14

SEAN KEEL AND JAMES McKERNAN

contracting this curve. Let jn C Fn indicate a section disjoint from a^ (all such sections are

linearly equivalent, and have self-intersection n).

Where no confusion arises, we will use the same notation to indicate a divisor, and its strict

transform under a birational transformation. For a singular point p G 5, let p(p) count the

number of exceptional divisor of S — S lying over p. On the other hand let p be the relative

Picard number of S — S. Clearly

pes

By a curve over a point p £ S, we mean a curve in the exceptional locus of S — S over p.

By a /j-curve on a normal surface T we mean a complete curve whose strict transform on T

has self-intersection k. Note that if the curve is ^-negative, and k 0, then by adjunction,

its strict transform is a smooth rational curve.

A Q-Weil divisor Y^ai^% 1S called a subboundary if a; 1, and a boundary if 0 a; 1.

It is called pure if a; 1.

Ylai^i 1S

called reduced if all the non-zero ai are equal to one.

When we express a Q-Weil divisor as Y^at^i w e assume, that the D{ are distinct and irre-

ducible.

By a component of the Q-Weil divisor A = Y^aiD%- w ^ h a^ 0, we mean a Di with a; 0.

The support of A, Supp(A), is the union of its components (with reduced structure).

By or we mean strict inequality.

We write

^T,aiDi

"^biDi if a* 6; for all i. We write

^2ai^i

S ^ A ^

m

addition

^ bi for some i.

A log resolution of a pair (X, D) of a variety X and a reduced divisor D is a birational map

/ : Y — X with divisorial exceptional locus E, such that Y, and all components of E -\- D

are smooth, and E + D has normal crossings. A log resolution of (X, A) is a log resolution

of (X. Supp(A)). (X, D) is called log smooth, if X and every component of D is smooth, and

(X, D) has normal crossings.

Throughout the paper, unless otherwise noted, by a divisor, we mean a Q-Weil divisor, and

by a component we mean an irreducible component.

We will make frequent, and occasionally unremarked use of the classification of two dimen-

sional quotient singularities, see Chapter 3 of [27]. For the readers convenience we recall the

most important facts in Appendix L. We also make frequent use of the formulas of Chapter 3

of [27] for the index and discrepancies. We will often refer to a cyclic singularity as a chain

singularity, as the resolution graph is in this case a chain. For a non-cyclic singularity, there is

exactly one vertex in the resolution graph meeting three edges. We call this the corresponding

exceptional divisor the central divisor. Three chains meet this vertex. We refer to these as

the branches. In a branch, we call the curve meeting the central divisor, the adjacent curve,