# The volume of a surface or orbifold pair

Jonathan Wahl  Department of Mathematics
The University of North Carolina
Chapel Hill, NC 27599-3250<EMAIL_ADDRESS>
###### Abstract.

A _surface pair_ $(X,C)$ is a germ of a normal surface singularity $(X,0)$ and
a sum $C=\sum c_{i}C_{i}$ of curves on $X$, with $c_{i}\in[0,1]$. An _orbifold
pair_ has $c_{i}=1/n_{i}$, as intersecting with a small sphere gives a
$3$-dimensional orbifold $(\Sigma,\gamma_{i},n_{i})$. There are natural
notions of morphism and log cover of surface pairs. We introduce a volume
$Vol(X,C)\in\mathbb{Q}_{\geq 0}$, computable from any log resolution,
analogous to that in our 1990 JAMS paper when $C=0$. Denoting
$\bar{C}=\sum(1-c_{i})C_{i}$, one has $(X,C)$ log canonical iff
$Vol(X,\bar{C})=0$. The main theorem (5.4-5.6) is that $Vol(X,C)$ is
“characteristic”: if $f:(X^{\prime},C^{\prime})\rightarrow(X,C)$ is a morphism
of degree $d$, then $Vol(X^{\prime},C^{\prime})\geq d\cdot Vol(X,C)$, with
equality if $f$ is a log cover. We prove (6.7) that
$Vol(X,\sum(1/n_{i})C_{i})=0$ iff the associated orbifold has finite or
solvable orbifold fundamental group, and these are classified. In (8.2) is
proved a key case of the DCC Volumes Conjecture: The set
$\\{Vol(X,\sum(1/n_{i})C_{i})|X\ \text{RDP}\\}$ satisfies the DCC, with
minimum non-$0$ volume 1/3528.

###### Key words and phrases:

surface singularity, surface pair, log canonical pair, orbifold pair, log
cover, orbifold fundamental group, singularity volume

###### 2000 Mathematics Subject Classification:

32S50, 14J17, 57M10, 57R18

##  1\. Introduction

Milnor and Thurston [6] define a _characteristic number_ to be a real-valued
function $\lambda$ on a class of compact $3$-manifolds with the property that
it multiplies by degree in an unramified covering. As an example, for the
class of compact hyperbolic $3$-manifolds, compatibility of the metrics in a
finite covering means _volume_ is a characteristic number. [6] defines a
$\lambda$ for all $3$-manifolds with a further important submultiplicative
property: if $f:M^{\prime}\rightarrow M$ has degree $d$, then
$\lambda(M^{\prime})\geq d\lambda(M)$, with equality if $f$ is a covering.
However, this invariant is extremely difficult to both define and to compute.

In our paper [10], we defined a characteristic number for the class of links
of normal surface singularities (NSS’s). Intersecting the germ of a NSS
$(X,0)$ with a small sphere yields its link $\Sigma$, a compact oriented
$3$-manifold. For a good resolution $(\tilde{X},E)\rightarrow(X,0)$, $\Sigma$
is the boundary of a tubular neighborhood of $E$ on $\tilde{X}$, and its
topology is completely determined by the weighted dual graph $\Gamma$ of $E$.

######  Theorem 1.1.

[10] For a NSS $(X,0)$, there is an invariant $Vol(X)$, computable from any
resolution graph $\Gamma$, for which:

  1. (1)

$Vol(X)\in\mathbb{Q}_{\geq 0}$ is a characteristic number for the link
$\Sigma$.

  2. (2)

If $f:(X^{\prime},0)\rightarrow(X,0)$ finite of degree $d$, one has

$Vol(X^{\prime})\geq d\cdot Vol(X),$

with equality if $f$ is unramified off $0$.

  3. (3)

$Vol(X)=0$ if and only if $(X,0)$ is log canonical if and only if the local
fundamental group $\pi_{1}(\Sigma)$ is finite or solvable.

  4. (4)

If $(X,0)$ is quasi-homogeneous and not log canonical, there is a natural
metric on the Seifert link $\Sigma$ whose volume is $4\pi^{2}Vol(X)$.

Using the Zariski decomposition $K_{\tilde{X}}+E=P+N$ of $\mathbb{Q}$-divisors
on any good resolution, we showed $-P\cdot P$ is a characteristic number. It
is now renamed as $Vol(X)$. In [1], Boucksom, de Fernex, and Favre define a
volume for higher dimensional isolated normal singularities, proving property
(2); the definition was much more complicated, requiring deep work of
Shokurov.

In analogy with results about the set of volumes of hyperbolic manifolds, the
following theorem was conjectured in [10] and proved by F. Ganter.

######  Theorem 1.2.

[2] The set of volumes $\\{Vol(X)|(X,0)\text{Gorenstein}\\}$ satisfies the
DCC, with smallest non-$0$ value $1/42$, for the Brieskorn singularity
$V(2,3,7)$.

The goal of the present work is to extend [10] to consideration of _surface
pairs_ $(X,C=\sum_{i=1}^{r}c_{i}C_{i})$, where $(X,0)$ is a NSS and the
“curve” $C$ consists of irreducible Weil divisors $C_{i}$ with
$c_{i}\in[0,1]\cap\mathbb{Q}$. An important special case are the _orbifold
pairs_ $(X,\sum_{i=1}^{r}(1/n_{i})C_{i})$, with $n_{i}\in\mathbb{N}$
(regrettably, called “singular pairs” in [7] and written
$(X,\sum_{i=1}^{r}n_{i}C_{i}$)). As remarked in [7], intersecting $(X,0)$ and
the $C_{i}$ with a small sphere gives an _orbifold_
$(\Sigma,\gamma_{1},\cdots,\gamma_{r},n_{1},\cdots,n_{r})$, where the
$\gamma_{i}$ are knots in $\Sigma$.

Surface pairs are studied via a _log resolution_ $\pi:(\tilde{X},\tilde{C}\cup
E)\rightarrow(X,C)$, in which $\pi^{-1}(\cup C_{i})$ has normal crossings.
Calculations can be made from an associated decorated graph $\Gamma^{*}$, as
in [7], showing the location of the proper transforms $\tilde{C_{i}}$ of the
$C_{i}$ and the associated weights $c_{i}$ (or $n_{i}$). Consider the Zariski
decomposition on $\tilde{X}$:

$K_{\tilde{X}}+E+\sum c_{i}\tilde{C}_{i}=P_{\tilde{C}}+N_{\tilde{C}}.$

In Section 3 are established basic properties:

######  Proposition (3.1).

  1. (1)

$P_{\tilde{C}}=0$ iff $(X,C)$ is log canonical.

  2. (2)

$-P_{\tilde{C}}\cdot P_{\tilde{C}}\equiv-P_{C}\cdot P_{C}\in\mathbb{Q}_{\geq
0}$ is independent of the log resolution.

  3. (3)

$C^{\prime}\leq C$ implies $-P_{C}\cdot P_{C}\leq-P_{C^{\prime}}\cdot
P_{C^{\prime}}$.

  4. (4)

$C=0$ implies $-P_{C}\cdot P_{C}=Vol(X)$.

  5. (5)

For $X$ log terminal and $C$ reduced, the log canonical threshold of $C$ is
the maximum $\epsilon$ with $-P_{\epsilon C}\cdot P_{\epsilon C}=0$

For calculations, it is easiest to use the _minimal orbifold log resolution_
of a pair, with the property that every $\tilde{C_{i}}$ intersects an end
curve of $\Gamma$, and each end intersects at most one $\tilde{C_{i}}$. In
this case, if $\Gamma^{*}$ is star-shaped, one can easily write a formula for
$-P_{C}\cdot P_{C}$ in terms of weights and lengths of chains (Proposition
$4.2$). It is important to be able to compute when all $c_{i}\geq 1/2$.

######  Theorem (4.6).

Consider $(X,\sum c_{i}C_{i})$ for which all non-$0$ $c_{i}$ are $\geq 1/2.$
If $\Gamma^{*}$ is not star-shaped, then $-P_{C}\cdot P_{C}$ can be computed
directly from the minimal orbifold log resolution, without performing a
Zariski decomposition.

In Section 5, we define maps and coverings of pairs. In [7], an orbifold
covering of $(\Sigma,\gamma_{i},n_{i})$ is a map of an orbifold to $\Sigma$
which is branched only over the $\gamma_{i}$, and with branching order $\leq
n_{i}$. That notion was adapted for maps of surface pairs, giving the

######  Definition (5.1).

A _map of surface pairs_ $f:(X^{\prime},C^{\prime})\rightarrow(X,C)$ is a
finite map $f:(X^{\prime},0)\rightarrow(X,0)$ for which $f^{*}(C)=C^{\prime}$.
A _log cover_ is a map of surface pairs which is unramified off the support of
$C^{\prime}$.

For any curve $C=\sum c_{i}C_{i}$, the _opposite curve_ is
$\bar{C}=\sum(1-c_{i})C_{i}$. The crucial definition is:

######  Definition (5.3).

For $(X,C=\sum c_{i}C_{i})$ a surface pair, the _volume of the pair_ is

$Vol(X,C)=-P_{\bar{C}}\cdot P_{\bar{C}}.$

The major result of this paper, proved primarily in 5.4 and 5.6, is

######  Theorem (5.4),(5.6).

Let $(X,C=\sum c_{i}C_{i})$ be a surface pair, with decorated resolution graph
$\Gamma^{*}$ and all $c_{i}\neq 0$.

  1. (1)

$Vol(X,C)\in\mathbb{Q}_{\geq 0}$ is computable from $\Gamma^{*}$.

  2. (2)

If $f:(X^{\prime},C^{\prime})\rightarrow(X,C)$ is a map of pairs of degree
$d$, then

$Vol(X^{\prime},C^{\prime})\geq d\cdot Vol(X,C)$,

with equality if $f$ is a log cover.

  3. (3)

If $C$ is reduced (all $c_{i}=1$), then $Vol(X,C)=Vol(X)$.

  4. (4)

For orbifold pairs $(X,\sum(1/n_{i})C_{i})$, $Vol(X,C)$ is a characteristic
number for the corresponding orbifold $(\Sigma,\gamma_{i},n_{i})$.

Recall that orbifold covers of an orbifold $(\Sigma,\gamma_{i},n_{i})$ are
controlled by an _orbifold fundamental group_ $\pi_{1}^{orb}(\Sigma)$ (6.3).
Using the basic properties above and the notion of _UALC (universal abelian
log cover)_ from [7], we prove a result analogous to the classification of log
canonical singularities:

######  Theorem (6.7).

The volume of an orbifold pair is $0$ if and only if the corresponding
orbifold fundamental group is finite or solvable.

Via computations of Section 4 and consideration of star-shaped graphs
$\Gamma^{*}$ with $3$ branches, one has:

######  Theorem (7.4).

The resolution graphs of all Volume $0$ orbifolds are classified.

An orbifold $(X,\sum(1/n_{i})C_{i})$ has volume $0$ iff
$(X,\sum(1-1/n_{i})C_{i})$ is log-canonical. Since J. Kollár has classified
log canonical surface pairs with all non-zero $c_{i}\geq 1/2$ ([4], (3.3)),
the last result could be deduced from his list. In fact, the approach of this
paper can yield his more general result as well ((7.5)-(7.8)).

Simple examples indicate that the set of volumes of Gorenstein orbifold pairs
does not satisfy the DCC. But the following seems to be a reasonable

######  DCC Volumes Conjecture.

The set of volumes of orbifold pairs $(X,C)$ on rational double points
satisfies the DCC, and has minimum non-$0$ value $1/3528=(2\cdot 42\cdot
42)^{-1}.$

It makes sense to consider first the non-trivial case of star-shaped
$\Gamma^{*}$.

######  Theorem (8.2).

Consider the set of volumes of Gorenstein orbifold pairs $(X,C)$ for which
$(X,0)$ is a rational double point and the minimal orbifold log resolution is
star-shaped. Then this set satisfies the DCC, with minimum non-$0$ volume
$1/3528$.

At present, we see no metric interpretation of $Vol(X,C)$ for an orbifold pair
when $(X,0)$ and the $C_{i}$ are quasi-homogeneous.

In a separate work, writing $C=\sum c_{i}C_{i}$, we shall discuss the behavior
of the function $-P_{C}\cdot P_{C}$ as a function of $(c_{1},\cdots,c_{r})$
(it is piecewise-quadratic). This is particularly relevant for $cC$, where $C$
is a reduced plane curve germ.

We are grateful to Enrique Artal Bartolo for several important comments, and
to the organizers (including NSF) of the 115AM Jaca Conference in June, 2023.

##  2\. Review of Zariski decomposition and [10]

Let $(\tilde{X},E)\rightarrow(X,0)$ be a good resolution of a normal surface
singularity. The adjoint homomorphism

$\text{Pic}\ \tilde{X}\rightarrow\bigoplus\mathbb{Q}\cdot
E_{i}\cong\mathbb{E}_{\mathbb{Q}},$

sends a line bundle $\mathcal{L}$ to the exceptional $\mathbb{Q}$-divisor
$\Sigma a_{i}E_{i}$ satisfying

$\mathcal{L}\cdot E_{j}=(\Sigma a_{i}E_{i})\cdot E_{j},\ \text{all}\ j.$

Denote the image of $\mathcal{L}$ by $L$, while $K$ denotes both the canonical
line bundle and its image; similarly, use the same notation for a curve
$\tilde{C}$ on $\tilde{X}$ and its image under the map. By abuse of notation,
we shall write equality ($=$) when we really mean numerical equivalence
($\equiv$) of exceptional $\mathbb{Q}$-divisors.

######  Proposition 2.1.

(Sakai [9], p. 408). Let $\L$ be an exceptional $\mathbb{Q}$-divisor. Then
there exists a unique Zariski-decomposition $L=P+N$ in
$\mathbb{E}_{\mathbb{Q}}$, where

  1. (1)

$P$ is nef, i.e., $P\cdot E_{i}\geq 0$, all $i$.

  2. (2)

$N$ is effective, i.e., $N=\Sigma a_{i}E_{i}$, with all $a_{i}\geq 0$.

  3. (3)

$P\cdot N=0$, i.e., $P\cdot E_{i}=0$ for all $E_{i}\subset\text{Supp}\ N$.

######  Remark 2.2.

The usual inductive approach to constructing $N$ from $L$ starts with letting
$\mathcal{A}_{1}=\\{j|\ L\cdot E_{j}<0\\}$, and defining $N_{1}=\Sigma
b_{j}E_{j}$ ($j\in\mathcal{A}_{1}$) so that $L\cdot E_{j}=N_{1}\cdot E_{j}$,
all $j\in\mathcal{A}_{1}$. If $P_{1}=L-N_{1}$ is nef, we are done; if not,
$\mathcal{A}_{2}$ and $N_{2}$ are needed. This would occur if a neighbor
$E_{k}$ of the support of $N_{1}$ dots to $0$ with $L$, which curve would have
to be included in $N_{2}$. So, an “improved” $N_{1}$ can be defined as
follows: If $L$ is not nef, consider the union of all $E_{i}$ such that
$L\cdot E_{i}\leq 0$, and pick out the connected components which contain a
curve with $L\cdot E_{j}<0$. Let $\mathcal{A}_{1}$ correspond to all the
$E_{i}$ in such components. Choose $N_{1}$ as above using this
$\mathcal{A}_{1}$, so that $P\cdot E_{k}=N_{1}\cdot E_{k}$ for all
$k\in\mathcal{A}_{1}$. Often, $P_{1}=L-N_{1}$ will be nef. If not, continue as
above to find $N_{2}$.

######  Remark 2.3.

The Zariski decomposition $P+N$ of $L$ can be constructed directly from the
knowledge of Supp $N$. For, $N$ is the linear combination of the curves in its
support satisfying $L\cdot E_{i}=N\cdot E_{i}$ for all such $E_{i}$.

We later use some results proved in [10]:

  1. (1)

Zariski decomposition is preserved under pullback of a generically finite and
proper surjective map $f:(\tilde{Y},F)\rightarrow(\tilde{X},E)$, since
$f^{*}:\mathbb{E}_{\mathbb{Q}}\rightarrow\mathbb{F}_{Q}$ preserves both
effective and nef divisors, and multiples intersection numbers by deg($f$).

  2. (2)

Suppose $\mathcal{L}$ is a line bundle on $\tilde{X}$, with $L=P+N$ the
associated Zariski decomposition of $\mathbb{Q}$-divisors. Then for $n>>0$,

$\text{dim}\
H^{0}(\tilde{X}-E,\mathcal{L}^{n})/H^{0}(\tilde{X},\mathcal{L}^{n})=n^{2}/2(-P\cdot
P)+O(n).$

  3. (3)

Suppose $\mathcal{L}^{\prime}=\mathcal{L}(-D)$, where $D$ is an effective
exceptional divisor. If $L^{\prime}=P^{\prime}+N^{\prime}$, then

$-P\cdot P\leq-P^{\prime}\cdot P^{\prime}.$

The third statement follows from the second because
$H^{0}(\tilde{X},\mathcal{L}^{\prime n})\subset
H^{0}(\tilde{X},\mathcal{L}^{n}).$

We need a slight generalization of the third statement.

######  Proposition 2.4.

Let $L=P+N$ and $L^{\prime}=P^{\prime}+N^{\prime}$ be Zariski decompositions
of two $\mathbb{Q}$-divisors. Assume there is an effective
$\mathbb{Q}$-divisor $D$ with $L^{\prime}=L-D.$ Then

$-P\cdot P\leq-P^{\prime}\cdot P^{\prime}.$

###### Proof.

Multiplying a $\mathbb{Q}$-divisor by an integer $k$ multiplies its $-P\cdot
P$ by $k^{2}$. So we can multiply $L$ and $D$ by $k$ and assume the divisors
$L$ and $L^{\prime}$ are integral and $D$ is integral and effective. Define
$\mathcal{L}=\mathcal{O}(L)$ and $\mathcal{L}^{\prime}=\mathcal{L}(-D)$ and
apply the third statement above. ∎

In [10], we considered the Zariski decomposition $K_{\tilde{X}}+E=P+N$ on a
good resolution $\tilde{X}$, of $(X,0)$, and showed $-P\cdot P\equiv Vol(X)$
is a non-negative rational number independent of the resolution. On the
minimal good resolution of $(X,0)$, it is straighforward to write down $N$,
thus enabling calculation of $Vol(X)$; this allowed recovery of Kawamata’s
classification in [3] of the log canonical singularities, where $P=0$ (i.e.,
$K_{\tilde{X}}+E$ is numerically effective). The key properties of $Vol(X)$
are listed in Theorem 1.1 in the Introduction.

##  3\. Surface pairs and their $-P\cdot P$

We consider a surface pair $(X,C)$, where $(X,0)$ is a NSS and
$C=\sum_{i=1}^{r}c_{i}C_{i}$ is a finite sum of irreducible Weil divisors,
with $c_{i}\in[0,1]\cap\mathbb{Q}$. If
$C^{\prime}=\sum_{i=1}^{r}c^{\prime}_{i}C_{i}$, write $C^{\prime}\leq C$ if
$c^{\prime}_{i}\leq c_{i}$ for all $i$. $C=0$ means all $c_{i}=0$; $C$
_reduced_ means all non-$0$ $c_{i}$ equal $1$. A _log resolution_ of $(X,C)$
is a good resolution $\pi:(\tilde{X},E)\rightarrow(X,0)$ for which
$E\cup\pi^{-1}(\cup C_{i})$ has strong normal crossings. Denote the proper
(sometimes called birational) transform of $C_{i}$ on $\tilde{X}$ by
$\tilde{C}_{i}$. As each $\tilde{C}_{i}$ dots non-negatively with every
$E_{j}$, and positively with at least one such, we conclude that
$\tilde{C}_{i}$ is numerically equivalent to a $\mathbb{Q}$-divisor all of
whose coefficients are negative rational numbers.

Writing

$K_{\tilde{X}}+\sum c_{i}\tilde{C}_{i}\equiv\sum a_{i}E_{i},$

the $a_{i}$ are the _discrepancies_ of the pair $(X,C)$. Often one assumes $K$
and the $C_{i}$ are $\mathbb{Q}$-Cartier, but this is not necessary.

$(X,C)$ is called _log canonical_ if all $a_{i}\geq-1$, _log terminal_ if all
$a_{i}>-1.$

As in [10], it is natural to consider instead the $\mathbb{Q}$-divisor

$K_{\tilde{X}}+E+\sum c_{i}\tilde{C}_{i}$

and its Zariski decomposition

$K_{\tilde{X}}+E+\sum c_{i}\tilde{C}_{i}=P_{\tilde{C}}+N_{\tilde{C}}.$

Then log canonicality is equivalent to $P_{\tilde{C}}=0$, and log terminality
to the additional requirement that $N_{\tilde{C}}$ is strictly effective.

######  Proposition 3.1.

The non-negative rational number $-P_{\tilde{C}}\cdot P_{\tilde{C}}$ is
independent of the log resolution of the pair $(X,C)$.

###### Proof.

(Cf.[10], (2.7)). Let $\pi:(X^{\prime},F)\rightarrow(\tilde{X},E)$ be the
blow-up along a point $q$ of $E$, adding an exceptional $-1$-curve $F_{1}$ to
$F$. Then one has

$K_{X^{\prime}}+F=\pi^{*}(K_{\tilde{X}}+E)+\delta F_{1},$

where $\delta=1$ if $q$ is a smooth point of $E$, and $\delta=0$ if $q$ is a
double point. Further, if $C^{\prime}_{i}$ is the proper transform of
$\tilde{C}_{i}$, then

$C^{\prime}_{i}=\pi^{*}(\tilde{C}_{i})-\epsilon_{i}F_{1},$

where $\epsilon_{i}=1$ if $q\in\tilde{C}_{i}$ and is $0$ otherwise. Putting
these together gives

$K_{X^{\prime}}+F+\sum c_{i}C^{\prime}_{i}=\pi^{*}(K_{\tilde{X}}+E+\sum
c_{i}\tilde{C}_{i})\ +\ (\delta-\sum\epsilon_{i}c_{i})F_{1}.$

Note the coefficient of $F_{1}$ on the right is non-negative in all cases
(e.g., equal $1-c_{i}\geq 0$ if $q\in\tilde{C}_{i}$, $0$ if $q$ is a double
point). Since $\pi^{*}(P_{\tilde{C}})\cdot F_{1}=0$ and Zariski decomposition
is preserved under pull-back, one has on $X^{\prime}$ a new Zariski
decomposition

$K_{X^{\prime}}+F+\sum
c_{i}C^{\prime}_{i}=\pi^{*}(P_{\tilde{C}})+(\pi^{*}(N_{\tilde{C}})+(\delta-\sum\epsilon_{i}c_{i})F_{1}).$

It follows that that $\pi^{*}(P_{\tilde{C}})=P_{C^{\prime}}$, whence the
assertion. ∎

######  Definition 3.2.

For a surface pair $(X,C)$, let _$-P_{C}\cdot P_{C}$_ be the quantity
$-P_{\tilde{C}}\cdot P_{\tilde{C}}$ computed from any log resolution.

It will be convenient to consider only log resolutions of $(X,C)$ for which
every $\tilde{C_{i}}$ intersects an end curve of the graph, with at most one
$\tilde{C_{i}}$ per end-curve. As in [7], (6.4), there exists a minimal such
log resolution (except when $E$ consists of one curve and two $\tilde{C_{i}}$
intersecting it at two points), called there the _minimal orbifold
resolution_. We can restrict our attention to this resolution. In this case,
the line bundle defined by $\tilde{C}_{i}$, which intersects only $E_{i}$, is
numerically equivalent to minus the effective $\mathbb{Q}$-divisor $e_{i}$
satisfying $e_{i}\cdot E_{j}=-\delta_{ij}$.

######  Proposition 3.3.

$C^{\prime}\leq C$ implies $-P_{C^{\prime}}\cdot P_{C^{\prime}}\leq-P_{C}\cdot
P_{C}$.

###### Proof.

Adding a multiple of $\tilde{C_{i}}$ is equivalent to subtracting an effective
divisor, so the result follows from Proposition 2.4. ∎

The log canonical pairs with $C$ reduced ($c_{i}=0$ or $1$) were classified
completely by Kawamata [3], as previously mentioned. J. Kollár has listed
([4],(3.3)) the examples with all $c_{i}\geq 1/2$, and we indicate how to
rederive this result in $((7.5)-(7.8))$ below. Calculations are much harder
for smaller non-$0$ values of $c_{i}$.

Because each $\tilde{C}_{i}$ is equivalent to a strictly negative divisor, one
concludes

######  Lemma 3.4.

  1. (1)

If $(X,C)$ is log canonical, and $C^{\prime}<C$, then $(X,C^{\prime})$ is log
terminal

  2. (2)

If $(X,0)$ is log terminal, $C$ arbitrary, then for sufficiently small
$\epsilon>0$, $(X,\epsilon C)$ is log terminal.

###### Proof.

If $D_{1}$ and $D_{2}$ are strictly positive divisors, then for all small
$\epsilon$, $D_{1}-\epsilon D_{2}$ is strictly positive. ∎

######  Definition 3.5.

If $(X,0)$ is log terminal, C a reduced curve, then the _log canonical
threshhold (or LCT) of $(X,C)$_ is the largest $\epsilon\leq 1$ for which
$(X,\epsilon C)$ is log canonical.

##  4\. Examples with simplest graphs

######  Lemma 4.1.

Suppose $(X,C)$ has a log resolution $\tilde{X}$ for which $E$ is a chain of
smooth rational curves, and $\tilde{C}_{i}$ ($i=1,2$) intersect the ends
transversally:

|  |  |
---|---|---|---
$\textstyle{\tilde{C_{1}}\leftarrow}$ $\bullet$ $\bullet$
$\bullet$$\textstyle{\rightarrow\tilde{C_{2}}}$

Then $(X,c_{1}C_{1}+c_{2}C_{2})$ is log-terminal, except that when
$c_{1}=c_{2}=1$ it is only log-canonical.

###### Proof.

If $D$ is an exceptional $\mathbb{Q}$-divisor satisyfing $D\cdot E_{i}\leq 0$
for all $i$, then $D$ is either $0$ or is strictly positive. Dotting
$K_{\tilde{X}}+E+\sum_{i=1}^{2}c_{i}\tilde{C}_{i}$ with an interior curve is
$0$; dotting with an end gives $c_{i}-1$ if the chain is not a single curve,
or $c_{1}+c_{2}-2$ in case $E$ is one curve. ∎

From now one we shall pretty much exclude this simple case from consideration.

To find Zariski decomposition on the minimal orbifold resolution
$(\tilde{X},\tilde{C}\cup E)\rightarrow(X,C)$, it is clear from [10] (2.3) one
must consider maximal strings of rational curves $E_{1},\cdots,E_{s}$
emanating from the graph $\Gamma$. Here $E_{i}\cdot E_{i+1}=1,\ 1\leq i<s$;
$E_{2},\cdots,E_{s}$ have no other intersections in $E$ (so $E_{s}$ is an end-
curve); and $E_{1}$ intersects exactly one other curve, either of positive
genus or valence at least $3$. Writing $a_{j}=-E_{j}\cdot E_{j}$, one has a
continued fraction $m/q=a_{1}-1/a_{2}-\cdots-1/a_{s}$; by minimality, all
$a_{j}\geq 2$, unless the string consists of one $-1$ curve (when $m=1$). The
expansion going in the other direction, starting from $a_{s}$, corresponds to
$m/q^{\prime}$, where $qq^{\prime}\equiv 1$ mod $m$. To an $m/q$ string we
associate effective $\mathbb{Q}$-cycles:

  * •

$D=\sum_{j=1}^{s}b_{j}E_{j}$, with $D\cdot E_{s}=-1$ and $D\cdot E_{j}=0,j<s$;
then $b_{1}=1/m$ and $b_{s}=q^{\prime}/m$.

  * •

$D^{\prime}=\sum_{j=1}^{s}b^{\prime}_{j}E_{j}$, with $D^{\prime}\cdot
E_{1}=-1$ and $D^{\prime}\cdot E_{j}=0,j>1$; then $b^{\prime}_{1}=q/m$ and
$b^{\prime}_{s}=1/m.$

Since $K+E$ dots to $-1$ with the end-curve $E_{s}$ and to $0$ with other
curves in the string, the $N$ in its Zariski decomposition must include the
cycle $D$.

Consider a star-shaped resolution graph, consisting of: a smooth central curve
$F$ of genus $g$ and self-intersection $-d$, and $t$ strings $S_{i}$ with
invariants $m_{i}/q_{i}$, emanating from $F$.

|  |  |  |  |
---|---|---|---|---|---
|  |  |  |  |
$\textstyle{\hbox to0.0pt{\vbox to0.0pt{\vss\hbox to0.0pt{\hss$\scriptstyle
m_{2}/q_{2}$\hskip
8.0pt}\vss}}\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$.. ..
. .. ..$\textstyle{\hbox to0.0pt{\vbox to0.0pt{\vss\hbox to0.0pt{\hskip
8.0pt$\scriptstyle
m_{t-1}/q_{t-1}$\hss}\vss}}\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hbox
to0.0pt{\vbox to0.0pt{\vss\hbox to0.0pt{\hss$\scriptstyle m_{1}/q_{1}$\hskip
8.0pt}\vss}}\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\scriptstyle-d$ $\circ$
$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hbox
to0.0pt{\vbox to0.0pt{\vss\hbox to0.0pt{\hskip 8.0pt$\scriptstyle
m_{t}/q_{t}$\hss}\vss}}\bullet}$$\textstyle{F}$

Define the familiar invariants

$\chi=2g-2+t-\sum_{i=1}^{t}(1/m_{i})$
$\epsilon=d-\sum_{i=1}^{t}(q_{i}/m_{i}).$

A $-1$ string (with $m_{i}=q_{i}=1$) collapsed to a point leaves $\chi$ and
$\epsilon$ unchanged. Recall $e>0$; $\chi<0$ exactly for quotient
singularities; $\chi=0$ only for the familiar log canonical star-shaped
graphs; and in the Gorenstein case there is an integer $t$ with $\chi=te.$. It
is easy to see that if $\chi>0$, then $\chi\geq 1/42$, which occurs for
$g=0,t=3,\\{m_{1},m_{2},m_{3}\\}=\\{2,3,7\\}$.

Noting that $F\cdot D^{\prime}_{i}=q_{i}/m_{i}$, one has that $F+\sum
D^{\prime}_{i}$ dots with $F$ to $-\epsilon$ and to $0$ with all other curves.
Thus $f$, the effective $\mathbb{Q}$-cycle dotting to $-1$ with $F$ and $0$
with other curves, satisfies

(3.5.1) $F+\sum_{i=1}^{t}D^{\prime}_{i}\equiv\epsilon f.$

One concludes that $f\cdot f=-1/\epsilon.$

Suppose now that $(X,\sum_{i=1}^{t}c_{i}C_{i})$ has star-shaped minimal
orbifold log resolution $(X,E)$ as above, where $\tilde{C_{i}}$ intersects the
end of the $i^{th}$ string. One has a decorated graph $\Gamma^{*}$, where
$\tilde{C_{i}}$ is represented by an arrow emerging from the end of the
string, with a $c_{i}$ marked over it (e.g., (4.5) below). Assume at least one
$c_{i}\neq 0$, so $t\geq 1$; $m_{i}=1$ implies $c_{i}\neq 0$; and $t\geq 3$ if
$g=0$.

######  Proposition 4.2.

With $(X,\sum_{i=1}^{t}c_{i}C_{i})$ as above, define

$\chi_{C}=\chi+\sum_{i=1}^{t}c_{i}/m_{i}.$

Then $(X,C)$ is

  1. (1)

log terminal iff $\chi_{C}<0$

  2. (2)

log canonical iff $\chi_{C}\leq 0$

  3. (3)

not log canonical iff $\chi_{C}>0$, in which case

$-P_{C}\cdot P_{C}=(\chi_{C})^{2}/\epsilon.$

###### Proof.

If $D_{j}$ is the cycle above associated to $S_{j}$, then $(1-c_{j})D_{j}$ is
effective and dots to $0$ with $K+E+\sum c_{i}\tilde{C}_{i}$; further,
$D_{j}\cdot F=b_{1}=1/m_{j}.$ With $f$ as above, one checks that

$K+E+\sum c_{i}\tilde{C}_{i}=(-\chi_{C})f+\sum(1-c_{i})D_{i},$

as seen by dotting each side with an end of $S_{i}$, an interior curve of
$S_{i}$, and with $F$. $f$ is an effective cycle, and is the only cycle on the
right whose support contains $F$.

The assertions about $\chi_{C}<0$ and $\chi_{C}\leq 0$ now follow, because the
right-hand side is effective (or $0$).

If $\chi_{C}>0$, then $(-\chi_{C})f$ is a positive cycle, so equals
$P_{\tilde{C}}$, the positive part of the Zariski decomposition of $K+E+\sum
c_{i}\tilde{C}_{i}$. As $f\cdot f=-1/\epsilon$, one has $-P_{C}\cdot
P_{C}=(\chi_{C})^{2}/\epsilon$.

∎

For log canonicality of $(X,C)$ $(C\neq 0)$ in star-shaped cases, one must
have $(X,0)$ log terminal, hence a quotient singularity (or smooth). Then
$-P_{C}\cdot P_{C}$, as a function of the $c_{i}$’s, is $0$ in the region
$\sum c_{i}/m_{i}\leq-\chi$ (and $0\leq c_{i}\leq 1$, all $i$), and given by
the quadratic polynomial $(\chi+\sum c_{i}/m_{i})^{2}$ outside that region.
Note log canonical examples can have many strings of $-1$ curves, but at most
$3$ whose determinant (i.e., $m_{i}$) is greater than $1$. One easily deduces
the following:

######  Corollary 4.3.

Assume $(X,C)$ as above has all $c_{i}$ equal to $0$ or $1$. If $(X,C)$ is log
canonical, then the graph is

|  |
---|---|---
|
$\textstyle{\hbox to0.0pt{\vbox to0.0pt{\vss\hbox to0.0pt{\hskip
6.0pt$\scriptstyle-2$\hss}\vss}}\bullet}$$\textstyle{\hbox to0.0pt{\vbox
to0.0pt{\vss\hbox to0.0pt{\hss$\scriptstyle-2$\hskip
4.0pt}\vss}}\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\bullet$ $\bullet$ $\bullet$$\textstyle{\rightarrow\tilde{C}}$

######  Corollary 4.4.

Assume $(X,0)$ is log terminal. Then excluding the example of Corollary 4.3,
the log canonical threshhold of the reduced curve
$C=\sum_{i=1}^{t^{\prime}}C_{i}$ (for some $t^{\prime}\leq t$) is

$[2-\sum_{i=1}^{t}(1-1/m_{i})]/\sum_{i=1}^{t^{\prime}}(1/m_{i}).$

The complete list of examples for which the log canonical threshhold is at
least $1/2$ can be determined from Proposition $7.7$ below..

######  Example 4.5.

Consider the pair $(\mathbb{C}^{2},\text{c}C)$, where $(C,0)$ has an ordinary
cusp singularity. Recalling that an unmarked vertex is a rational $-2$ curve,
the decorated graph for the minimal orbifold log resolution is

|  |
---|---|---
$\textstyle{\hbox to0.0pt{\vbox to0.0pt{\vss\hbox to0.0pt{\hskip
6.0pt$\scriptstyle$\hss}\vss}}\bullet}$$\textstyle{c}$ $\scriptstyle-3$
$\bullet$ $\bullet$ $\scriptstyle-1$ $\bullet$ $\scriptstyle\tilde{C}$

Since $\chi=-5/6$, one has log canonicality only up to $(5/6)C$. For $5/6\leq
c\leq 1$, $-P_{C}\cdot P_{C}$ equals $(c-5/6)^{2}$.

Returning to the general case, the minimal orbifold log resolution
$(\tilde{X},E+\tilde{C})$ of $(X,C)$ has maximal strings if $C\neq 0$, hence
associated cycles $D_{j}$ and $D^{\prime}_{j}$. The effective divisor
$N_{1}=\sum(1-c_{j})D_{j}$ is perpendicular to $P_{1}=K+E+\sum
c_{j}\tilde{C}_{j}-N_{1}$, so is a first step in constructing the negative
part of the Zariski decomposition. If $P_{1}$ is nef, then $P_{1}+N_{1}$ is in
fact this decomposition. Otherwise, one needs to find an $N_{2}$, as described
in Remark $2.2$.

######  Theorem 4.6.

Let $(\tilde{X},E+\tilde{C})$ be the minimal orbifold log resolution of
$(X,C=\sum c_{i}C_{i})$, which is not star-shaped. Suppose that all non-$0$
$c_{i}$ are $\geq 1/2$. Then

  1. (1)

$N\equiv\sum(1-c_{i})D_{i}$ is the negative part of the Zariski decomposition
of $K+E+\sum c_{i}\tilde{C}_{i}$.

  2. (2)

If $(X,C)$ is log canonical, then the graph of $E$ is

|  |  |  |
---|---|---|---|---
|  |  |
$\textstyle{\hbox to0.0pt{\vbox to0.0pt{\vss\hbox to0.0pt{\hskip
6.0pt$\scriptstyle$\hss}\vss}}\circ}$$\textstyle{\hbox to0.0pt{\vbox
to0.0pt{\vss\hbox to0.0pt{\hskip 6.0pt$\scriptstyle$\hss}\vss}}\circ}$ $\circ$
$\bullet$ $\bullet$ $\bullet$ $\bullet$ $\circ$,

Each $\circ$ is either a $-1$ curve with an arrow sticking out, corresponding
to a $\tilde{C}_{i}$ with coefficient $c_{i}=1/2$; or, a $-2$ exceptional
curve.

###### Proof.

We prove that $P=K+E+\sum c_{i}\tilde{C}_{i}-N$ is nef. By construction, it
dots to $0$ with every curve in a string (in particular, $P\cdot N=0$). Next,
consider a curve $F$ not in a string but intersecting $S_{1},\cdots,S_{t}$,
with invariants $m_{i}/q_{i},1\leq i\leq t$. Assume $F$ has genus $g$ and
valency $t^{\prime}$ in the graph, with $t^{\prime}>t$ because the star-shaped
case is excluded. Then

$P\cdot F=2g-2+t^{\prime}-\sum_{i=1}^{t}(1-c_{i})/m_{i}.$

If $g>0$, then $P\cdot F>0$. If $t^{\prime}\geq t+2$, then $P\cdot F\geq 0$.
So consider the case $g=0$ and $t^{\prime}=t+1$. Further, $t>1$, else $F$
would be part of a string. Now examine

$P\cdot F=-1+\sum_{i=1}^{t}(1-(1-c_{i})/m_{i}).$

The case $m_{i}=1$ corresponds to a $-1$ string, so $c_{i}\geq 1/2$ by
assumption. Then for this term, $1-(1-c_{i})/1\geq 1/2$. If $m_{i}\geq 2$,
then the corresponding term is also $\geq 1/2$. As $t\geq 2$, one has $P\cdot
F\geq 0$.

If $F$ is not in or adjacent to a string, clearly $P\cdot F\geq 0$. Thus, $P$
is nef.

Assume now that $P=0$. Returning to $F$ as above intersecting $t$ strings, one
must have $g_{F}=0$. As noted, each $(1-c_{i})/m_{i}\leq 1/2$, so $P\cdot F=0$
implies $t^{\prime}<t+2$. Thus $t^{\prime}$ must equal $t+1$, and $t\geq 2$
(else $F$ would be part of a string). One concludes that

$1=\sum_{i=1}^{t}(1-(1-c_{i})/m_{i}).$

Since each term in the sum is at least $1/2$, we conclude that $t=2$ and each
term equals $1/2$. Thus, either $m_{i}=1$ and $c_{i}=1/2$, or $m_{i}=2$ and
$c_{i}=0$.

If $F$ is a curve which is not on a string and does not intersect any, then
$P\cdot F=0$ implies that $g_{F}=0$ and $t^{\prime}=2$. Thus the resolution
graph is as claimed.

∎

######  Remark 4.7.

The $5$ possible configurations of the $\circ$’s are described from left to
right via $1$’s and $2$’s, e.g., specifically $1111$; $1112$; $1122$; $1212$;
and $1222$.

##  5\. maps and log covers of pairs

An orbifold pair $(X,\sum(1/n_{i})C_{i})$ gives rise on the boundary to an
orbifold
$\tilde{\Sigma}=(\Sigma,\gamma_{1},\cdots,\gamma_{r},n_{1},\cdots,n_{r})$. An
orbifold cover of $\tilde{\Sigma}$ is a finite map to $\Sigma$ which is
unramified except over the $\gamma_{i}$, over which it is branched with
ramification order $\leq n_{i}$. (These facts are recalled in [7],§$1$).
Analogously, Definition $2.1$ of [7] says a _log cover of orbifold pairs_
$f:(Y,C^{\prime})\rightarrow(X,C=\sum_{i=1}^{r}(1/n_{i})C_{i})$ is a finite
map of normal germs $f:Y\rightarrow X$ so that

  1. (1)

$f^{-1}(|C|)=|C^{\prime}|$, and $f$ is an unramified covering off this set.

  2. (2)

Writing $f^{*}C_{i}=\sum_{j=1}^{s_{i}}m_{ij}C^{\prime}_{ij}$, one has
$m_{ij}|n_{i}$, all $j$.

  3. (3)

$C^{\prime}=\sum_{i=1}^{r}\sum_{j=1}^{s_{i}}(m_{ij}/n_{i})C^{\prime}_{ij}$.

The conditions $(2)$ and $(3)$ say that $f^{*}(C)=C^{\prime}.$ We extend the
above definition to

curves $(X,C=\sum c_{i}C_{i})$, where $c_{i}\in[0,1]\cap\mathbb{Q}$; if
$f:Y\rightarrow X$ has $f^{*}(C_{i})=\sum_{j=1}^{s_{i}}m_{ij}C^{\prime}_{ij}$,
define $C^{\prime}=f^{*}(C)=\sum c_{i}m_{ij}C^{\prime}_{ij}$ as before, but
only if all coefficients $c_{i}m_{ij}\leq 1$.

######  Definition 5.1.

A _map of pairs_ is a finite morphism $f:(Y,C^{\prime}=\sum
c^{\prime}_{j}C^{\prime}_{j})\rightarrow(X,C=\sum c_{i}C_{i})$ for which
$f^{*}(C)=C^{\prime}$. A _log cover of pairs_ is a map of pairs
$f:(Y,C^{\prime})\rightarrow(X,C)$ which is unramified off
$f^{-1}(\text{Supp}\ C)$.

From a finite $f:Y\rightarrow X$ there is a natural way to make maps and
covers of pairs.

######  Proposition 5.2.

[7],(2.2) Suppose $f:Y\rightarrow X$ is finite, and $C_{1},\cdots,C_{r}$
irreducible curves on $X$. Writing
$f^{*}(C_{i})=\sum_{j=1}^{s_{i}}m_{ij}C^{\prime}_{ij}$ and
$n_{i}=lcm(m_{ij})$, define $C=\sum(1/n_{i})C_{i}$ and
$C^{\prime}=\sum_{i=1}^{r}\sum_{j=1}^{s_{i}}m_{ij}/n_{i}\ C^{\prime}_{ij}$.
Then

  1. (1)

$f:(Y,C^{\prime})\rightarrow(X,C)$ is a map of pairs.

  2. (2)

If $f$ is unramified off $f^{-1}(\cup C_{i})$, then $f$ is a log cover of
pairs.

Section $6$ below recalls basic ways to construct log covers of a given pair
$(X,C)$.

We need a simple but important definition. If $C=\sum_{i=1}^{r}c_{i}C_{i}$,
with all $c_{i}>0$, we define the _opposite curve_ $\bar{C}$ of $C$ by

$\bar{C}=\sum_{i=1}^{r}(1-c_{i})C_{i}.$

######  Definition 5.3.

The _characteristic number of a pair_ $(X,C)$ is

$\text{ch}(X,C)\equiv-P_{\bar{C}}\cdot P_{\bar{C}}.$

This quantity is computed from a log resolution
$(\tilde{X},E)\rightarrow(X,C)$ via the Zariski decomposition

$K_{\tilde{X}}+E+\sum_{i=1}^{r}(1-c_{i})\tilde{C}_{i}=P_{\bar{C}}+N_{\bar{C}}.$

Note that characteristic numbers satisfy:

  1. (1)

If $C^{\prime}\leq C$ on $X$, then
$\text{ch}(X,C)\leq\text{ch}(X,C^{\prime})$.

  2. (2)

If $C$ is reduced (i.e., all $c_{i}=1$), then $\text{ch}(X,C)=-P_{X}\cdot
P_{X}=Vol(X)$.

######  Theorem 5.4.

Suppose $f:(Y,C^{\prime})\rightarrow(X,C)$ is a log cover of pairs of degree
$d$. Then

$(-P_{\bar{C^{\prime}}}\cdot P_{\bar{C^{\prime}}})=d(-P_{\bar{C}}\cdot
P_{\bar{C}}).$

###### Proof.

We follow the proof in [10], Theorem 2.8. Let $(\tilde{X},E)\rightarrow(X,C)$
be the minimal log resolution so that every $\tilde{C}_{i}$ intersects a
rational end-curve $E_{i}$ of $E$, and $E_{i}$ intersects at most one such
curve. Denote by $\tilde{C}$ the reduced sum of the $\tilde{C_{i}}$. Let
$f^{\prime}:Y^{\prime}\rightarrow\tilde{X}$ be the normalization of
$\tilde{X}$ in the quotient field of $(Y,0)$. This finite map is unramified
off the inverse image of $E\cup\tilde{C}$, and $Y^{\prime}$ has only cyclic
quotient singularities. Above the intersection point $P_{i}=\tilde{C}_{i}\cap
E$ are $s_{i}$ cyclic quotient singularities (or smooth points), at points
$P_{ij}$ on some Weil divisor $\tilde{C}^{\prime}_{ij}$ on $Y^{\prime}$. A
local equation defining $\tilde{C}_{i}$ on $\tilde{X}$ has a zero of
multiplicity $m_{ij}$ along $\tilde{C}^{\prime}_{ij}$.

Let $\tilde{Y}\rightarrow Y^{\prime}$ be the result of first minimally
resolving all the cyclic quotient singularities on $Y^{\prime}$, and then
insuring that the proper transforms $\tilde{C}^{\prime\prime}_{ij}$ of the
$\tilde{C}^{\prime}_{ij}$ intersect transversally a unique end of the reduced
exceptional fibre $F$ of the map $\tilde{Y}\rightarrow Y$.

Letting $\tilde{f}:\tilde{Y}\rightarrow Y^{\prime}\rightarrow\tilde{X}$ be the
induced map, we conclude that

$\tilde{f}^{*}(\tilde{C}_{i})=\sum(m_{ij}\tilde{C}^{\prime\prime}_{ij}+F_{ij}),$

where $F_{ij}$ is a (not necessarily reduced) effective exceptional cycle
collapsed to a point on $Y^{\prime}$.

Since $\tilde{f}$ is proper and generically finite and unramified off the
inverse image of the normal crossings divisor $E+\tilde{C}$ , it follows
(e.g., [10], proof of (2.8)) that

$\tilde{f}^{*}(K_{\tilde{X}}+E+\tilde{C})=K_{\tilde{Y}}+F+\tilde{C}^{\prime\prime},$

where as before $\tilde{C}^{\prime\prime}$ denotes the reduced sum of the
$\tilde{C}^{\prime\prime}_{ij}$. Therefore,

$\tilde{f}^{*}(K_{\tilde{X}}+E+\sum_{i=1}^{r}(1-c_{i})\tilde{C}_{i})=K_{\tilde{Y}}+F+\sum_{i=1}^{r}\sum_{j=1}^{s_{i}}(1-c_{i}m_{ij})\tilde{C}^{\prime\prime}_{ij}-\sum_{i=1}^{r}\sum_{j=1}^{s_{i}}c_{i}F_{ij}.$

As the left hand side is $\tilde{f}^{*}(P_{\bar{C}}+N_{\bar{C}})$, we can
rewrite as

$\tilde{f}^{*}(P_{\tilde{C}})+(\tilde{f}^{*}(N_{\tilde{C}})+\sum_{i,j}c_{i}F_{ij})=K_{\tilde{Y}}+F+\sum_{i,j}(1-c_{i}m_{ij})\tilde{C}^{\prime\prime}_{ij}.$

The Zariski decomposition of the right-hand side is
$P_{\bar{C^{\prime}}}+N_{\bar{C^{\prime}}}$. We claim that the left-hand side
is also a Zariski decomposition. First, the purportedly negative part is
effective. Next, $\tilde{f}$ factors through $Y^{\prime}$, so that
$\tilde{f}^{*}$ of any $\mathbb{Q}$-divisor on $\tilde{X}$ dots to $0$ with
all components of each cycle $F_{ij}$. So, the two summands are orthogonal.
Therefore, $\tilde{f}^{*}(P_{\bar{C}})=P_{\bar{C}^{\prime}}$. Self-
intersections multiply by the degree of $\tilde{f}$ under pull-back, hence the
result claimed.

∎

######  Corollary 5.5.

The characteristic number of a pair multiplies by degree in a log cover.

######  Theorem 5.6.

Suppose $f:(Y,C^{\prime})\rightarrow(X,C)$ is a map of pairs of degree $d$.
Then

$(-P_{\bar{C^{\prime}}}\cdot P_{\bar{C^{\prime}}})\geq d(-P_{\bar{C}}\cdot
P_{\bar{C}}).$

###### Proof.

We again follow the proof of [10], Theorem $2.9$. Suppose
$\tilde{f}:\tilde{Y}\rightarrow\tilde{X}$ is as in the proof of Theorem 5.5
above, but we may also assume that both branch and ramification loci have
strong normal crossings divisors. Let $R\subset\tilde{X}$ denote the reduced
sum of those components of the branch locus of $\tilde{f}$ which are not among
the curves in $E+\tilde{C}$; let $R^{\prime}\subset\tilde{Y}$ be the
corresponding reduced sum of components of the inverse image of $R$. Since
$\tilde{f}$ is proper and generically finite and unramified off the inverse
image of the normal crossings divisor $E+\tilde{C}+R$ , it follows that

$\tilde{f}^{*}(K_{\tilde{X}}+E+\tilde{C}+R)=K_{\tilde{Y}}+F+\tilde{C}^{\prime\prime}+R^{\prime}.$

Therefore,

$\tilde{f}^{*}(K_{\tilde{X}}+E+\sum_{i=1}^{r}(1-c_{i})\tilde{C}_{i}+R)=K_{\tilde{Y}}+F+R^{\prime}+\sum_{i,j}(1-c_{i}m_{ij})\tilde{C}^{\prime\prime}_{ij}-\sum_{i,j}c_{i}F_{ij}.$

Note $\tilde{f}^{*}(R)-R^{\prime}=R^{\prime\prime}+Z,$ where
$R^{\prime\prime}$ is effective and supported on $R^{\prime}$, and $Z$ is an
effective exceptional divisor. Thus

$\tilde{f}^{*}(K_{\tilde{X}}+E+\sum_{i=1}^{r}(1-c_{i})\tilde{C}_{i})+(Z+\sum_{i,j}c_{i}F_{ij})+R^{\prime\prime}=K_{\tilde{Y}}+F+\sum_{i,j}(1-c_{i}m_{ij})\tilde{C}^{\prime\prime}_{ij}.$

Consider the following $\mathbb{Q}$-divisors and their Zariski decompositions:

$L=\tilde{f}^{*}(K_{\tilde{X}}+E+\sum_{i=1}^{r}(1-c_{i})\tilde{C}_{i})=\tilde{f}^{*}(P_{\bar{C}})+\tilde{f}^{*}(N_{\bar{C}})$
$L^{\prime}=L+(Z+\sum_{i,j}c_{i}F_{ij})=\tilde{f}^{*}(P_{\bar{C}})+(\tilde{f}^{*}(N_{\bar{C}})+Z+\sum_{i,j}c_{i}F_{ij})$
$L^{\prime\prime}=L^{\prime}+R^{\prime\prime}=K_{\tilde{Y}}+F+\sum_{i,j}(1-c_{i}m_{ij})\tilde{C}^{\prime\prime}_{ij}=P_{\bar{C^{\prime}}}+N_{\bar{C^{\prime}}}.$

The Zariski decomposition of $L^{\prime}$ follows because
$Z+\sum_{i,j}c_{i}F_{ij}$ is effective and is collapsed to points by
$\tilde{f}$, hence dots to $0$ with $\tilde{f}^{*}(P_{\bar{C}})$.

So, the $-P\cdot P$ invariant for both $L$ and $L^{\prime}$ is

$-\tilde{f}^{*}(P_{\bar{C}})\cdot\tilde{f}^{*}(P_{\bar{C}})=-(\text{deg}\
\tilde{f})(P_{\bar{C}}\cdot P_{\bar{C}})=d(-P_{\bar{C}}\cdot P_{\bar{C}}),$

while the $-P\cdot P$ invariant for $L^{\prime\prime}$ is

$-(P_{\bar{C^{\prime}}}\cdot P_{\bar{C^{\prime}}}).$

Replacing a $\mathbb{Q}$-divisor $L$ by an integral mulitple $mL$ multiplies
the $-P\cdot P$ invariant by $m^{2}$. Thus, to prove the desired inequality of
the theorem, it suffices to work with $mL^{\prime}$ and $mL^{\prime\prime}$
for an integer $m$ making the two $\mathbb{Q}$-divisors integral, thus
corresponding to line bundles $\mathcal{L}^{\prime}=\mathcal{O}(mL^{\prime})$
and
$\mathcal{L}^{\prime\prime}=\mathcal{O}(mL^{\prime\prime})=\mathcal{L}^{\prime}(mR^{\prime\prime})$
on $\tilde{Y}$. Using these line bundles and the result recalled in x.x, it
suffices to show that for all $n>>0$,

$\text{dim}\ H^{0}(\tilde{Y}-F,\mathcal{L}^{\prime
n})/H^{0}(\tilde{Y},\mathcal{L}^{\prime n})\leq\text{dim}\
H^{0}(\tilde{Y}-F,\mathcal{L}^{\prime\prime
n})/H^{0}(\tilde{Y},\mathcal{L}^{\prime\prime n}).$

But this inequality follows from the assertion

$H^{0}(\tilde{Y}-F,\mathcal{L}^{\prime n})\cap
H^{0}(\tilde{Y},\mathcal{L}^{\prime\prime
n})=H^{0}(\tilde{Y},\mathcal{L}^{\prime n}).$

This equality is true by local considerations simply because
$\mathcal{L}^{\prime\prime}$ differs from $\mathcal{L}^{\prime}$ by the
effective divisor $mR^{\prime\prime}$ containing no components of $F$; if a
section of $\mathcal{L}^{\prime n}$ over $\tilde{Y}-F$ acquired a pole over
$F$, it could not extend to a holomorphic section of
$\mathcal{L}^{\prime\prime n}$ (which only allows new poles over
$R^{\prime\prime}$).

∎

As with the $-P\cdot P$’s in [10], these “characteristic” quantities behave
like _volumes_ : multiplicative by degree in unramified coverings (properly
defined), and submultiplicative for arbitrary maps.

##  6\. log covers and volumes of orbifold pairs

Since the characteristic number $\text{ch}(X,C)$ behaves like a volume for
orbifold pairs, we make the basic

######  Definition 6.1.

Vol$(X,\sum(1/n_{i})C_{i})=\text{ch}(X,C)=-P_{\bar{C}}\cdot P_{\bar{C}}$.

In this language, the Theorems of the last chapter take the form

######  Theorem 6.2.

Suppose
$f:(X^{\prime},\sum(1/n^{\prime}_{j})C^{\prime}_{j})\rightarrow(X,\sum(1/n_{i})C_{i})$
is a map of degree $d$ of orbifold pairs. Then

$Vol(X^{\prime},C^{\prime})\geq d\cdot Vol(X,C),$

with equality if $f$ is a log cover.

Note that when $C^{\prime}$ is reduced (i.e., all $n^{\prime}_{j}=1$), then
$Vol(X^{\prime},C^{\prime})=Vol(X^{\prime})$.

There is an analogue of the fundamental group of manifolds for describing
orbifold covers of orbifolds (recalled in [7], §$1$).

######  Definition 6.3.

Consider an orbifold
$\tilde{\Sigma}=(\Sigma,\gamma_{1},\cdots,\gamma_{r},n_{1},\cdots,n_{r}).$ The
_orbifold fundamental group_ is

$\pi_{1}^{orb}(\tilde{\Sigma})\equiv\pi_{1}(\Sigma-\bigcup_{i}\gamma_{i})/<\mu_{1}^{n_{1}},\cdots,\mu_{r}^{n_{r}}>,$

where $\mu_{i}$ is represented by the boundary of a small transverse disc to
$\gamma_{i}$.

One has the usual Galois correspondence between subgroups of finite index of
$\pi_{1}^{orb}(\tilde{\Sigma})$ and finite orbifold covers of
$\tilde{\Sigma}$. There is a _universal orbifold cover_ of $\tilde{\Sigma}$ as
well as a _universal abelian orbifold cover_ (or UAOC) corresponding to the
abelianization of $\pi_{1}^{orb}$. Recall the

######  Proposition 6.4.

([7],(1.3)) Let $\tilde{\Sigma}=(\Sigma,\gamma_{i},n_{i})$ be an orbifold for
which $\Sigma$ is a $\mathbb{Q}$HS.

  1. (1)

The UAOC $\tilde{\Sigma}^{ab}$ of $\tilde{\Sigma}$ is a finite orbifold cover.

  2. (2)

$\tilde{\Sigma}^{ab}=\Sigma^{ab}$ is a manifold, for which all weights
$n_{j}^{ab}=1$.

  3. (3)

$\pi_{1}^{orb}(\tilde{\Sigma}^{ab})=\pi_{1}(\Sigma^{ab})$.

  4. (4)

The covering group $G=H_{1}^{orb}(\tilde{\Sigma},\mathbb{Z})$ sits in an exact
sequence

$0\rightarrow\mathbb{Z}/(n_{1})\oplus\cdots\oplus\mathbb{Z}/(n_{r})\rightarrow
G\rightarrow H_{1}(\Sigma;\mathbb{Z})\rightarrow 0.$

When the orbifold arises from an orbifold pair $(X,\sum(1/n_{i})C_{i})$, then
log covers give orbifold covers, and one has the

######  Theorem 6.5.

([7], (3.1)) Suppose $(X,C)$ is an orbifold pair for which the link $\Sigma$
is a $\mathbb{Q}$HS. Then there exists a finite _universal abelian log cover
(or UALC) $(X^{\prime},C^{\prime})\rightarrow(X,C)$_, with $C^{\prime}$
reduced, which induces the UAOC on the boundary.

Since $C^{\prime}$ is reduced in the last result,
$Vol(X^{\prime},C^{\prime})=Vol(X^{\prime})$, and we conclude

######  Corollary 6.6.

Suppose $(X,\sum(1/n_{i})C_{i})$ is an orbifold pair for which the link
$\Sigma$ is a $\mathbb{Q}$HS, and let
$f:(X^{\prime},C^{\prime})\rightarrow(X,C)$ denote the UALC. Then

$Vol(X^{\prime})=Vol(X,C)\cdot|H_{1}(\Sigma,\mathbb{Z})|\cdot\Pi n_{i}.$

We are now ready to characterize topologically the orbifold pairs of Volume
$0$, analogous to the aforementioned theorem that $(X,0)$ is log canonical if
and only if it has Volume $0$ if and only if its local fundamental group is
finite or solvable. The complete list of these pairs is given in the next
Section.

######  Theorem 6.7.

An orbifold pair $(X,\sum(1/n_{i})C_{i})$ has Volume $0$ if and only if the
corresponding orbifold fundamental group is finite or solvable.

###### Proof.

Suppose $(X,\sum(1/n_{i})C_{i})$ has $\mathbb{Q}$HS link. Then by the
Corollary it has Volume $0$ iff the UALC $(X^{\prime\prime},C^{\prime\prime})$
has Volume $0$, which is the same as $Vol(X^{\prime\prime},0)=0$. Further,
since
$\pi_{1}^{orb}(\tilde{\Sigma^{\prime\prime}})=\pi_{1}(\Sigma^{\prime\prime})$
is a finite index normal subgroup of $\pi_{1}^{orb}(\tilde{\Sigma})$, the
finiteness or solvability of one of these groups is equivalent to that
property for the other. But $\pi_{1}(\Sigma^{\prime\prime})$ is finite or
solvable if and only if the volume of $(X^{\prime\prime},0)$ is $0$. This
proves the result in these cases.

If $(X,C)$ has volume $0$, then $(X,\sum(1-1/n_{i})C_{i})$ is log canonical,
hence ($3.4.1$) $(X,0)$ is log terminal, hence a quotient singularity; one is
in the $\mathbb{Q}$HS case, so the result about the orbifold fundamental
groups follows.

If $\pi_{1}^{orb}(\tilde{\Sigma})$ is finite or solvable, the same is true for
its quotient $\pi_{1}(\Sigma)$, so $(X,0)$ is log canonical. The only log
canonical $(X,0)$ which don’t have $\mathbb{Q}$HS links are simple elliptic
and cusp singularities (which have infinite solvable local fundamental
groups). Rather than handle these two special cases, we prove a more general
result below. It asserts that every orbifold pair $(X,C)$ has a log cover
$(X^{\prime},C^{\prime})$, with $C^{\prime}$ reduced. Then
$\pi_{1}^{orb}(\tilde{\Sigma^{\prime}})=\pi_{1}(\Sigma^{\prime})$ is a finite
index subgroup of $\pi_{1}^{orb}(\tilde{\Sigma})$, hence is itself finite or
solvable. Therefore, $(X^{\prime},C^{\prime})$ has volume $0$, and thus so
does $(X,C)$. ∎

As in [7], adjoining $n^{th}$ roots and normalizing gives cyclic log covers.

######  Proposition 6.8.

([7],(2.4)) Let $C\subset X$ be a reduced Weil divisor, $h$ a regular function
with zero divisor $(h)=C+nD$, some effective $D$. Then adjoining an $n^{th}$
root of $h$ to the function field of $(X,0)$ and normalizing gives a log-cover
$(X^{\prime},C^{\prime})\rightarrow(X,(1/n)C)$, where $C^{\prime}$ is reduced.

Imitating the proof from [7] of Theorem $6.5$ above, one can drop the
$\mathbb{Q}$HS restriction and prove the

######  Theorem 6.9.

For any orbifold pair $(X,C=\sum(1/n_{i})C_{i})$, there exists a log cover
$f:(Y,C^{\prime})\rightarrow(X,C)$ with $C^{\prime}$ reduced. In particular,

$Vol(X,C)=Vol(Y)/\text{deg}f.$

###### Proof.

We rely on the following (Cf. [8], (2.2)):

######  Lemma 6.10.

For any $d$ in the divisor class group $D=\text{Cl}(X,0)$ and
$n\in\mathbb{N}$, there exist $t\in\text{Tor}(D)$ and $u\in D$ so that
$d=t+nu.$

###### Proof.

For any good resolution $(\tilde{X},E)\rightarrow(X,0)$ one has the well-known
exact sequence of Mumford

$H^{1}(\tilde{X},\mathbb{Z})\rightarrow
H^{1}(\tilde{X},\mathcal{O}_{\tilde{X}})\xrightarrow{\alpha}D\xrightarrow{\beta}\mathbb{E}^{*}/\mathbb{E}\rightarrow
0.$

As $\beta(d)$ is torsion, $\beta(md)=0$ for some $m$, hence one can write
$md=\alpha(e)$. As $H^{1}(\tilde{X},\mathcal{O}_{\tilde{X}})$ is a divisible
group, one can define $t=d-\alpha((1/m)e)\in D$, a torsion element. Then
$d=t+n\alpha((1/mn)e).$ ∎

Returning to the Theorem, write the class of $C_{i}$ as
$[C_{i}]=t_{i}+n_{i}[D_{i}]$, for torsion elements $t_{i}$ and classes
$[D_{i}]$. Use the $t_{i}$ to successively form unramified coverings of
$(X,0)$, giving an unramified map $g:(X^{\prime},0)\rightarrow(X,0)$ for which
each $g^{*}(t_{i})$ is trivial in $\text{Cl}(X^{\prime},0)$. Thus
$g^{*}[C_{i}]-n_{i}g^{*}[D_{i}]$ is trivial, hence
$g^{*}(C_{i})-n_{i}g^{*}(D_{i})$ is a principal divisor. As in the proof of
Theorem $3.1$ of [7], there is a regular function $h_{i}$ on $X^{\prime}$
defining an effective divisor of the form $g^{*}(C_{i})+n_{i}F_{i}$, for some
curve $F_{i}$. Then, successively taking $n_{i}^{th}$ roots of the functions
$h_{i}$ and normalizing gives the desired result. ∎

######  Remark 6.11.

The decomposition in Lemma $6.10$ is not unique when $(X,0)$ does not have
$\mathbb{Q}$HS link, hence the log cover in Theorem $6.7$ need not be unique
(nor a Galois cover).

##  7\. Orbifold pairs with volume 0

An orbifold pair $(X,\sum(1/n_{i})C_{i})$ has Volume $0$ exactly when
$(X,\sum(1-1/n_{i})C_{i})$ is log canonical. Theorem $4.6(2)$ gives the only
examples for which the minimal log resolution is not star-shaped. (A
coefficient $c_{i}=1/2$ means orbifold weight $2$.) Lemma $4.1$ covers
rational chains (they are all log canonical, with arbitrary weights on the
ends). It only remains to consider star-shaped graphs with $t\geq 3$ branches.

Assume the determinants of the branches are $m_{1},m_{2},\cdots,m_{t}$, with
weights $n_{1},\cdots,n_{t}$ (recall we set $n_{j}=1$ if there is no curve
$\tilde{C_{j}}$ on the minimal log resolution.) Then $c_{i}=1-1/n_{i}$, and
the inequality $\chi_{C}\leq 0$ of Proposition $4.2$ translates easily to

$\sum_{i=1}^{t}(1-(1/n_{i}m_{i}))\leq 2.$

Note $m_{i}=1$ implies $n_{i}\geq 2$ , by minimality of the resolution; so
$n_{i}m_{i}\geq 2$, hence summands are at least $1/2$, whence $t\leq 4$. One
easily concludes:

######  Proposition 7.1.

The volume $0$ orbifold pairs with $t=4$ have $n_{i}m_{i}=2$, all $i$, with
graph

|  |
---|---|---
$\textstyle{\hbox to0.0pt{\vbox to0.0pt{\vss\hbox to0.0pt{\hskip
6.0pt$\scriptstyle$\hss}\vss}}\circ}$ $\circ$ $\bullet$
$\circ$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox
to0.0pt{\vbox to0.0pt{\vss\hbox to0.0pt{\hskip
6.0pt$\scriptstyle$\hss}\vss}}\circ}$

Each $\circ$ is either a $-1$ curve with an arrow sticking out, corresponding
to a $\tilde{C}_{i}$ with weight $n_{i}=2$; or, an unweighted $-2$ exceptional
curve.

One is left with the case $t=3$, for which the above inequality becomes

$\sum_{i=1}^{3}1/n_{i}m_{i}\geq 1.$

Therefore,

######  Proposition 7.2.

The volume $0$ orbifold pairs with $t=3$ are those whose ordered set of
triples $\\{n_{1}m_{1}\leq n_{2}m_{2}\leq n_{3}m_{3}\\}$ equals one of the
following:

  * •

$\\{2\leq 2\leq k\\},k\geq 2$

  * •

$\\{2\leq 3\leq k\\},3\leq k\leq 6$

  * •

$\\{2\leq 4\leq 4\\}$

  * •

$\\{3\leq 3\leq 3\\}$

In enumerating all the examples of the $n_{i}$ and $m_{i}$ from the list
above, recall that at least one $n_{i}\geq 2$; an $m_{i}>2$ allows for several
chains $m_{i}/q_{i}$; self-intersection of the central curve is arbitrary as
long as the graph is negative-definite.

######  Example 7.3.

For the ordered triple $\\{2,3,4\\}$, we specify all possible ordered sets of
weights $(n_{1},n_{2},n_{3})$ (from which the $m_{i}$ are determined). These
are: $(1,1,4),(1,1,2)$, $(1,3,1),(2,1,1),(1,3,4),(1,3,2)$,
$(2,1,2),(2,1,4),(2,3,1),(2,3,2),(2,3,4)$. (Alternatively, one could list
possible triples of $m_{i}$’s.)

We summarize in the following

######  Theorem 7.4.

Suppose $(X,\sum(1/n_{i})C_{i})$ is an orbifold pair of volume $0$. Then the
graph of the minimal log-resolution is one of the following:

  1. (1)

a string of rational curves in Lemma 4.1, with arbitrary weights on ends

  2. (2)

graph in Corollary 4.3, with arbitrary weight on the end

  3. (3)

graph in Theorem 4.6(2)

  4. (4)

star-shaped graph of valency four in Proposition 7.1

  5. (5)

any star-shaped graph of valency three whose multiplicities and weights are
given in Proposition 7.2.

Kollár ([4],p.125)lists all log canonical pairs $(X,\sum c_{i}C_{i})$ for
which all $c_{i}$ are $0$ or $\geq 1/2$. If all $c_{i}$ are of the form
$1-1/n_{i}$, we say the pair comes from a volume $0$ orbifold; these have all
been enumerated in Theorem 7.4. The few examples with a $c_{i}=1$ ((4.1) and
(4.3)), which correspond to orbifold cases for which $\tilde{C}_{i}$ can have
arbitrary weight, can be set aside.

######  Lemma 7.5.

Let $(X,\sum c_{i}C_{i})$ be a log canonical pair for which all $c_{i}$ are
$0$ or $\geq 1/2$. Write $c_{i}=1-1/\bar{n_{i}}$, where $\bar{n_{i}}$ equals
$1$ or is $\geq 2$. Let $n_{i}=\lfloor\bar{n_{i}}\rfloor$. Then
$(X,\sum(1/n_{i})C_{i})$ is an orbifold pair of volume $0$, with the same
graph but smaller weights..

###### Proof.

$(X,\sum(1-1/n_{i})C_{i})$ is less than or equal to $(X,\sum c_{i}C_{i})$,
hence is itself a log canonical pair. ∎

######  Lemma 7.6.

Let $(X,\sum c_{i}C_{i})$ be a log canonical pair for which all $c_{i}$ are
$0$ or $\geq 1/2$. Unless the minimal log resolution is star-shaped with $3$
branches, it is a volume $0$ orbifold pair.

###### Proof.

The preceding lemma shows any such pair with possibly lowered weights is an
orbifold pair. But Theorem 7.4 shows that examining all cases except for the
star-shaped $t=3$, the allowed $c_{i}$ for the graphs are $0$ or $1/2$. ∎

Lowering the weights on a star-shaped $t=3$ case leaves the $m_{i}$’s alone,
and we conclude:

######  Proposition 7.7.

Suppose $(X,\sum c_{i}C_{i})$ is a log canonical pair with all non-$0$
$c_{i}\geq 1/2$, whose minimal orbifold resolution is star-shaped with three
branches. The possibilities for the determinants $(m_{1},m_{2},m_{3})$ of the
branches are:

  1. (1)

$\\{1,1,1\\}$

  2. (2)

$\\{1,1,m\\},m\geq 2$

  3. (3)

$\\{1,2,m\\},m\geq 2$

  4. (4)

$\\{1,3,m\\},m\in\\{3,4,5,6\\}$

  5. (5)

$\\{1,4,4\\}$

  6. (6)

$\\{2,2,m\\},m\geq 2$

  7. (7)

$\\{2,3,3\\}$

######  Example 7.8.

Consider $(\mathbb{C}^{2},c_{1}L_{1}+c_{2}L_{2}+c_{3}L_{3})$, where the
$L_{i}$ are lines through the origin. The minimal log resolution is star-
shaped with a central $-4$, and the three branches are $-1^{\prime}s$ with
weights the $c_{i}$. If all $c_{i}\geq 1/2$, it is log canonical iff
$c_{1}+c_{2}+c_{3}\leq 2$. The orbifold examples are clearly sparsely
distributed among these.

To work out each of the other examples, one should separate cases based on
whether an $m_{i}>1$ corresponds to a $c_{i}=0$ or $c_{i}\geq 1/2$.

Since $\overline{1/2C}=1/2C$ for a reduced curve, one can ask for which plane
curves $C$ is $Vol(\mathbb{C}^{2},(1/2)C)=0$.

######  Proposition 7.9.

Let $C=\\{f(x,y)=0\\}$ be reduced. Then $(\mathbb{C}^{2},(1/2)C)$ has volume
$0$ if and only if $f$ is one of the following (up to equisingularity):

  1. (1)

$y$

  2. (2)

$xy$

  3. (3)

$y^{2}-x^{2n}$, $n\geq 2$

  4. (4)

$y^{2}-x^{2n+1}$, $n\geq 1$

  5. (5)

$x(y^{2}-x^{2n})$, $n\geq 2$

  6. (6)

$x(y^{2}-x^{2n+1})$, $n\geq 2$

  7. (7)

$y^{3}+x^{4}$

  8. (8)

$y(y^{2}+x^{3})$

  9. (9)

$y^{3}+x^{5}$

  10. (10)

$y^{3}+x^{6}$

  11. (11)

$y^{4}+x^{4}$

  12. (12)

$(y+x^{2})(y^{2}+x^{2n+4})$

  13. (13)

$(y+x^{2})(y^{2}+x^{2n+3})$,

  14. (14)

$x(y+x)(y^{2}+x^{2n})$

  15. (15)

$x(y+x)(y^{2}+x^{2n+1})$

###### Proof.

By Proposition 6.8, the Volume for the double cover $\\{z^{2}=f(x,y)\\}$ is
twice the Volume of the pair $(\mathbb{C}^{2},(1/2)C)$; hence one is log
canonical if and only if the other is. But the log canonical double points in
$\mathbb{C}^{3}$ are well-known; they are either rational double points, two
types of simple elliptic singularities (which have a modulus), and two types
of cusp singularities. Equations of the last type can be found on page 1290-91
of [5], with dual graph type called _No_.

∎

##  8\. The set of volumes of rdp orbifold pairs

As remarked in the Introduction, F. Ganter proved that the set of volumes of
Gorenstein NSS’s satisfies the DCC, with minimum non-$0$ value $1/42$
(achieved for the Brieskorn singularity $V(2,3,7)$). However, there is no
simple generalization for Gorenstein orbifold pairs. Example 8.8 below gives a
family $(X_{m},C_{m})$ of Gorenstein (even minimally elliptic) pairs for which
the volumes $Vol(X_{m},C_{m})$ are strictly decreasing (here, the volumes
$Vol(X_{m})$ are strictly increasing). It therefore makes sense to formulate a
version for which the $X$ are Gorenstein log terminal, i.e. RDP’s.

######  DCC Volumes Conjecture.

The set of volumes of RDP orbifold pairs $(X,C=\sum(1/n_{i})C_{i})$ satisfies
the DCC, with minimum non-$0$ value $1/3528=1/(2\cdot 42^{2})$.

######  Example 8.1.

The orbifold pair

$(X,C)=(\mathbb{C}^{2},(1/2)\\{x=0\\}+(1/3)\\{y=0\\}+(1/7)\\{x+y=0\\})$

has minimal log resolution

|  |
---|---|---
|
$\scriptstyle 1-1/7$ $\textstyle{\uparrow}$$\textstyle{\hbox to0.0pt{\vbox
to0.0pt{\vss\hbox to0.0pt{\hskip
6.0pt$\scriptstyle-1$\hss}\vss}}\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\scriptstyle 1-1/2$ $\scriptstyle-1$ $\bullet$ $\scriptstyle-4$ $\bullet$
$\scriptstyle-1$ $\bullet$ $\scriptstyle 1-1/3$

where the arrows correspond to the curves in $C$. The UALC is given by
adjoining $u,v,w$ satisfying $u^{2}=x,v^{3}=y,w^{7}=x+y$; this is the
hypersurface $\\{u^{2}+v^{3}=w^{7}\\}$, of volume $1/42$. Therefore,
$Vol(X,C)=1/(2\cdot 3\cdot 7)\cdot(1/42).$ Since $\\{\pm\text{Id}\\}$ acts
freely on $(X,C)$, the quotient $(Y,C^{\prime})$ is an $A_{1}$ singularity
with weights along three lines. The minimal log resolution is the same as the
one above, except that the self-intersection of the central curve is $-5$. One
has $Vol(Y,C^{\prime})=(1/2)Vol(X,C)=(2\cdot 42\cdot 42)^{-1}=1/3528.$

We will verify the Conjecture in the star-shaped case (as done in [10],(3.5)
for volumes of singularities, before [2] succeeded for the general case).

######  Theorem 8.2.

The set of volumes of RDP orbifold pairs with star-shaped minimal log
resolution satisfies the DCC, with minimum non-$0$ volume $1/3528.$

Here the log-resolution graph has some number $s$ of $-1$-curves attached to a
central vertex, and blowing those down gives an RDP, with $r\leq 3$ branches
emerging from the center. The new central vertex is either a $-2$ (in which
case the graph now is a familiar ADE) or a $-1$ (which can happen only for
$r\leq 2$). The RDP consists either of $r$ chains of determinant
$m_{1},\cdots,m_{r}$ connected to the central vertex, with $r$ is $1,2$, or
$3$; or ($r=0$) one simply has $s\geq 3$ $-1$-curves emerging from a central
$-(s+1)$ or $-(s+2)$. To exclude Volume $0$ examples, one has $r+s\geq 3$.

Write the orbifold pair as
$(X,C=\sum_{i=1}^{r}(1/n_{i})C_{i}+\sum_{j=1}^{s}(1/n_{j}^{\prime})C_{j}^{\prime})$,
where the $C_{i}$’s are at the end of the chain corresponding to $n_{i}$, and
the $C^{\prime}_{j}$’s are at the ends of the $-1$ curves emerging from the
central vertex. Note $n_{i}\geq 1$, $m_{i}\geq 2$ and $n_{j}^{\prime}\geq 2$.
One has the negative quantity $\chi=-2+\sum_{i=1}^{r}(1-1/m_{i})$, while
positive volume of the pair requires positivity of
$\chi_{\bar{C}}=\chi+\sum_{i=1}^{r}(1-1/n_{i})/m_{i}+\sum_{j=1}^{s}(1-1/n^{\prime}_{j})$.
We rewrite as

($*$)
$\chi_{\bar{C}}=-2+\sum_{i=1}^{r}(1-1/n_{i}m_{i})+\sum_{j=1}^{s}(1-1/n^{\prime}_{j}).$

######  Lemma 8.3.

$\chi_{\bar{C}}>0$ implies $\chi_{\bar{C}}\geq 1/42$, achieved when $r+s=3$,
all $n_{i}=1$, and $\\{m_{i},n_{j}^{\prime}\\}=\\{2,3,7\\}$.

###### Proof.

The expression $-2+\sum_{i=1}^{u}(1-1/k_{i})$, where all $k_{i}\geq 2$, is
clearly negative if $u\leq 2$ and $\geq 1/2$ if $u\geq 5$. For $u=4$, the
minimum positive value is $1/6$, occurring for the $k_{i}^{\prime}s$ equal
$2,2,2,3$. For $u=3$, the expression is $1-1/k_{1}-1/k_{2}-1/k_{3}$, and it is
easy to check the smallest positive value is $1/42$, for the $k_{i}^{\prime}s$
$2,3,7$. This result then implies the assertion of the Lemma. ∎

######  Lemma 8.4.

The minimum positive value of $Vol(X,C)$ is $1/3528$.

###### Proof.

The volume equals $\chi_{\bar{C}}^{2}/e$, where $e=|\chi/t|$ is computed from
the entries in the chains and the self-intersection of the central vertex. By
the last result, the numerator is $\geq(1/42)^{2}$. The denominator is
$\leq|\chi|\leq 2$, so the whole expression is $\geq(1/42)^{2}/2=1/3528$. The
Example above shows this case does occur. ∎

######  Lemma 8.5.

Given an infinite sequence of graphs whose volumes are strictly decreasing,
there is an infinite subsequence with fixed $s$ and with fixed $r$.

###### Proof.

There are only $4$ values for $r$, so at least one of them occurs infinitely
often. Since $e\leq 2$, $Vol(X,C)\geq\chi_{\bar{C}}/2\geq(-2+s/2)/2$, so $s$
must be bounded in the sequence. Therefore, at least one value occurs
infinitely often, and we take such a subsequence. ∎

To conclude the DCC among a set of volumes, use the likely well-known

######  Proposition 8.6.

Suppose $\mathbb{N}^{k}$ is given a partial order using the weight function
$W(n_{1},n_{2},\cdots,n_{k})=\sum_{i=1}^{k}1/n_{i}$. Then $\mathbb{N}^{k}$
satisfies the ACC (ascending chain condition): every subset
$Z\subset\mathbb{N}^{k}$ contains an element of maximal weight.

###### Proof.

The case $k=1$ being obvious, proceed by induction on $k$. For $1\leq i\leq
k$, denote by $\pi_{i}:\mathbb{N}^{k}\rightarrow\mathbb{N}^{k-1}$ projection
off the $i^{th}$ component. By induction, $\pi_{i}(Z)$ contains an element of
maximal weight; let
$\alpha^{i}=(\alpha^{i}_{1},\alpha^{i}_{2},\cdots,\alpha^{i}_{k})\in Z$ be an
element which maps to it.

The assertion of the proposition is proved if there are only finitely many
elements of $Z$ whose weight is greater than or equal to all $W(\alpha^{i})$.
Suppose $\beta\in Z$ has $W(\beta)\geq W(\alpha^{i})$ for all $i$, and write
$\beta=(\beta_{1},\cdots,\beta_{k})$. We claim that
$\beta_{i}\leq\alpha^{i}_{i}$, all $i$; this would imply the finiteness
assertion. Now,

$W(\beta)=1/\beta_{i}+\sum_{j\neq i}1/\beta_{j}\geq
W(\alpha_{i})=1/\alpha^{i}_{i}+\sum_{j\neq i}(1/\alpha^{i}_{j}).$

By the maximality assumption for $\alpha^{i}$, the sum for $j\neq i$ of
$1/\beta_{j}$ is less than or equal to the corresponding sum of
$1/\alpha^{i}_{j}$, hence $1/\beta_{i}\geq 1/\alpha^{i}_{i}$, as claimed. ∎

######  Lemma 8.7.

For fixed $r$ and $s$, the set of positive expressions $\chi_{\bar{C}}$
satisfies the DCC.

###### Proof.

By $(*)$, the $\chi_{\bar{C}}$ is $r+s-2$ minus a sum of $r+s$ reciprocals of
integers, so apply the Proposition. ∎

If volumes $\chi_{\bar{C}}^{2}/e$ are strictly decreasing, to use Lemma 8.7
requires understanding the $e$’s in the sequence. We handle separately the
cases of various $r$.

For $r=0$, one has $\chi=-2$ and $e=1$ or $2$; one of these cases occurs
infinitely often, so Lemma 8.7 can be used.

For $r=3$, we break up into the cases of the $n_{i}$’s. For
$(2,3,3),(2,3,4),(2,3,5)$, there is only one $e$, so the result follows from
Lemma 8.7. For $(2,2,n)$, one has $e=1/n$. But since $\chi_{\bar{C}}\geq
1/42$, in an infinite decreasing family of this type we have $Vol(X,C)\geq
n\cdot(1/42)^{2}$, whence $n$ must be bounded above. So, restrict to a
subsequence with a fixed $n$, hence fixed $e$, and then apply Lemma 8.7 to the
resulting expression.

In case $r=2$, one has $\chi=-(1/n_{1}+1/n_{2})$ (with $n_{i}\geq 2$ by
minimality) so $1/e=|t|/(1/n_{1}+1/n_{2})$. In this case $Vol(X,C)\geq|t|\cdot
1/3528$, whence $|t|$ is bounded in any decreasing family; restricting to a
subsequence we can assume it is fixed, so that one is examining the volumes
$\chi_{\bar{C}}^{2}/(1/n_{1}+1/n_{2})$. In the infinite sequence, the
expression $1/n_{1}+1/n_{2}$ is bounded above and below, hence there is an
infinite subsequence which is strictly monotone or constant. Pass to this
subsequence in the family of decreasing volumes. By Proposition 8.6, the
subsequence cannot be strictly increasing. If the sequence becomes constant,
then one has $e$ fixed in the family; but Lemma 8.7 then implies a
contradiction to an infinite sequence of decreasing volumes. Finally, if the
subsequence is strictly decreasing, then the numerator $\chi_{\bar{C}}^{2}$
must be strictly decreasing in order for the quotient to be strictly
decreasing. Now apply Lemma 8.7.

The same argument applies in the case $r=1$; $|\chi|$ in the denominator of
the volume has the form $1+1/n$ or $1+1/(2n+1)$, so in a subsequence its value
either increases or decreases or stays the same.

######  Example 8.8.

Let $X_{m}$ denote (for $m\geq 7$) the quasi-homogeneous minimally elliptic
singularity whose minimal good resolution graph has a $-1$ rational central
curve surrounded by $3$ rational curves, of self-intersection respectively
$-2$, $-3$, $-m$, with $m\geq 7$. Then $\chi_{m}=e_{m}=1/6-1/m$, hence
$Vol(X_{m})=1/6-1/m$ strictly increases as $m$ increases. Next let $C_{m}$ be
a curve of weight $2$ on $X_{m}$ for which of the resolution $\tilde{C_{m}}$
is a smooth curve passing transversally through the central curve at a fourth
point. The orbifold log resolution (with an unmarked bullet a $-2$) is

|
---|---
$\textstyle{\hbox to0.0pt{\vbox to0.0pt{\vss\hbox to0.0pt{\hskip
6.0pt$\scriptstyle-m$\hss}\vss}}\bullet}$ $\bullet$ $\bullet$ $\scriptstyle-1$
$\bullet$ $\scriptstyle 1-1/2$ $\textstyle{\hbox to0.0pt{\vbox
to0.0pt{\vss\hbox to0.0pt{\hskip
6.0pt$\scriptstyle-3$\hss}\vss}}\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Therefore $\chi_{\bar{C_{m}}}=\chi_{m}+(1-1/2)$, and so
$Vol(X_{m},C_{m})=(\chi_{m}+1/2)^{2}/(\chi_{m})$, which can be rewritten as

$8/3+2(4m+3)/m(m-6).$

We see that the volumes of the pairs $(X_{m},C_{m})$ strictly decrease as $m$
increases.

## References

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