a curve the solution is elementary, \m(Kx + D)\ is empty for all m 0 iff U = A1
or P 1 .
In dimension two the problem is already surprisingly subtle, and has received considerable
attention. An import special case was settled by Miyanishi and Tsunoda, [31],[32], and further
results have been obtained by Zhang, [36]. Here our main goal is a complete, and self-contained
solution (throughout the paper, everything takes place over C):
1.1 Theorem. Let U be a smooth quasi-projective variety of dimension at most two. Then
\m(Kx + D)\ is empty for all m 0 iffU is dominated by images of A1.
We say U is dominated by images of a curve C, if there is a dense open subset V of
U and for every point p of V, we can find a non-constant morphism / : C U such that
p f(C). We say U is dominated by rational curves, if it is dominated by images of P 1 .
Received by the editor September 20, 1997
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