RATIONAL CURVES ON QUASI-PROJECTIVE SURFACES
and the other end of the chain, the opposite curve.
We will say that a chain singularity is almost D u Val if its resolution graph has only one
vertex of weight other than 2, that of weight 3 and occurring at the end of the chain, that is the
chain has form (2, 2, 2 , . . . , 3) (where we also allow (3)).
By the index of a quotient singularity, we mean the order of the local fundamental group.
We will make frequent use of the main results of the (log) Minimal Model Program: the cone
theorem, the contraction theorem, and the log abundance theorem. The first two of these are
known in all dimensions, and the last up to dimension three. However we only use them in
dimension two. For proofs in this case see . For overviews of the general theory (as well
as further references) see , , . We will also make use of the standard definitions and
notations of the program, as in .
We will say a pair (Ar, D) of a normal variety and a reduced divisor is log uniruled if X° is
covered by rational curves meeting D at most once (that is through a general point of x there is
a complete rational curve, contained in the smooth locus, and meeting D at most once). We will
say X is log uniruled if (X, 0) is log uniruled, or equivalently, if X° is uniruled, that is through
a general point of X, there is a complete rational curve, contained in the smooth locus.
§3 GORENSTEIN DEL PEZZO SURFACES
In this section we collect a number of results on Gorenstein log del Pezzo surfaces that we
will use at various points in the paper. Especially useful will be (3.6-8) which together give a
complete and simple picture of rank one Gorenstein log del Pezzo surfaces whose smooth locus
is algebraically simply connected.
Notation: Throughout the section S denotes a rank one Gorenstein log del Pezzo.
In  possible singularities of S are classified. We will make frequent use of this classification,
which we refer to as the list. We write for example S(A\ 4- A3 -f A5) for a rank one Gorenstein
del Pezzo with those singularities, and for example S(Ai -f A3 4- A$) for its desingularisation.
For the reader's convenience, here is a copy of the list:
Au Ai + A2, A4i 2Ai + A3, D
, A1 + A5, 3A2, E6j
$Ai + D4, A7, AX + DQ, E7, AX+2A3, A2+A5,
, 2Al + D6, Es, Ax + E7, Ax + A7, 2A4, As,
Ai + A2 + A5, A2 + E6, A3 + Db, 4A2, 2AX + 2A3, 2D4.
By , the following are the subset of possibilities with 5° algebraically simply connected, and
any rank one Gorenstein del Pezzo with these singularities has algebraically simply connected