16

SEAN KEEL AND JAMES M^KERNAN

smooth locus:

(3.2) Au Al + A2, A4i D

5

, Ee, E7, E8.

We will abuse notation and call this the simply connected list.

We note that in general S is not determined by its singularities. However, for the simply

connected list, this is almost the case. See (3.10) below.

The /iT-negative MMP for Gorenstein surfaces is almost as simple as that for smooth surfaces.

3.3 Lemma. Let g : T — W be a birational KT-contraction, of relative Picard number one,

with (irreducible) exceptional divisor E7 such that T is Gorenstein along E. Along E7 T is either

smooth, or has a unique singularity. In the latter case the singularity is an Ar singularity, with

i^T + E log terminal, and the singularity is removed by the contraction. Furthermore the induced

map

7r"y(7l0)

—

TT^9(W°)

is an isomorphism.

Proof. On the minimal desingularisation, E is a — 1-curve. Contractibility considerations give

the description of the possible singularities. The last claim follows from (7.3). •

Gorenstein P r o r a t i o n s (of relative Picard number one) are also quite simple. Here we state

the possibilities. We will prove a more precise result in §11, see (11.5.4).

3.4 Lemma. Suppose n : T — C is a

P1

-fibration, of relative Picard number one, and G C T

is a fibre, contained in the Du Val locus ofT. One of the following holds:

(1) T is smooth along G.

(2) There are exactly two singularities, A\ points, along G. KT + G is It.

(3) There is a unique singularity, a Dn point along G.

Proof. Let T — T be the minimal desingularisation, and let //, : T — W be a relative minimal

model of T — T, thus W — C is smooth, and h is a composition of blow ups at smooth

points. Because TT has relative Picard number one, each blow up in h is at a point along the — 1-

curve of the previous blow up. Now one checks easily that the proposition gives all possibilities

with Du Val singularities. For more details see (11.5). •

3.5 Lemma. If T is a Gorenstein del Pezzo surface, then

|7r"5(T°)|

8.

Proof. Note if T' — T is a degree n cover, etale in codimension one, then T' is again a

Gorenstein del Pezzo, and KT, — nKT 8. Since KT is a positive integer, the result follows. •

Recall that throughout the paper, a proper ^-negative curve on a normal surface T is called

a —1-curve, if C C T is a —1-curve in the usual sense. Note when T is Gorenstein, such a C is

a —1-curve iff C is rational, KT • C — — 1 and C is smooth.