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THE END CURVE THEOREM FOR NORMAL COMPLEX SURFACE SINGULARITIES THE END CURVE THEOREM FOR NORMAL COMPLEX
SURFACE SINGULARITIES
WALTER D. NEUMANN AND JONATHAN WAHL
Abstract. We prove the End Curve Theorem, which states that a normal sur-
face singularity (X, o) with rational homology sphere link is a splice-quotient
singularity if and only if it has an end curve function for each leaf of a good reso-
lution tree.
An end-curve function is an analytic function (X, o) (C, 0) whose zero set
intersects in the knot given by a meridian curve of the exceptional curve corre-
sponding to the given leaf.
A splice-quotient singularity (X, o) is described by giving an explicit set of
equations describing its universal abelian cover as a complete intersection in C
t
,
where t is the number of leaves in the resolution graph for (X, o), together with an
explicit description of the covering transformation group.
Among the immediate consequences of the End Curve Theorem are the previously
known results: (X, o) is a splice quotient if it is weighted homogeneous (Neumann
1981), or rational or minimally elliptic (Okuma 2005).
We consider normal surface singularities whose links are rational homology spheres
(QHS for short). The QHS condition is equivalent to the condition that the resolution
graph of a minimal good resolution be a rational tree, i.e., is a tree and all
exceptional curves are genus zero.
Among singularities with QHS links, splice-quotient singularities are a broad gen-
eralization of weighted homogeneous singularities. We recall their denition briey
here and in more detail in Section
1
. Full details can be found in [
20
].
Recall rst that the topology of a normal complex surface singularity (X, o) is
determined by and determines the minimal resolution graph . Let t be the number
of leaves of . For i = 1, . . . , t, we associate the coordinate function x
i
of C
t
to the
ith leaf. This leads to a natural action of the discriminant group D = H
1
() by
diagonal matrices on C
t
(see Section
1
).
Under two (weak) conditions on , called the semigroup and congruence con-
ditions, one can write down an explicit set of t 2 equations in the variables x
i
, which
2000 Mathematics Subject Classication. 32S50, 14B05, 57M25, 57N10.
Key words and phrases. surface singularity, splice quotient singularity, rational homology sphere,
complete intersection singularity, abelian cover, numerical semigroup, monomial curve, linking
pairing.
Research supported under NSF grant no. DMS-0456227 and NSA grant no. H98230-06-1-011.
Research supported under NSA grant no. FA9550-06-1-0063.
1 2
WALTER D. NEUMANN AND JONATHAN WAHL
denes an isolated complete intersection singularity (V, 0) and which is invariant un-
der the action of D. Moreover the resulting action of D on V is free away from 0, and
(X, o) = (V, 0)/D is a normal surface singularity whose minimal good resolution graph
is . This (X, o) is what we call a splice quotient singularity. Since the covering trans-
formation group for the covering map V X is D = H
1
(X {o}) =
1
(X {o})
ab
,
the covering (V, 0) (X, o) (branched only at the singular points) may be called
the universal abelian cover of (X, o). In particular, for a splice quotient singularity,
one can write down explicit equations for the universal abelian cover just from the
resolution graph, i.e., from the topology of the link.
The link of the singularity (X, o) can be expressed as the boundary of a plumbed
regular neighborhood N of the exceptional divisor E = E
1
� � � E
n
in the minimal
good resolution
X of (X, o). Then each meridian curve of an E
i
gives a knot K
i
in .
A meridian curve means the boundary of a small transverse disk to the exceptional
divisor E
i
. If E
i
is the exceptional curve corresponding to a leaf of we call K
i
an
end knot. A (germ of a) smooth complex curve on
X which intersects E transversally
on such a leaf curve (and hence which cuts out an end-knot on ) is called an end
curve; we also use this name for the image curve in X.
If (X, o) is a splice-quotient singularity as described above, then some power z
i
= x
d
i
of the coordinate function x
i
on V is well dened on X = V /D. The zero set in
resp. X of z
i
is the end knot resp. end curve corresponding to the ith leaf of (the
degree of vanishing may be > 1). We say that the end knot or end curve is cut out
by the function z
i
and that z
i
is an end curve function.
Our main result is
End Curve Theorem. Let (X, o) be a normal surface singularity with QHS link .
Suppose that for each leaf of the resolution diagram there exists a corresponding end
curve function z
i
: (V, o) (C, 0) which cuts out an end knot K
i
(or end curve)
for that leaf. Then (X, o) is a splice quotient singularity and a choice of a suitable
root x
i
of z
i
for each i gives coordinates for the splice quotient description.
An immediate corollary (conjectured in [
20
] and rst proved by Okuma [
23
]) is that
rational singularities and most minimally elliptic singularities (the few with nonQHS
link must be excluded) are splice-quotients. Another direct corollary is the result of
[
16
], that a weighted homogeneous singularity with QHS link has universal abelian
cover a Brieskorn complete intersection. The special case of the End Curve Theorem,
when the link is an integral homology sphere (so that D is trivial), was proved in our
earlier paper [
21
].
We rst proved the End Curve Theorem in summer of 2005, but it has taken a
while to write up in what we hope is an understandable form. In the meantime,
Okuma resp. N�
emethi and Okuma in [
24
,
13
,
14
] (see also Braun and N�
emethi [
2
])
have used this to compute the geometric genus p
g
of any splice-quotient, and to prove
for splice-quotients the Casson invariant conjecture [
17
] for singularities with ZHS THE END CURVE THEOREM FOR NORMAL COMPLEX SURFACE SINGULARITIES
3
links (in which case D = {1} so V = X), as well as the N�
emethi-Nicolaescu extension
[
11
] of the Casson Invariant Conjecture to singularities with QHS links.
These results of N�
emethi and Okuma give topological interpretations of analytic
invariants; this is analogous to the fact that for rational singularities some of the
important analytic invariants are topologically determined. As happens for rational
singularities, the set of resolution graphs that belong to splice quotient singularities
is closed under the operations of taking subgraphs and of decreasing the intersection
weight at any vertex [
14
]. It is worth noting, however, that rational singularities did
not have explicit analytic descriptions before splice quotients were discovered; even
the fact that their universal abelian covers are complete intersections was unexpected
until it was conjectured in [
20
] (see also [
22
]).
Of course, unlike rationality, the property of being a splice-quotient is not topo-
logically determinedfor example, splice quotients, as quotients of Gorenstein sin-
gularities, are necessarily QGorenstein, which is generally a very special property
within a topological type. Even more, equisingular deformations of very simple
splice quotients need not be of this type (see Example
10.4
).
We once over-optimistically conjectured that QGorenstein singularities with QHS
links would have complete intersection universal abelian covers, and although this
is false in general [
12
], we see it is true for a large class of singularities. There is a
natural arithmetic analog. A standard dictionary that developed out of proposals
of Mazur and others pairs 3manifolds with number elds, knots with primes, and so
on. A natural analog of universal abelian covers of QHS links belonging to complete
intersections would be that the ring of integers of the Hilbert class eld of a number
eld K be a complete intersection over Z. This is true, proved by de Smit and Lenstra
[
5
]. The analogy between splice singularities and Hilbert class elds is enticing, since
it is a signicant open problem to compute Hilbert class elds, while the explicit
splice singularity description is easily computed from the resolution diagram.
We summarise the proof of the End Curve Theorem in Section
2
after rst recalling
the theory of splice quotient singularities in Section
1
. We complete the proof in
Section
8
. Some applications and examples are discussed in the nal section
10
.
Some of the ingredients in our proof could be of independent interest. We need an
extension to the equivariant reducible case of the theory of numerical semigroups and
monomial curves developed by Delorme, Herzog, Kunz, Watanabe and the authors
[
4
,
8
,
9
,
21
,
28
]. The necessary parts of this theory are developed in sections
3 7
. Some
topological results about knots in Qhomology spheres and their linking numbers and
Milnor numbers are collated in Section
9
.
1. Splice quotient singularities
We recall here the detailed construction of splice-quotient singularities. For full
details see [
20
]. 4
WALTER D. NEUMANN AND JONATHAN WAHL
Let ( �
X, o) (C
N
, o) be a normal surface singularity whose link = �
X S
2N 1
is
a QHS. Equivalently the minimal good resolution resolves the singularity by a tree of
rational curves. Let be the resolution graph. In some cases we can construct directly
from singularities which have the same link as �
X (but might well be analytically
distinct).
We denote by A() the intersection matrix of the exceptional divisor (we say in-
tersection matrix of ); this is the