# On the geography of $3$-folds via asymptotic behavior of invariants

Yerko Torres-Nova<EMAIL_ADDRESS>Facultad de Matemáticas, Pontificia
Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna
4860, Santiago, Chile.

###### Abstract.

Roughly speaking, the problem of geography asks for the existence of varieties
of general type after we fix some invariants. In dimension $1$, where we fix
the genus, the geography question is trivial, but already in dimension $2$ it
becomes a hard problem in general. In higher dimensions, this problem is
essentially wide open. In this paper, we focus on geography in dimension $3$.
We generalize the techniques that compare the geography of surfaces with the
geography of arrangements of curves via asymptotic constructions. This
involves resolutions of singularities and a certain asymptotic behavior of the
associated Dedekind sums and continued fractions. We discuss the general
situation with emphasis on dimension $3$, analyzing the singularities and
various resolutions that show up, and proving results about the asymptotic
behavior of the invariants we fix.

Keywords: Complex algebraic geometry, Geography of threefolds, Chern numbers.
MSC2020: 14A99, 14C17, 14J30, 14J40.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Asymptoticity for a non-singular branch locus
  4. 4 Asymptoticity of invariants
  5. 5 Applications to the geography of 3-folds
  6. 6 Discussion

##  1\. Introduction

We work with normal projective varieties $X$ over the complex numbers
${\mathbb{C}}$. As usual, when $\dim X=1$, $2$, or $d\geq 3$ we say that $X$
is a curve, a surface, or a $d$-fold respectively. The main purpose of this
paper is to study the geography problem for $3$-folds of general type.
Following the philosophy of Hirzebruch and Sommese [Hir83][Som84] studying the
slopes of Chern numbers $c_{1}^{2}/c_{2}$ for minimal surfaces of general
type, in [Hun89] the author proposed the study of slopes

$[c_{1}^{3},c_{1}c_{2},c_{3}]\in{\mathbb{P}}^{2}_{{\mathbb{Q}}}$

for non-singular minimal $3$-folds of general type. This, allows us to work by
charts, in this case, in the charts $c_{1}^{3}\neq 0$ or $c_{1}c_{2}\neq 0$.

Since minimal models may admit singularities in higher dimensions, we want to
extend our problem, but we do not directly have the notion of Chern numbers.
If $X$ is a non-singular $3$-fold, then we have Chern numbers

$c_{1}^{3}(X)=-K_{X}^{3},\quad c_{1}c_{2}(X)=24\chi({\mathcal{O}}_{X}),\quad
c_{3}(X)=e(X).$

From the birational geometry of $3$-folds, the triple
$(K_{X}^{3},\chi({\mathcal{O}}_{X}),e(X))\in{\mathbb{Q}}\times{\mathbb{Z}}^{2}$
is invariant between minimal models of the same birational class [Mat10,
12.1.2] [Kol91, Th.3.2.2]. Then we are extending naturally to the singular
case by taking slopes
$[-K_{X}^{3},\chi({\mathcal{O}}_{X}),e(X)]\in{\mathbb{P}}^{2}_{{\mathbb{Q}}}$.
There are several inequalities for minimal $3$-folds, most of them for
Gorenstein minimal $3$-folds, since we have well-behaved dualities on the
cohomology groups. For example, in this case
$\chi(\omega_{X})=-\chi({\mathcal{O}}_{X})$. The most important for us are the
Miyaoka-Yau inequality $-K_{X}^{3}\geq 72\chi({\mathcal{O}}_{X})$ ([Miy87]
[GKPT19]), and the classical general type inequalities
$K_{X}^{3},\chi({\mathcal{O}}_{X})<0$ where the last one is just for the
Gorestein case. The search for Noether’s type inequalities is an active
research area, see [CH06] [Hu18] [CCJ20].

Previous results in this research line are the following. In [Hun89], the
author studied slopes of non-singular $3$-folds with ample canonical class in
the chart $c_{1}c_{2}\neq 0$. In [CL01] is proved that the Chern slopes
$[c_{1}^{3},c_{1}c_{2},c_{3}]$ of non-singular $3$-folds with ample canonical
bundle define a bounded region by finding inequalities for the coordinate
$c_{3}/c_{1}c_{2}$. For a more general result, let $N(d)$ be the number of
partitions of $d$. In [DS22], it is proved that Chern slopes
$[c_{1}^{d},\ldots,c_{d}]\in{\mathbb{P}}^{N(d)-1}_{{\mathbb{Q}}}$ of non-
singular $d$-folds with ample canonical bundle define a bounded region in the
chart $c_{1}^{d}\neq 0$. See [TN23, p.14-18] to see an updated map in the
chart $c_{1}c_{2}\neq 0$ for non-singular $3$-folds with ample canonical
class. The problem is that we do not know if those regions are optimal as we
have for surfaces, i.e., the slopes $c_{1}^{2}/c_{2}$ are dense in the
interval $[1/5,3]$.

###  1.1. Results of this work

When working with slopes, it turns out that we can apply the tool of $n$-th
root coverings to investigate their behavior. The main reason is that, as $n$
grows to infinity, the slopes erase the effects of the degree $n$ of the
cover. This allows us to control the slopes from the base of the cover, and
this is the main technique to get density results for surfaces

Consider the following data. Let $Z$ be a non-singular $d$-fold, and let
$D_{1},\ldots,D_{r}$ be distinct prime divisors in $Z$. Assume that
$D_{red}:=D_{1}+\ldots+D_{r}$ is a simple normal crossing divisor (SNC). Let
$n>1$ be a positive integer, and let $0<\nu_{i}<n$ be a collection of $r$
integers coprime to $n$. Assume that there exists a line bundle
${\mathcal{L}}$ on $Z$ such that

${\mathcal{L}}^{\otimes
n}\simeq{\mathcal{O}}_{Z}\left(\sum_{i=1}^{r}\nu_{i}D_{i}\right).$ (1)

Then, there exists an $n$-th root cover $h_{n}\colon Y_{n}\to Z$ branched
along $D_{red}$, where $Y_{n}$ is a $d$-fold (Section 2.2). These are the
$n$-th root covers developed by Esnault and Viehweg (Cf. [EV92]). For curves,
the prime divisors $D_{1},\ldots,D_{r}$ are distinct points on $Z$. Since
points are isolated, we have $Y_{n}$ as a non-singular curve. By the Riemann-
Hurwitz formula, we have

$\frac{c_{1}(Y_{n})}{n}=\frac{nc_{1}(Z)-(n-1)r}{n}=c_{1}(Z)-r+\frac{r}{n}=\bar{c}_{1}(Z,D)+\frac{r}{n}.$

Here $\bar{c}_{1}(Z,D)$ is the first logarithmic Chern number of the pair
$(Z,D_{red})$ (see Section 2.1). Hence, if we fix the points $D_{red}$ and we
consider partitions $\nu_{1}+\ldots+\nu_{r}=n$ with $n\gg 0$ a prime number,
then we asymptotically have $c_{1}(Y_{n})\approx n\bar{c}_{1}(Z,D)$.

######  Question 1.1.

Does this asymptotic phenomenon happen in higher dimensions?

In Section 3, we prove that this phenomenon occurs in any dimension for
logarithmic morphisms of degree $n$ with branch locus as a disjoint collection
of non-singular distinct prime divisors $D_{1},\ldots,D_{r}$. As an
application, for $n$-root cover we have $Y_{n}$ as a non-singular projective
variety, thus in Corollary 3.2 we get.

######  Theorem A.

Assume we have $n$-th root covers $h\colon Y_{n}\to Z$ branched at
$D=\sum_{j}\nu_{j}D_{j}$. If $D_{red}$ is non-singular, then for each
partition $i_{1}+\ldots+i_{m}=d$, the Chern numbers satisfy,

$\frac{c_{i_{1}}\ldots
c_{i_{m}}(Y_{n})}{n}\to\bar{c}_{i_{1}}\ldots\bar{c}_{i_{m}}(Z,D),$

as $n\to\infty$ for prime numbers.

∎

However, this theorem is restrictive for us in terms of geography. It is not
easy to get the necessary hypothesis to construct a minimal $d$-fold of
general type. On the other hand, we do not drop the possibility of having
applications in other contexts. Also, this is a cornerstone of our research
and opens the following discussion.

For $\dim Z\geq 2$, if the branch divisor $D_{red}=D_{1}+\dots+D_{r}$ has
singularities, then $Y_{n}$ have rational singularities [Vie77]. In order to
have well-behaved invariants, we can choose a (partial) resolution of
singularities. For $\dim Z=2$, the asymptoticity of Chern numbers was proved
in [Urz09] for _random_ $n$-th root covers. Let us explain briefly what random
means (see Section 2.5 for more details). First, in dimension two, each
singularity of $Y_{n}$ is a cyclic surface singularity of type
$\frac{1}{n}(q,1)$ for some $0<q<n$. Thus, we use the Hirzebruch-Jung
algorithm (see Section 2.4.1) to resolve these singularities in a minimal way.
For us, there are two important quantities, (1) the _length of the resolution_
, i.e., the number of steps of the algorithm, and (2) the Dedekind sums,

$d(q,1,n)=\sum_{i=1}^{n-1}\left(\left(\frac{iq}{n}\right)\right)\left(\left(\frac{i}{n}\right)\right),$

where $((\cdot)):{\mathbb{R}}\to{\mathbb{R}}$ is the saw-tooth function (see
Section 2.4.2). We get a resolution of singularities $X_{n}\to Y_{n}$.
However, the Chern numbers $c_{1}^{2}$ and $c_{2}$ depend on the lengths and
the Dedekind sums coming from all cyclic singularities resolved. To guarantee
asymptoticity, we have to consider _asymptotic arrangements_. Indeed, for each
prime number $n\geq 17$, there exists a set $O_{n}\subset\\{1,\ldots,n\\}$
(Section 2.5) such that for each $q\in O_{n}$, the lengths and Dedekind sums
are bounded by $c\sqrt{n}$ for a constant $c>0$. Thus we say that $D_{red}$ is
an asymptotic arrangement if satisfies :

  * •

For prime numbers $n\gg 0$, there exists multiplicities $0<\nu_{j}<n$ such
that the singularity over $D_{j}\cap D_{k}$ of $Y_{n}$ is of type
$\frac{1}{n}(q_{jk},1)$ with $q_{jk}\in O_{n}$.

  * •

For each $n$ there are line bundles ${\mathcal{L}}\in\mbox{Pic}(Z)$ such that

${\mathcal{L}}^{\otimes
n}\simeq{\mathcal{O}}_{Z}\left(\sum_{j=1}^{r}\nu_{j}D_{j}\right).$

The main set-up is when there exists $H\in\mbox{Pic}(Z)$ such that
$D_{j}\simeq H$ for each component of $D_{red}$. Observe that the condition
(1) is satisfied as

$D=\nu_{1}D_{1}+\ldots+\nu_{r}D_{r}\sim(\nu_{1}+\ldots+\nu_{r})H\sim nH,$

thus we can construct $n$-th root covers. In [Urz09] was proved that for a
random partition $\nu_{1}+\ldots+\nu_{r}=n$, the probability of $D$ being an
asymptotic arrangement tends to $1$ as $n$ grows. In this way, for $n\gg 0$ we
take random asymptotic partitions, and we get a family of random surfaces
$X_{n}$ with

$c_{1}^{2}(X_{n})\approx n\bar{c}_{1}^{2}(Z,D),\quad c_{2}(X_{n})\approx
n\bar{c}_{2}(Z,D).$

Now we can discard the SNC property for the branch divisor for those one
having only normal crossings. We consider a _$\log$ resolution_ $\gamma\colon
Z^{\prime}\to Z$ such that the reduced divisor defined by $\gamma^{*}D_{red}$
is SNC. If such a divisor turns out to be an asymptotic arrangement, then we
have the asymptotic result for Chern numbers as above. For this see [TN23, p.
57]), and see [Urz16] to connect this result with minimal models. A direct
application is a relation between Chern slopes of simply connected surfaces of
general type and Chern slopes of arrangements of lines [EFU22]. In this way,
in higher dimensions, we have several issues with achieving an analog
asymptotic result. For example, the singularities of $Y_{n}$ are not cyclic,
and the choice of a right (partial) resolution of $Y_{n}$ with good behavior
as $n$ grows is a challenging problem.

######  Question 1.2.

Is there an analog of the asymptotic results in dimension two for dimension
three?

For instance, if this question has a positive answer, then we would be able to
study the geography of $3$-folds using arrangements of planes in
${\mathbb{P}}^{3}$. The first result in this direction is in Section 4.1,
where we find that the Chern number $c_{1}c_{2}=24\chi$ is _asymptotic_ and
independent of the chosen resolution.

######  Theorem B.

Let $Z$ be any non-singular projective $3$-fold, and let
$\\{D_{1},\ldots,D_{r}\\}$ be an asymptotic arrangement. For prime numbers
$n\gg 0$ there are projective non-singular $3$-folds covers $X_{n}\to Z$ of
degree $n$ such that

$\frac{c_{1}c_{2}(X_{n})}{n}\to\bar{c}_{1}\bar{c}_{2}(Z,D),$

as $n\to\infty$

∎

However, the canonical volume and the topological characteristic depend on the
chosen resolution. In this way, the first issue is that the singularities of
$Y_{n}$ are of order (multiplicity) $n^{2}$, too big. This means that to
connect with a non-singular model, at a bad choice of resolution we could have
big exceptional data with respect to $n$, and so we would lose the asymptotic
property. In Section 4.2, we introduce a prototype of _first step_ , i.e., by
toric methods we construct a _local cyclic resolution_ In this way, we get
singularities of multiplicity lower than $n$, and of cyclic quotient type. We
are interested in cyclic quotient singularities since they are $\log$-terminal
and have a well-known algorithm to resolve them: the Fujiki-Oka continuous
fraction (Cf. [Ash19]). In Section 4.3 we globalize this local cyclic
resolution, and we get a cyclic resolution $X_{n}\to Y_{n}$ having the desired
asymptotic property. The following is a summary of Theorem 4.2 (using
Corollary 2.16), Theorem 4.18, and Theorem 4.24.

######  Theorem C.

Let $Z$ be any non-singular $3$-fold, and let $\\{D_{1},\ldots,D_{r}\\}$ be an
asymptotic arrangement. For prime numbers $n\gg 0$ there are $3$-folds
$X_{n}\to Z$ with at most cyclic quotient singularities of order lower than
$n$ such that

$\frac{-K_{X_{n}}^{3}}{n},\frac{24\chi({\mathcal{O}}_{X_{n}})}{n},\frac{e(X_{n})}{n}\approx\bar{c}_{1}^{3}(Z,D),\bar{c}_{1}\bar{c}_{2}(Z,D),\bar{c}_{3}(Z,D),$

where $\bar{c}_{i}(Z,D)$ are the Chern classes of the arrangement.

∎

It is important to note that the above is an embedded
${\mathbb{Q}}$-resolution in the language of [ABMMOG12], introduced as an
efficient resolution without _useless data_. In Section 5.1, as a by-product
of the computations to get C, we construct minimal non-singular $3$-folds of
general type using as a base $3$-folds $Z\hookrightarrow{\mathbb{P}}^{4}$ with
$3$ hyperplane sections. This allows us to prove.

######  Corollary 1.3.

For $d>5$ and prime numbers $n\gg 0$ there are minimal non-singular $3$-folds
$X_{n}$ of general type over $Z$ with slopes

$\frac{c_{1}^{3}(X_{n})}{c_{1}c_{2}(X_{n})}\approx\frac{(d-2)^{3}-1}{(d-2)(d-1)^{2}},\quad\frac{c_{3}(X_{n})}{c_{1}c_{2}(X_{n})}\approx\frac{(d-5)(d^{2}+2d+6)}{(d-2)(d-1)^{2}},$

as $n\to\infty$. In particular, as the degree of $Z$ grows, the slopes have
limit point $(1,1)$.

∎

In Section 5.2, we see our resolution in terms of pairs
$(X_{n},\tilde{D}_{red})\to(Z,D_{red})$. As a corollary, we prove that for
asymptotic arrangements of hyperplane sections on a minimal $3$-fold of
general type, the resolution preserves the bigness of the $log$-canonical
divisor $K_{X_{n}}+\tilde{D}_{red}$.

######  Corollary 1.4.

Let $Z\hookrightarrow{\mathbb{P}}^{d}$ be a minimal non-singular projective
$3$-fold of general type, and let $\\{H_{1},\ldots,H_{r}\\}$ be a collection
of hyperplane sections in general position. Then, for prime numbers $n\gg 0$
there are finite morphisms of degree $n$
$(X_{n},\tilde{D}_{red})\to(Z,D_{red})$ such that:

  1. (1)

$X_{n}$ is of $\log$-general type, i.e., $K_{X_{n}}+\tilde{D}_{red}$ is big
and nef,

  2. (2)

$K_{X_{n}}^{3}>0$,

  3. (3)

$X_{n}$ has cyclic singularities ($\log$-terminal) of order lower than $n$,
and

  4. (4)

the slopes $(-K^{3}/24\chi,e/24\chi)$ of $X_{n}$ are arbitrarily near to
$(2,1/3)$.

We point out that our cyclic resolution preserves the positivity of the
canonical volume, i.e., we start with $K_{Z}^{3}>0$ and we get
$K_{X_{n}}^{3}>0$. So in future work, if we are able to control the MMP of the
chosen resolution, then we will have the asymptotic results with $K_{X_{n}}$
nef, so for $n\gg 0$, the varieties $X_{n}$ will be minimal of general type.
For us, the goal is that the asymptotic behavior of the slopes of $X_{n}$
coincides with the slopes of its minimal models. For this, we need a _good
terminalization_ of the cyclic singularities obtained. In Section 6 we discuss
what means the word _good_.

Acknowledgments: I am grateful to my Ph.D. thesis advisor Giancarlo Urzúa for
his time, guidance, and support throughout this work. The results in this
paper are part of my Ph.D. thesis at the Pontificia Universidad Católica de
Chile. I would also like to thank Jungkai Alfred Chen for his hospitality
during my stay at the National Center for Theoretical Sciences in Taiwan.
Special thanks to Pedro Montero and Maximiliano Leyton, for many comments and
suggestions to improve this work. I was funded by the Agencia Nacional de
Investigación y Desarrollo (ANID) through the Beca Doctorado Nacional 2019.

##  2\. Preliminaries

###  2.1. Logarithmic properties

For a non-singular projective variety $Z$ of dimension $d$, let
$A(Z)=\bigoplus_{e\geq 0}A^{e}(Z)$ be its Chow ring, where $A^{e}(Z)$ are the
Chow groups of $e$-cycles. We denote its Chern classes by $c_{e}(Z)\in
A^{e}(Z)$, i.e. the Chern classes of its tangent bundle ${\mathcal{T}}_{Z}$
[Har77, Appendix A]. We have that $c_{1}(Z)=-K_{Z}$, where $K_{Z}$ is the
canonical class. We define the Chern numbers of $Z$ as the degree of the top-
intersection of its Chern classes

$c_{i_{1}}(Z)\ldots c_{i_{m}}(Z)\in A^{d}(Z),\quad i_{1}+\ldots+i_{m}=d.$

In the following, if the context is understood, we abuse notation using the
symbol $c_{i_{1}}\ldots c_{i_{m}}(Z)$ for Chern numbers or more simply the
notation $c_{i_{1}}\ldots c_{i_{m}}$.

We have $c_{1}^{d}=(-1)^{d}K_{Z}^{d}$, and since we work over ${\mathbb{C}}$,
it is well-known that $c_{d}=e(Z)$, the topological Euler characteristic.
These numbers are codified into the Todd class of ${\mathcal{T}}_{Z}$ by the
following formal sum

$\mbox{td}({\mathcal{T}}_{Z})=1+\frac{c_{1}}{2}+\frac{c_{1}^{2}+c_{2}}{12}+\frac{c_{1}c_{2}}{24}-\frac{c_{1}^{4}-4c_{1}^{2}c_{2}-3c_{2}^{2}-c_{1}c_{3}+c_{4}}{720}+\ldots.$

As a consequence of the Hirzebruch-Riemann-Roch Theorem, we have the
_Noether’s identities_ , i.e., the analytic Euler characteristic are equal to
the $d$-th summand of $\mbox{td}({\mathcal{T}}_{Z})$. For example,

$\chi({\mathcal{O}}_{Z})=\dfrac{c_{1}c_{2}}{24},\quad\text{when }d=3.$

######  Definition 2.1.

A _simple normal crossing_ (SNC) divisor $D=\sum_{j=1}^{r}D_{j}$ is a reduced
effective divisor with distinct non-singular components $D_{j}$ satisfying the
following condition: for each $p\in D$ there are local coordinates
$x_{1},\ldots,x_{d}$ on $Z$ such that the equation defining $D$ on $p$ is
$x_{1}\ldots x_{e}=0$, with $e\leq d$.

From [Iit77] we introduce the following sheaf on $Z$.

######  Definition 2.2.

For a SNC divisor $D$, the _sheaf of $\log$-differentials along $D$_, denoted
by $\Omega^{1}_{Z}(\log D)$, is the ${\mathcal{O}}_{Z}$-submodule of
$\Omega^{1}_{Z}\otimes{\mathcal{O}}_{Z}(D)$ described as follows. Let $p\in Z$
be a point.

  * (i)

If $p\not\in D$, then $(\Omega^{1}_{Z}(\log D))_{p}=\Omega^{1}_{Z,p}$.

  * (ii)

If $p\in D$, we choose local coordinates $x_{1},\ldots,x_{d}$ on $Z$ with
$x_{1}\ldots x_{e}=0$ defining $D$ on $p$. Then, $(\Omega^{1}_{Z}(\log
D))_{p}$ is generated as ${\mathcal{O}}_{Z,p}$-module by

$\frac{dx_{1}}{x_{1}},\ldots,\frac{dx_{e}}{x_{e}},dx_{e+1},\ldots,dx_{d}.$

If $D=\sum_{j=1}^{r}\nu_{j}D_{j}$ is a divisor on $Z$, whose associated
reduced divisor $D_{red}=\sum_{j}D_{j}$ is a SNC divisor, then for simplicity
we set

$\Omega_{Z}^{1}(\log D):=\Omega_{Z}^{1}(\log D_{red}).$

In the rest of this section, we assume $D=\sum_{j=1}^{r}\nu_{j}D_{j}$ as a
divisor with $D_{red}$ a SNC divisor.

######  Definition 2.3.

The _$\log$ -Chern classes_ of a pair $(Z,D)$ are defined as

$\bar{c}_{i}(Z,D)=c_{i}(\Omega_{Z}(\log D)^{\vee}).$

The _$\log$ -Chern numbers_ of a pair $(Z,D)$ are defined as the degree of
top-dimensional intersections

$\bar{c}_{i_{1}}\ldots\bar{c}_{i_{m}}:={\bar{c}}_{i_{1}}(Z,D)\ldots{\bar{c}}_{i_{m}}(Z,D),\quad
i_{1}+\ldots+i_{m}=d.$

We set $\Omega^{e}_{Z}(\log D):=\bigwedge^{e}\Omega_{Z}^{1}(\log D)$ for any
$1\leq e\leq d$. In this way, $\Omega^{d}_{Z}(\log
D)={\mathcal{O}}_{Z}(K_{Z}+D_{red})$, i.e.,
${\bar{c}}_{1}(Z,D)=c_{1}(Z)-D_{red}$, and it is known that,

${\bar{c}}_{d}=e(Z)-e(D_{red}),$

see [Iit78, Prop. 2].

######  Lemma 2.4.

We have a natural exact sequence

$0\to\Omega^{1}_{Z}\to\Omega^{1}_{Z}(\log
D)\to\bigoplus_{i=1}^{r}{\mathcal{O}}_{D_{i}}\to 0,$

which is known as the _residual exact sequence_.

###### Proof.

See [EV92, Proposition 2.3]. ∎

From the residual exact sequence, we can compute the Chern polynomial through
the identity

$c_{t}(\Omega_{Z}^{1}(\log
D))=c_{t}(\Omega_{Z}^{1})\prod_{j=1}^{r}\left(\sum_{e=0}^{d}D_{j}^{e}t^{e}\right).$
(2)

Let $0\leq e\leq d$, and let $i_{1}+\ldots+i_{m}=e$ be any partition of
positive integers. By convention, for the case $e=0$ we assume the existence
of a unique partition, i.e., $i_{1}=0$. We introduce the following notation,

$D^{[i_{1},...,i_{m}]}:=\sum_{j_{1}<\ldots<j_{m}}D_{j_{1}}^{i_{1}}\ldots
D_{j_{m}}^{i_{m}},\quad D^{[0]}=1.$

Examples of this notation are,

$D^{[e]}=\sum_{j=1}^{r}D_{j}^{e},\quad D^{[1,1]}=\sum_{j<k}D_{j}D_{k},$

and in general,

$\prod_{j=1}^{r}\left(\sum_{e=0}^{d}D^{e}\right)=\sum_{e=0}^{d}\left(\sum_{i_{1}+\ldots+i_{m}=e}D^{[i_{1},\ldots,i_{m}]}\right).$
(3)

######  Corollary 2.5.

We have the identity,

$\bar{c}_{d}(Z,D)=c_{d}(Z)+\sum_{e=1}^{d}(-1)^{e}c_{d-e}(Z)\left(\sum_{i_{1}+\ldots+i_{m}=e}D^{[i_{1},\ldots,i_{m}]}\right).$

###### Proof.

Using identity (2), and the previous identity (3) we can compute $\bar{c}_{d}$
as the degree $d$ element of the expression

$\left(\sum_{e=0}^{d}c_{e}(Z)\right)\left(\sum_{e=0}^{d}\left(\sum_{i_{1}+\ldots+i_{m}=e}D^{[i_{1},\ldots,i_{m}]}\right)\right).$

∎

######  Corollary 2.6.

Assume $D$ is non-singular, i.e., its non-singular components are pairwise
disjoint. Then for each $1\leq e\leq d$ we have
$\bar{c}_{e}(Z,D)=c_{e}(Z)+R_{e}(D)$ where

$R_{e}(D)=\sum_{\begin{subarray}{c}k+l=e\\\ k\neq
e\end{subarray}}(-1)^{l}D^{[l]}c_{k}(Z),$

for each $e=1,\ldots,d$.

###### Proof.

Since $D$ is non-singular, then $D_{i}D_{j}=0$ for all $i\neq j$. Thus, from
the identity (2) , we get

$\sum_{e=0}^{d}(-1)^{e}\bar{c}_{e}(Z,D)t^{e}=\left(\sum_{e=0}^{d}(-1)^{e}c_{e}(Z)t^{e}\right)\left(\sum_{e=0}^{d}D^{[e]}t^{e}\right).$

Using the Cauchy product formula for polynomials we have

$(-1)^{e}\bar{c}_{e}(Z,D)=\sum_{k+l=e}(-1)^{k}c_{k}(Z)D^{[l]},$

and from this, the formula follows. ∎

######  Corollary 2.7.

Consider a non-singular $3$-fold $Z$, and $D=\sum_{j}\nu_{j}D_{j}$ on $Z$. We
have

${\bar{c}}_{2}(Z,D)=c_{2}(Z)-D_{red}(c_{1}-D_{red})-D^{[1,1]}.$

Thus, the logarithmic Chern numbers for $3$-folds are,

$\bar{c}_{1}^{3}=c_{1}^{3}-3c_{1}^{2}D_{red}+3c_{1}([D^{[2]}+2D^{[1,1]})+D^{[3]}+3(D^{[1,2]}+D^{[2,1]})+6D^{[1,1,1]}$
$\bar{c}_{1}\bar{c}_{2}=c_{1}c_{2}-D_{red}(c_{1}^{2}+c_{2})+c_{1}(2D^{[2]}+3D^{[1,1]})-D_{red}(D^{[2]}+D^{[1,1]}).$
$\bar{c}_{3}=c_{3}-c_{2}D_{red}+c_{1}(D^{[2]}+D^{[1,1]})-\left(D^{[3]}+D^{[1,2]}+D^{[2,1]}+D^{[1,1,1]}\right).$

###### Proof.

Using identity (2) for $d=3$, and looking for the degree $2$ terms we obtain
$\bar{c}_{2}(Z,D)$. The other formulas are direct computations using the above
lemmas. ∎

######  Example 2.8.

For $Z$ a non-singular curve, set $D=\nu_{j}P_{1}+\ldots+\nu_{r}P_{r}$ where
$P_{i}\in Z$ are points. So for the unique $\log$-Chern number we have

$\bar{c}_{1}=c_{1}-r=-(2g(X)-2+r).$

Let $Z$ be a non-singular surface, and $D=\sum_{j=1}^{r}\nu_{j}D_{j}$ with
$D_{j}$ non-singular curves. Then

$\bar{c}_{1}^{2}=c_{1}^{2}-2c_{1}D_{red}+D_{red}^{2},\bar{c}_{2}=c_{2}+t_{2}+2\sum_{j=1}^{r}(g(D_{j})-2),$

Where $t_{2}$ is the number of nodes of $D$. See [Urz09, Prop. 3.1]. As in the
case of nodes for surfaces, let us denote the number of triple points of $D$
by $t_{3}$. Then we can rewrite

$\bar{c}_{3}=c_{3}-c_{2}D_{red}+c_{1}(D^{[2]}+D^{[1,1]})-\left(D^{[3]}+D^{[1,2]}+D^{[2,1]}+t_{3}\right).$

######  Example 2.9.

Let $Z\hookrightarrow{\mathbb{P}}^{m}$ be a non-singular projective $3$-fold.
Let $H_{1},\ldots,H_{r}\sim H$ hyperplane sections defining an SNC
arrangement. We have

$\bar{c}_{1}^{3}=c_{1}^{3}(Z)-r^{3}\deg(Z)-3rc_{1}^{2}H+3c_{1}(Z)H^{2}$
$\bar{c}_{1}\bar{c}_{2}=c_{1}c_{2}(Z)-rH(c_{1}^{2}+c_{2})(Z)+\left(2r+3\binom{r}{2}\right)c_{1}(Z)H^{2}-\deg(Z)r\left(r+\binom{r}{2}\right),$
$\bar{c}_{3}=c_{3}(Z)-rHc_{2}(Z)+\left(r+\binom{r}{2}\right)c_{1}(Z)H^{2}-\deg(Z)\left(r+2\binom{r}{2}+\binom{r}{3}\right).$

Thus as $r$ grows, we have

$\lim_{r\to\infty}\frac{\bar{c}_{1}^{3}}{\bar{c}_{1}\bar{c}_{2}}=\lim_{r\to\infty}\frac{r^{3}}{r\binom{r}{2}}=2,\
\ \text{and}\ \
\lim_{r\to\infty}\frac{\bar{c}_{3}}{\bar{c}_{1}\bar{c}_{2}}=\lim_{r\to\infty}\frac{\binom{r}{3}}{r\binom{r}{2}}=\frac{1}{3}.$

Logarithmic Chern classes are well-behaved under logarithmic morphisms.

######  Definition 2.10.

Let $Z$ be a non-singular projective variety and $D$ an effective divisor with
$D_{red}$ as SNC divisor. A surjective morphism $h\colon Y\to Z$ between non-
singular projective varieties is called a _$\log$ -morphism_, if
$D^{\prime}_{red}=(h^{*}D)_{red}$ is a SNC divisor.

######  Lemma 2.11.

For any $\log$-morphism $h\colon(Y,D^{\prime})\to(Z,D)$, we have an injection

$h^{*}\Omega_{Z}(\log D)\hookrightarrow\Omega_{Y}(\log D^{\prime}).$

Moreover, if $h$ is finite and ramified at $D$, then we have isomorphism
outside the singularities of $D$.

###### Proof.

See [Vie82, Lemma 1.6]. ∎

###  2.2. $n$-th root covers

In this section, we follow [EV92, Sec. 3]. Consider the following building
data $(Z,D,n,{\mathcal{L}})$ where

  1. (1)

$Z$ is a non-singular projective variety of dimension $d$,

  2. (2)

$D=\sum_{j=1}^{r}\nu_{j}D_{j}$ is an effective divisor on $Z$, with
$D_{red}=\sum_{j=1}^{r}D_{j}$ a SNC divisor,

  3. (3)

$n\geq 2$ is a prime number, and

  4. (4)

${\mathcal{L}}$ a line bundle on $Z$ such that
${\mathcal{O}}_{Z}(D)\simeq{\mathcal{L}}^{\otimes n}$.

With this building data, we construct

$f^{\prime}\colon
Y^{\prime}_{n}=\mbox{Spec}_{Z}\bigoplus_{i=0}^{n-1}{\mathcal{L}}^{-i}\to Z,$

and the normalization $f\colon Y_{n}\to Y^{\prime}_{n}\to Z$ will be called
the _$n$ -th root covering_ associated to the building data
$(Z,D,n,{\mathcal{L}})$. Since the morphisms are finite of degree prime, both
$Y^{\prime}_{n}$ and $Y_{n}$ are projective and irreducible. The morphism $f$
is branched at $D$, and $Y_{n}$ has its singularities over the intersection of
components of $D$. Thus, $Y_{n}$ is non-singular if $D_{j}\cap
D_{k}=\emptyset$ for all $j,k$. On the other hand, if $D_{j_{1}}\cap\ldots\cap
D_{j_{e}}\neq\emptyset$, then the intersection is defined by a local equation

$z_{j_{1}}^{\nu_{j_{1}}}\ldots z^{\nu_{j_{e}}}_{j_{e}}=0,$

where $z_{1},\ldots,z_{d}$ are local parameters for $Z$ on $p$. Thus the
singularity of $Y_{n}$ over $D_{j_{1}}\cap\ldots\cap D_{j_{e}}$ is locally
analytically isomorphic to the normalization of

$\mbox{Spec}\left(\frac{{\mathbb{C}}[z_{1},\ldots,z_{d},t]}{t^{n}-z_{j_{1}}^{\nu_{j_{1}}}\ldots
z^{\nu_{j_{e}}}_{j_{e}}}\right).$

######  Definition 2.12.

A _partial resolution of singularities_ of $Y_{n}$ is a projective,
surjective, birational morphism $g:X\to Y_{n}$ with $X$ a projective normal
variety having at most rational singularities. This last means that for any
resolution of singularities $g^{\prime}\colon X^{\prime}\to X$ we have
$R^{i}g^{\prime}_{*}{\mathcal{O}}_{X^{\prime}}=0$ for $i>0$. As usual, we omit
the word _partial_ if $X$ is non-singular.

Since the degree $n$ of $f$ is a prime number, we have
$f^{*}D_{j}=nD^{\prime}_{j}$, where $D_{j}^{\prime}=(f^{*}D_{j})_{red}$. Thus,
for any partial resolution $h\colon X\to Y_{n}\to Z$ we must have a
ramification formula

$h^{*}D_{j}=nD^{\prime}_{j}+\Delta_{j},$

where $\Delta_{j}$ is a divisor supported in the exceptional divisors of $h$
over $D_{j}$.

######  Theorem 2.13.

For the $n$-th root covering $f\colon Y_{n}\to Z$ we have

  1. (1)

The morphism $f$ is flat.

  2. (2)

The variety $Y_{n}$ has rational singularities.

  3. (3)

We have the following decomposition on eigenspaces

$f_{*}{\mathcal{O}}_{Y_{n}}=\bigoplus_{i=0}^{n-1}{\mathcal{O}}_{Z}(-L^{(i)}),\quad
L^{(i)}=-iL+\sum_{j}\left\lfloor\frac{i\nu_{j}}{n}\right\rfloor D_{j}.$

Indeed, $Y_{n}=\mbox{Spec}f_{*}{\mathcal{O}}_{Y_{n}}$. Therefore $f:Y_{n}\to
Z$ is an affine morphism.

  4. (4)

For any partial resolution of singularities $g\colon X\to Y_{n}$, the
composition $h=f\circ g$ satisfies the following ${\mathbb{Q}}$-numerical
equivalence

$K_{X}\sim_{{\mathbb{Q}}}h^{*}\left(K_{Z}+\frac{n-1}{n}D_{red}\right)+\Delta,$

where $\Delta$ is a divisor supported on the exceptional divisor of $g$.

###### Proof.

For (1), (2), and (3) see [EV92, Ch. 3]. For (4), we use the fact that
$\operatorname{codim}(\mbox{Sing}(Y_{n}))\geq 2$. In this way, there are non-
singular open sets $U\hookrightarrow X\to V\hookrightarrow Z$ avoiding the
singularities of $D$. From Lemma 2.11, we get $h^{*}\Omega^{d}_{U}(\log
D)=\Omega^{d}_{V}(\log D^{\prime})$ where $D^{\prime}$ is the strict transform
of $D$. In terms of divisors,

$h^{*}(K_{V}+D_{red})=K_{U}+D^{\prime}_{red}\sim_{{\mathbb{Q}}}K_{U}+\frac{h^{*}(D_{red})}{n},$

from where the result holds locally. After extending globally the exceptional
term $\Delta$ appears. ∎

The following corollary will be useful in Section 5. First recall that on a
variety $X$ a curve $C$ is _$K_{X}$ -negative, $K_{X}$-positive or
$K_{X}$-trivial_ if its intersection with the canonical divisor is negative,
positive, or zero.

######  Corollary 2.14.

The ${\mathbb{Q}}$-divisor
$h^{*}\left(\frac{n-1}{n}\sum_{i=1}^{r}D_{i}\right)+\Delta$ is an effective
${\mathbb{Z}}$-divisor. Thus, if $K_{Z}$ is nef, then the $K_{X_{n}}$-negative
curves of $X_{n}$ are contained in the support of $h^{*}D$.

###### Proof.

The first assertion is a direct consequence of 4.5. Now assume that $K_{Z}$ is
nef, and let $C$ be a curve in $X$. If $C$ is not contained in the support of
$h^{*}D$, then

$K_{X_{n}}\,C=\left(h^{*}\left(K_{Z}+\frac{n-1}{n}\sum_{i=1}^{r}D_{i}\right)+\Delta\right).C>0,$

since $h^{*}K_{Z}$ is nef (projection formula), and effectiveness of
$h^{*}\left(\frac{n-1}{n}\sum_{i=1}^{r}D_{i}\right)+\Delta$. Thus, if $C$ is
negative must lie in the support of $h^{*}D$.
∎

######  Lemma 2.15.

Let $Y$ be a normal variety and $g:X\to Y$ a proper, surjective, birational
morphism. Assume that $X$ has rational singularities. Then
$g_{*}{\mathcal{O}}_{X}={\mathcal{O}}_{Y}$ and $R^{i}g_{*}{\mathcal{O}}_{X}=0$
for all $i>0$ if and only if $Y$ has rational singularities.

###### Proof.

See [Vie77, Lemma 1].∎

######  Corollary 2.16.

For any partial resolution of singularities $g:X\to Y_{n}$, we have
$\chi({\mathcal{O}}_{X})=\sum_{i=0}^{n-1}\chi({\mathcal{O}}_{Z}(-L^{(i)}))$,
i.e., the analytic Euler characteristic of $X$ is independent of the chosen
partial resolution.

###### Proof.

Let $g^{\prime}\colon X^{\prime}\to X$ be a resolution of singularities. Since
$Y_{n}$ has rational singularities, by Lemma 2.15, we must have
$h_{*}{\mathcal{O}}_{X^{\prime}}=g_{*}{\mathcal{O}}_{X}={\mathcal{O}}_{Y}$,
and $R^{i}h_{*}{\mathcal{O}}_{X^{\prime}}=R^{i}g_{*}{\mathcal{O}}_{X}=0$ for
all $i>0$. Thus,
$\chi({\mathcal{O}}_{X^{\prime}})=\chi({\mathcal{O}}_{X})=\chi({\mathcal{O}}_{Y_{n}})$.
Then, we assume that $g$ is just a resolution of singularities, so

$H^{i}(X,{\mathcal{O}}_{X})\cong H^{i}(Y_{n},g_{*}{\mathcal{O}}_{X})\cong
H^{i}(Y_{n},{\mathcal{O}}_{Y_{n}}),\quad i\geq 0.$

Since $f$ is an affine morphism, also we have
$H^{i}(Y_{n},{\mathcal{O}}_{Y_{n}})\cong H^{i}(Z,f_{*}{\mathcal{O}}_{Y_{n}})$
for all $i\geq 0$. Thus, we have
$\chi({\mathcal{O}}_{X})=\sum_{i=0}^{n-1}\chi({\mathcal{O}}_{Z}(-L^{(i)}))$. ∎

Finally, we give a state about the connectedness of a partial resolution.

######  Proposition 2.17.

Any partial resolution $g:X\to Y_{n}$ is irreducible.

###### Proof.

Since $X$ is normal we reduce the proof to show that $X$ is connected [Sta18,
Tag. 0347]. From Corollary 2.16 we know that

$h^{0}({\mathcal{O}}_{X})=1+\sum_{i=1}^{n-1}h^{0}({\mathcal{O}}_{Z}(-L^{(i)})).$

If $Y$ is not connected, then $h^{0}({\mathcal{O}}_{X})>1$, so there exists a
$i\geq 1$ such that $h^{0}({\mathcal{O}}_{Z}(-L^{(i)}))\geq 1$. We have,

$-nL^{(i)}\sim\sum_{j=1}^{r}\\{i\nu_{j}\\}_{n}D_{j}.$

So, we choose curves $\Gamma_{j}$ on $Z$ such that $D_{j}\,\Gamma_{j}>0$.
Thus, we get a system of equations $A\nu\equiv 0\mod n$ where
$\nu=[\nu_{1},...,\nu_{r}]^{T}$ and $A=(D_{j}\Gamma_{k})_{jk}$. Since $n$ is
prime, $\nu\equiv 0\mod n$, and since $0<\nu_{j}<n$, we get a contradiction. ∎

###  2.3. Toric picture

In this section, for toric varieties we mainly follow the notation of [CLS11].

Let $n>0$ be a prime number and $0\leq\nu_{1},\ldots,\nu_{d}<n$ integers.
Choose a $\nu_{k}\neq 0$, and let $0\leq q_{1},\ldots,q_{d}<n$ be integers
such that $\nu_{j}+q_{j}\nu_{k}\equiv 0$ modulo $n$. In particular,
$q_{k}=n-1$. As usual set $N={\mathbb{Z}}^{d}$ and
$N_{{\mathbb{R}}}\cong{\mathbb{R}}^{d}$ with canonical basis
$e_{1},\ldots,e_{d}$, and $M=N^{\vee}\cong{\mathbb{Z}}^{d}$ with
$M_{{\mathbb{R}}}={\mathbb{R}}^{d}$. Consider the semigroup

$S=\left\langle e_{1},\ldots,e_{k-1},\sum_{j\neq
k}q_{j}e_{j}+ne_{k},e_{k+1},\ldots,e_{d},\sum_{j\neq
k}\frac{\nu_{j}+q_{j}\nu_{k}}{n}e_{j}+\nu_{k}e_{k}\right\rangle_{{\mathbb{N}}}.$

Since

$\sum_{j\neq k}\frac{\nu_{j}+q_{j}\nu_{k}}{n}e_{j}+\nu_{k}e_{k}=\sum_{j\neq
k}\frac{\nu_{j}}{n}e_{j}+\frac{\nu_{k}}{n}\left(\sum_{j\neq
k}q_{j}e_{j}+ne_{k}\right),$

we have that the saturation [CLS11, p. 27] of $S$ is
$S^{sat}=\sigma^{\vee}\cap{\mathbb{Z}}^{d}$ where

$\sigma^{\vee}=C\left(e_{1},\ldots,e_{k-1},\sum_{j\neq
k}q_{j}e_{j}+ne_{k},e_{k+1},\ldots,e_{d-1},e_{d}\right)\subset
M_{{\mathbb{R}}}$

is the simplicial $d$-cone defined by those elements. It is the dual cone of

$\sigma=C\left(ne_{1}-q_{1}e_{k},\ldots,ne_{k-1}-q_{k-1}e_{k},e_{k},ne_{k+1}-q_{k+1}e_{k},\ldots,ne_{d}-q_{d}e_{k}\right)\subset
N_{{\mathbb{R}}}.$

Observe that $\mbox{mult}(\sigma)=n^{d-1}$ and $\mbox{mult}(\sigma^{\vee})=n$.
Let $P_{\sigma}$ the fundamental parallelepiped of $\sigma$, i.e., the points
of $\sigma$ with coordinates in $[0,1)$ respect its generators. Direct
computations show that every element of $v\in P_{\sigma}\cap{\mathbb{Z}}^{d}$
can written as

$v=\frac{\sum_{i\neq j}v_{i}(ne_{i}-q_{i}e_{j})+\left\\{\sum_{i\neq
j}v_{i}q_{i}\right\\}_{n}e_{j}}{n},\quad 0\leq v_{i}<n.$ (4)

######  Proposition 2.18.

The toric variety associated with the semigroup $S$ is

$\mbox{Spec}({\mathbb{C}}[S])=\mbox{Spec}\left(\frac{{\mathbb{C}}[x_{1},\ldots,x_{d},t]}{(t^{n}-x_{1}^{\nu_{1}}\ldots
x_{d}^{\nu_{d}})}\right).$

Moreover, its normalization correspond with
$\mbox{Spec}({\mathbb{C}}[\sigma^{\vee}\cap{\mathbb{Z}}^{d}])$.

###### Proof.

For simplicity, we will prove the result in the case $k=d$. Since
$Norm(\mbox{Spec}({\mathbb{C}}[S]))=\mbox{Spec}({\mathbb{C}}[S^{sat}])$ we
will prove the first. Take the surjective morphism of semigroups
$\phi\colon{\mathbb{N}}^{d+1}\mapsto S$ such that

$\phi(e_{j})=e_{j},\quad\phi(e_{d})=\sum_{i=1}^{d-1}q_{j}e_{j}+ne_{d},\quad\phi(e_{d+1})=\sum_{j=1}^{d-1}\frac{\nu_{j}+q_{j}\nu_{d}}{n}e_{j}+\nu_{d}e_{d}.$

It induces a surjective morphism of coordinate rings
$f\colon{\mathbb{C}}[x_{1},\ldots,x_{d},t]\to{\mathbb{C}}[S]$, and by [CLS11]
in Proposition 1.1.9 it is known that

$\mbox{ker}(f)=(x_{1}^{a_{1}}\ldots
x_{d}^{a_{d}}t^{a_{d+1}}=x_{1}^{b_{1}}\ldots
x_{d}^{b_{d}}t^{b_{d+1}}:\phi(a)=\phi(b),a,b\in{\mathbb{N}}^{d+1}).$

If we set $x_{j}=a_{j}-b_{j}$, the condition $\phi(a)=\phi(b)$ gives equations

$\left\\{\begin{array}[]{ccc}x_{1}+q_{1}x_{d}+\dfrac{\nu_{1}+q_{1}\nu_{d}}{n}x_{d+1}&=&0\\\
\ldots&\ldots&\ldots\\\
x_{d-1}+q_{d-1}x_{d}+\dfrac{\nu_{d-1}+q_{d-1}\nu_{d}}{n}x_{d+1}&=&0\\\
nx_{d}+\nu_{d}x_{d+1}&=&0\end{array}\right.$

We can assume $x_{d+1}=nc$ with $c>0$, then $x_{d}=-\nu_{d}c$, and the
equations reduces to

$\left\\{\begin{array}[]{ccc}x_{1}&=&-\nu_{1}c\\\ \ldots&\ldots&\ldots\\\
x_{d-1}&=&-\nu_{d-1}c\end{array}\right.\Rightarrow\left\\{\begin{array}[]{cccc}b_{j}&=&a_{j}+\nu_{j}c,&1\leq
j\leq d\\\ a_{d+1}&=&b_{d+1}+nc&\end{array}\right.$

So $\mbox{ker}(f)$ is generated by elements of the form

$x_{1}^{a_{1}}\ldots x_{d}^{a_{d}}t^{b_{d+1}}((x_{1}^{\nu_{1}}\ldots
x_{d}^{\nu_{d}})^{c}-(t^{n})^{c}),$

and the result follows. ∎

######  Corollary 2.19.

The normalization of the affine varieties $t^{n}=x_{1}^{\nu_{1}}\ldots
x_{d}^{\nu_{d}}$ and $t^{n}=x_{j}\prod_{j\neq k}x_{i}^{n-q_{j}}$ are
isomorphic.

###### Proof.

Observe that the cones defining both varieties are the same, equal to
$\sigma$. ∎

######  Remark 2.20.

We point out the following. Assume that $j=e$ and
$\nu_{e+1}=\ldots=\nu_{d}=0$. Thus $q_{e+1}=\ldots=q_{d}=0$, and we have a
toric description of the normalization of the varieties
$t^{n}=x_{1}^{\nu_{1}}...x_{e}^{\nu_{e}}$ embedded in ${\mathbb{A}}^{d}$ for
any $1\leq e\leq d$.

######  Remark 2.21.

It is known that the cone $\sigma^{\vee}$ defines a toric variety isomorphic
to the $d$-dimensional quotient cyclic singularity of type

$\frac{(n-q_{1},\ldots,n-q_{k-1},1,n-q_{k+1},\ldots,n-q_{d})}{n}.$

We have
$\mbox{Spec}(\sigma\cap{\mathbb{Z}}^{3})\cong{\mathbb{C}}^{d}/\langle\phi\rangle$,
where $\phi\colon{\mathbb{C}}^{d}\to{\mathbb{C}}^{d}$ is defined by

$\phi\colon(z_{1},\ldots,z_{d})\mapsto(\zeta^{n-q_{1}}z_{1},\ldots,\zeta
z_{j},\ldots,\zeta^{n-q_{d}}z_{d}),$

with $\zeta^{d}=1$ (Cf. [Ash15]). In this way, quotient cyclic singularities
are geometrically dual to the singularities of $n$-th root covers. We will
denote a cyclic singularity of this type by
$C_{q_{1},...,\hat{q_{k}},...,q_{d}}$. In dimension $2$, it occurs the
accident that singularities of $n$-th root covers are also cyclic quotient
singularities.

###  2.4.  Hirzebruch-Jung algorithm and Dedekind sums

####  2.4.1. Planar cones and Hirzebruch-Jung algorithm

Set $N\cong{\mathbb{Z}}^{d}$ and $M=N^{\vee}$. A _planar cone_ $\tau$ in
$N_{{\mathbb{R}}}$ is a cone of dimension $2$, i.e., it is generated by two
rays defined by primitive generators $v_{0},v_{s+1}\in N$ ($s$ will have sense
soon). Assume that $\mbox{mult}(\tau)=n$. It is known that if $n>1$, then
there exists some $v\in\tau\cap N$ such that $v_{0},v$ generate $\tau\cap N$,
i.e., $|\det(v_{0},v)|=1$. If $v=c_{1}v_{0}+c_{2}v_{s+1}$, with
$c_{i}\in{\mathbb{Q}}_{\geq 0}$, then

$\det(v_{0},v)=nc_{2}=1\Leftrightarrow c_{2}=1/n.$

On the other hand, since $\det(v,v_{s+1})\in{\mathbb{N}}$, we have
$c_{1}\in\frac{1}{n}{\mathbb{N}}$. If we set $q=nc_{1}$, we say that $\tau$
_is of type $(n,q)$ in direction $v_{0}$ to $v_{s+1}$_, or type
$(n,q^{\prime})$ in the opposite direction, where $q^{\prime}$ is the inverse
modulo $n$ of $q$.

Assume that $n,q$ are coprime, then consider the Hirzebruch-Jung algorithm of
division for $n/q$, i.e., a pair of sequences

$m_{0}=n>m_{1}=q>\ldots>m_{s}=1>m_{s+1}=0,$
$n_{0}=0<n_{1}=1<\ldots<n_{s}=q^{\prime}<n_{s+1}=n,$

which are related by

$m_{\alpha+1}=k_{\alpha}m_{\alpha}-m_{\alpha-1}$
$n_{\alpha+1}=k_{\alpha}n_{\alpha}-n_{\alpha-1},$

where $k_{1},\ldots,k_{s}$ are integers satisfying $k_{\alpha}\geq 2$. Usually
we denote $n/q=[k_{1},\ldots,k_{s}]$. These sequences define the Hirzebruch-
Jung continuous fraction as

$\frac{n}{q}=k_{1}-\cfrac{1}{k_{2}-\cfrac{1}{\ddots-\frac{1}{k_{s}}}}.$

######  Remark 2.22.

Observe that sequence $n_{\alpha}$ is the sequence $m_{\alpha}$ for
$n/q^{\prime}$, i.e., if the pair $(m_{\alpha}^{\prime},n_{\alpha}^{\prime})$
is the resolution of $n/q^{\prime}$ then
$(m_{\alpha}^{\prime},n_{\alpha}^{\prime})=(n_{s+1-\alpha},m_{s+1-\alpha})$.

######  Lemma 2.23.

For each $\alpha$, we have the following relations

  1. (1)

$m_{\alpha}n_{\alpha+1}-m_{\alpha+1}n_{\alpha}=n,$

  2. (2)

$\mbox{gcd}(m_{\alpha},m_{\alpha+1})=1,$

  3. (3)

$\mbox{gcd}(m_{\alpha},n_{\alpha})=1.$

∎

The non-singular resolution of the planar cone $\tau$ is a refinement by
adding the rays defined recursively by

$v_{\alpha}=\frac{m_{\alpha}v_{\alpha-1}+v_{s+1}}{m_{\alpha-1}}=\frac{m_{\alpha}v_{0}+n_{\alpha}v_{s+1}}{n},\quad
1\leq\alpha\leq s.$

See Figure 1. Each cone $C(v_{\alpha},v_{\alpha+1})$ is non-singular, since

$\det(v_{\alpha},v_{\alpha+1})=\frac{1}{n}(m_{\alpha}n_{\alpha+1}-m_{\alpha+1}n_{\alpha})=1.$

From Remark 2.22 observe that we have a dual non-singular resolution given by
the sequence

$v_{\alpha}^{\prime}=\frac{m^{\prime}_{\alpha}v_{\alpha-1}^{\prime}+v_{0}}{m^{\prime}_{\alpha-1}}=\frac{m^{\prime}_{\alpha}v_{s+1}+n^{\prime}_{\alpha}v_{0}}{n},\quad
1\leq\alpha\leq s.$

from where we have $v_{\alpha}=v^{\prime}_{s+1-\alpha}$.

Figure 1. Resolved planar cone

Let us change the ${\mathbb{Z}}$-base of $N$ such that $e_{1}=v$ and
$e_{2}=v_{0}$. So, we have $v_{s+1}=ne_{1}-qe_{2}$. We have,

$\tau^{\vee}=C(e_{1},pe_{1}+ne_{2})\oplus\bigoplus_{i=3}^{d-3}{\mathbb{R}}w_{i}=C(w_{1},w_{2})\oplus{\mathbb{R}}^{d-2},$

where the $w_{i}\in M$ are such that $\langle v_{0},w_{i}\rangle=\langle
v_{s+1},w_{i}\rangle=0$ and $\langle\cdot,w\rangle\geq 0$ on $\tau$ for any
$w\in C(w_{1},w_{2})$. Thus, in terms of toric varieties,
$X_{\tau}=C_{q}\times({\mathbb{C}}^{\times})^{d-2}$, where
$C_{q}=\mbox{Spec}({\mathbb{C}}[C(w_{1},w_{2})\cap M])$ is the cyclic quotient
surface singularity of type $\frac{1}{n}(q,1)$ (Remark 2.21). Thus, the
constructed resolution is a blow-up $h:\mbox{Bl}_{{\mathfrak{a}}}(X_{\tau})\to
X_{\tau}$, where ${\mathfrak{a}}={\mathfrak{m}}\otimes{\mathbb{C}}[x_{3}^{\pm
1},...,x_{d}^{\pm 1}]$ and

${\mathfrak{m}}=\bigoplus_{w\in C(w_{1},w_{2})\cap(M\setminus 0)}\chi^{m}.$

For details see [CLS11, 11.3.6]. In particular,
$\mbox{Bl}_{{\mathfrak{a}}}(X_{\tau})=\mbox{Bl}_{{\mathfrak{m}}}(X_{\tau})\times({\mathbb{C}}^{\times})^{d-2}$.
We can give a explicit description of ${\mathfrak{m}}$ noting that the
projection $C_{q}\to{\mathbb{A}}^{2}$ is given by the surjection
${\mathbb{C}}[\chi^{w_{1}},\chi^{w_{2}}]\to{\mathbb{C}}[C_{q}]$.

####  2.4.2. Dedekind sums

Consider the _sawtooth function_ $((x))\colon{\mathbb{R}}\to{\mathbb{R}}$ is
defined as

$((x))=\left\\{\begin{array}[]{cc}x-\lfloor
x\rfloor-1/2&x\in{\mathbb{R}}\setminus{\mathbb{Z}}\\\
0&x\in{\mathbb{Z}}\end{array}\right..$

Observe that is an odd periodic function of period $1$. For $n\geq 3$ prime
and $a_{1},\ldots,a_{d}\in{\mathbb{Z}}$ define the _Dedekind sum_ of dimension
$d$ by

$d(a_{1},\ldots,a_{d},n)=\sum_{i=1}^{n-1}\left(\left(\frac{ia_{1}}{n}\right)\right)\cdots\left(\left(\frac{ia_{d}}{n}\right)\right).$

By periodicity, we can reduce $a_{i}\in{\mathbb{Z}}$ to $0\leq a_{i}<n$, then

$d(a_{1},\ldots,a_{d},n)=d(\\{a_{1}\\}_{n},\ldots,\\{a_{d}\\}_{n},n).$

where $\\{a_{i}\\}_{n}$ is the residue modulo $n$ of $a_{i}$. Since $((x))$ is
an odd function, we always have

$d(-a_{1},\ldots,-a_{d},n)=(-1)^{d}d(a_{1},\ldots,a_{d},n),\quad\forall
x\in{\mathbb{Z}},\quad(x,n)=1,$

i.e., $d(a_{1},...,a_{d},n)=0$ for any odd dimension $d$. Therefore, the non-
trivial Dedekind sums are those of even dimension. We can rewrite this sum as

$d(a_{1},\ldots,a_{d},n)=\frac{1}{n^{d}}\sum_{i=1}^{n-1}\left(\\{ia_{1}\\}_{n}-\frac{n}{2}\right)\cdots\left(\\{ia_{d}\\}_{n}-\frac{n}{2}\right).$

######  Lemma 2.24.

We have the following relations,

$\sum_{i=1}^{n-1}\\{ia\\}_{n}\\{ib\\}_{n}=n^{2}d(a,b,n)+\frac{n^{2}(n-1)}{4},$
$\sum_{i=1}^{n-1}\\{ia\\}_{n}^{2}\\{ib\\}_{n}=\sum_{i=1}^{n-1}\\{ia\\}_{n}\\{ib\\}_{n}^{2}=n^{3}d(a,b,n)+\frac{n^{2}(n-1)(2n-1)}{12}$
$\sum_{i=1}^{n-1}\\{ia\\}_{n}\\{ib\\}_{n}\\{ic\\}_{n}=\frac{n^{3}}{2}\left(d(a,b,n)+d(a,c,n)+d(b,c,n)\right)+\frac{n^{3}(n-1)}{8},$

###### Proof.

We have an identity

$\sum_{i=1}^{n-1}\\{ia\\}_{n}^{k}=\sum_{i=1}^{n-1}i^{k},$

for each $k\geq 0$ integer. Then, we use repeatedly this identity in the
following expressions. The first formula follows from,

$n^{2}d(a,b,n)=\sum_{i=1}^{n-1}\left(\\{ia\\}_{n}-\frac{n}{2}\right)\left(\\{ib\\}_{n}-\frac{n}{2}\right),$

and using that
$0=\sum_{i=1}^{n-1}\left(\\{ia\\}_{n}-\frac{n}{2}\right)\left(\\{ib\\}_{n}-\frac{n}{2}\right)\left(\\{ic\\}_{n}-\frac{n}{2}\right),$
we get the other two.
∎

A well-known result due to Barkan [Bar77] (see Holzapfel [Hol88]) is the
following formula relating the length $s$ of the continued fraction and
Dedekind sums,

######  Theorem 2.25.

Let $n$ be a prime number, and $q$ be an integer such that $0<q<n$. Let
$n/q=[k_{1},\ldots,k_{s}]$. Then

$d(1,q,n)+s=\sum_{\alpha=1}^{s}(k_{\alpha}-2)+\frac{q+q^{\prime}}{n}.$

∎

######  Remark 2.26.

In [Zag73] was studied the trigonometric version of the Dedekind sum treated
here.

###  2.5. Asymptotic resolution in dimension 2

Let $Z$ be a non-singular projective surface, and let $D$ be an effective
divisor with SNC reduced divisor. Assume the necessary hypothesis to construct
the normal $n$-th root cover $Y_{n}\to Z$ along $D$ (Section 2.2). We have to
choose a resolution of singularities $h\colon X_{n}\to Y_{n}\to Z$, and the
Chern numbers $c_{1}^{2},c_{2}$ of $X_{n}$ will depend on this resolution. The
singularities of $Y_{n}$ over each point of $D_{j}\cap D_{k}$ are analytically
isomorphic to the normalization of

$\mbox{Spec}\left(\frac{{\mathbb{C}}[x,y,t]}{t^{n}-x^{n-q_{jk}}y}\right),$

where $\nu_{j}+q_{jk}\nu_{k}\equiv 0$ modulo $n$. This singularity is a cyclic
quotient singularity of type $\frac{1}{n}(q_{jk},1)$, and the singular point
can be resolved by some weighted blow-ups. The exceptional data will be a
chain of non-singular rational curves $\\{E_{1},\ldots,E_{s}\\}$ with
$E_{j}E_{j+1}=1$ and $E_{j}^{2}=-k_{j}$, where the $k_{j}\geq 2$ are the
integers that define the negative regular continued fraction

$\frac{n}{q_{jk}}=k_{1}-\cfrac{1}{k_{2}-\cfrac{1}{\ddots-\frac{1}{k_{s}}}},$

usually called Hirzebruch-Jung continued fraction. See Section 2.4.1 for
explicit computations. The number $s$ is called the _length_ of the resolution
and we denoted it by $\ell(q_{jk},n)$. In this way, we resolve all
singularities of $Y_{n}$ obtaining a morphism $g\colon X_{n}\to Y_{n}$, with
composition $h\colon X_{n}\to Z$.

In dimension $2$, for the chosen resolution $X_{n}$ Dedekind sums and lengths
appear in the following formulas [Urz09],

$\chi({\mathcal{O}}_{X_{n}})=n\chi({\mathcal{O}}_{Z})-\frac{p^{2}-1}{12n}D^{[2]}-\frac{p-1}{4}e(D)+\sum_{j<k}d(1,q_{jk},n)D_{j}D_{k}$
$c_{2}(X_{n})=nc_{2}(Z)-(n-1)e(D)+\sum_{j<k}\ell(q_{jk},n)D_{j}D_{k}.$

Then, we can recover a formula for $c_{1}^{2}$ by Noether’s identity. In
[Gir03] and [Gir06], Girstmair proved that the lengths and the values of
Dedekind sums have a particular asymptotical behavior.

######  Theorem 2.27 (Girstmair).

For $n\geq 17$ there exists a set $O_{n}\subset\\{0,\ldots,n\\}$ such that for
any $q\in O_{n}$ we have $|d(1,q,n)|\leq 3\sqrt{n}+5,\ell(q,n)\leq
3\sqrt{n}+2$. Moreover $|\\{0,\ldots,n\\}\setminus
O_{n}|\leq\sqrt{n}\log(4n)$.

∎

######  Remark 2.28.

Observe that from the Barkan-Holzapfel relation (Theorem 2.25), and combining
it with the results of Girstmair, we have an asymptotic behavior for the
coefficients of the Hirzebruch-Jung continued fraction in the following sense:
For a prime number $n\gg 0$, and integers $q\in O_{n}$ with
$n/q=[k_{1},\ldots,k_{s}]$, we have

$\sum_{\alpha=1}^{s}(k_{\alpha}-2)\leq 6\sqrt{n}+7.$

######  Definition 2.29.

A collection of prime divisors $\\{D_{1},\ldots,D_{r}\\}$ on a non-singular
$d$-fold $Z$ is an _asymptotic arrangement_ if $D_{red}=D_{1}+\ldots+D_{r}$ is
SNC, and for prime numbers $n\gg 0$:

  1. (1)

There are multiplicities $0<\nu_{j}<n$, such that for any $j<k$ with
$D_{j}\cap D_{k}\neq\emptyset$, we have $q_{jk}\in O_{n}$, the unique integer
such that $\nu_{j}+q_{jk}\nu_{k}\equiv 0$ modulo $n$.

  2. (2)

There are line bundles ${\mathcal{L}}$ such that
${\mathcal{O}}_{Z}\left(\sum_{j=1}^{r}\nu_{j}D_{j}\right)\simeq{\mathcal{L}}^{\otimes
n}$.

######  Example 2.30.

Inside the proof of [Urz09, Th. 6.1], it was proved that for any large prime
number $n$ there exist a partition

$\nu_{1}+\ldots+\nu_{r}=n,$

with $q_{jk}\in O_{n}$ such that $\nu_{j}+q_{jk}\nu_{k}\equiv 0$ modulo $n$.
We call it an _asymptotic partition_ of $n$. Indeed, the probability of a
partition of $n$ to be asymptotic tends to $1$ as $n$ grows to infinity. Thus,
any collection of hyperplanes $\\{H_{1},\ldots,H_{r}\\}$ on ${\mathbb{P}}^{d}$
defining a SNC divisor, is itself an asymptotic arrangement with

$D=\nu_{1}H_{1}+\ldots+\nu_{r}H_{r}=(\nu_{1}+\ldots+\nu_{r})H=nH,$

where $H$ is a general hyperplane section on ${\mathbb{P}}^{d}$.

##  3\. Asymptoticity for a non-singular branch locus

###  3.1. Logarithmic asymptoticity

Let $A(Z)_{{\mathbb{R}}}=A(Z)\otimes_{{\mathbb{Z}}}{\mathbb{R}}$ be the
extended Chow ring of a fixed non-singular variety $Z$. For each $n\geq 1$
assume the existence of finite $\log$-morphisms $h_{n}\colon Y_{n}\to Z$
between non-singular varieties of the same dimension $d\geq 1$ with
$\deg(h_{n})=n$ (Definition 2.10). We have a morphism of extended Chow rings
$h_{n}^{*}\colon A(Z)_{{\mathbb{R}}}\to A(Y_{n})_{{\mathbb{R}}}$ for each $n$.
Since $h_{n}$ is flat, we have that $h_{n}^{*}(A^{e}(Z))\subset A^{e}(Y_{n})$
[Har77, III.9.6], then the same applies for the extended ring.

######  Theorem 3.1.

For $n\geq 1$ assume the existence of finite $\log$-morphisms $h_{n}\colon
Y_{n}\to Z$ ramified at a non-singular divisor $D$ whose reduced form is a SNC
divisor. Let $D_{1},...,D_{r}$ be the components of $D$, and $D_{j}^{\prime}$
the reduced preimage of each $D_{j}$. Assume $h_{n}^{*}D_{j}=nD^{\prime}_{j}$.
Then, we have,

$\lim_{n\to\infty}\frac{c_{i_{1}}\ldots
c_{i_{m}}(Y_{n})}{n}=\bar{c}_{i_{1}}\ldots\bar{c}_{i_{m}}(Z,D).$

###### Proof.

The proof of the theorem is based on proving the following,

$c_{e}(Y_{n})=h_{n}^{*}(\bar{c}_{e}(Z,D))+\frac{h_{n}^{*}(D^{[1]}\bar{c}_{e-1}(Z,D))}{n}\in
h_{n}^{*}(A^{e}(Z)_{{\mathbb{R}}}),$ (5)

for all $e\geq 0$. Then, for any partition $i_{1}+...+i_{m}=d$, we have

$\frac{c_{i_{1}}...c_{i_{m}}(Y_{n})}{n}=\bar{c}_{i_{1}}...\bar{c}_{i_{m}}(Z,D)+\sum_{\begin{subarray}{c}0\leq
e<m\\\
(j_{1},...,j_{m})\end{subarray}}\frac{\bar{c}_{j_{1},...,j_{e},j_{e+1}-1,...,j_{m}-1}(Z,D)\,D^{[m-e]}}{n^{m-e}},$

where the sum runs over each $(j_{1},...,j_{m})$ a permutation of
$\\{i_{1},...,i_{m}\\}$. The result follows directly since the combinatorial
quantities obtained do not depend on $n$. We proceed by induction on the
dimension $0\leq e\leq d$. The trivial case for $e=0$ is
$c_{0}(Y_{n})=h^{*}\bar{c}_{0}(Z,D)=1,$ assuming by convention
$\bar{c}_{-1}(Z,D)=0.$ Since $D$ is non-singular, by Lemma 2.11 we have
$\bar{c}_{e}(Y_{n},D)=h_{n}^{*}(\bar{c}_{e}(Z,D))$. Thus, by Corollary 2.6 we
get

$\displaystyle c_{e}(Y_{n})$
$\displaystyle=h_{n}^{*}(\bar{c}_{e}(Z,D))-R_{e}(D^{\prime}).$

Since $h_{n}^{*}D_{j}=nD^{\prime}_{j}$, we have,

$\displaystyle c_{e}(Y_{n})$
$\displaystyle=h_{n}^{*}(\bar{c}_{e}(Z,D))-\sum_{k=0}^{e-1}(-1)^{e-k}\frac{h_{n}^{*}(D^{[e-k]})}{n^{e-k}}c_{k}(Y_{n}).$

Assuming the induction hypothesis (5) for $k\leq e-1$ we get,

$\displaystyle c_{e}(Y_{n})$
$\displaystyle=h_{n}^{*}(\bar{c}_{e}(Z,D))-\sum_{k=0}^{e-1}(-1)^{e-k}\frac{h_{n}^{*}(D^{[e-k]})}{n^{e-k}}\left(h^{*}(\bar{c}_{k}(Z,D))+\frac{h_{n}^{*}(D^{[1]}\bar{c}_{k-1}(Z,D))}{n}\right)$
$\displaystyle=h_{n}^{*}(\bar{c}_{e}(Z,D))-\sum_{k=0}^{e-1}\left((-1)^{e-k}\frac{h_{n}^{*}(D^{[e-k]}\bar{c}_{k}(Z,D))}{n^{e-k}}-(-1)^{e-k-1}\frac{h_{n}^{*}(D^{[e-k-1]}\bar{c}_{k-1}(Z,D))}{n^{e-k-1}}\right)$
$\displaystyle=h_{n}^{*}(\bar{c}_{e}(Z,D))+\frac{h_{n}^{*}(D^{[1]}\bar{c}_{e-1}(Z,D))}{n^{1}},$

where the last step is by a telescopic argument using $\bar{c}_{-1}(Z,D)=0$. ∎

The main situation to apply Theorem 3.1 is the case of $n$-th root covers. In
this case, take a non-singular SNC divisor $D_{1}+\ldots+D_{r}$ on $Z$, and we
restrict our attention to prime numbers $n\geq 2$. For each $n$, assume the
existence of $L$ and $0<\nu_{j}<n$ such that $D=\sum_{j}\nu_{j}D_{j}\sim nL$
(Section 2.2). Construct the non-singular covers $h_{n}\colon Y_{n}\to Z$
along each $D$. Then $h_{n}^{*}D_{j}=nD_{j}^{\prime},$ and we get.

######  Corollary 3.2.

Under the above hypothesis, the $n$-th root covers $Y_{n}$ satisfy,

$\frac{c_{i_{1}}\ldots
c_{i_{m}}(Y_{n})}{n}\to\bar{c}_{i_{1}}\ldots\bar{c}_{i_{m}}(Z,D),$

as $n\to\infty$ for prime numbers $n\gg 0$.

∎

For our purposes in geography, this result has a disadvantage, the difficulty
in finding good pairs $(Z,D)$ whose covers $Y_{n}$ are minimal of general
type. For example, from Theorem 2.13 would be enough $K_{Z}$ big and nef, and
$D$ ample with many components. However, at least the condition $K_{Z}$ big
seems difficult to assure since most of varieties with arbitrary collections
of disjoint divisors appear to be a fiber space. In [BPS16] was proved the
following: Assume that $Z$ has a collection of disjoint divisors
$\\{D_{j}\\}_{j\in J}$, if $|J|\gg 0$, then there exists a surjective morphism
from $Z$ to a curve such that every $D_{j}$ is contained in a fiber. Thus,
generically any variety $Z$ of $\dim Z\geq 2$ having collections of disjoint
divisors is a fiber space $Z\to V$ over some variety $V$.

######  Remark 3.3.

We can extend Corollary 3.2 to Abelian covers, i.e., to the case
$G_{n}={\mathbb{Z}}/n_{1}{\mathbb{Z}}\oplus\ldots\oplus{\mathbb{Z}}/n_{k}{\mathbb{Z}}$
a sequence of Abelian groups of order $n=n_{1}\ldots n_{k}$ with each $n_{j}$
a prime number with $n_{j}\neq n_{k}$. See [Par91] or [Gao11]. In this case,
the Abelian covers $Y_{n}\to X$ depend on a data $D^{i}\sim n_{i}L_{i}$ with
$D^{i}_{red}$ a SNC divisor for $i=1,\ldots,k$. Then the SNC divisor to take
is $D_{red}=(D^{1}+\ldots+D_{k})_{red}$. The ramification numbers for each
component $D_{i}^{j}$ of $D^{i}$ are given by
$h^{*}D_{j}^{i}=\frac{n}{\mbox{gcd}(n_{i},\nu_{j}^{i}(n))}{D^{\prime}_{j}}^{i}=n{D^{\prime}_{j}}^{i}$,
where $\nu_{j}^{i}(n)$ is the multiplicity of $D_{j}^{i}$ in $D^{i}$. From
here, we leave to the reader the analog asymptotic result. However, we can
ask: How can this argument be extended to any Galois cover?

By the above discussion, in the rest of this paper, we will study the above
results for the case of $3$-folds when $D$ has its components with non-empty
intersections. Thus $Y_{n}$ will have singularities.

##  4\. Asymptoticity of invariants

###  4.1. Asymptoticity of $\chi$ for $3$-folds

Consider a data $(Z,D,n,{\mathcal{L}})$ as in Section 2.2, with $Z$ a non-
singular projective $3$-fold. Let $h\colon X_{n}\to Y_{n}\to Z$ be any
resolution of singularities of the branched $n$-th root cover $Y_{n}$ along
the effective divisor $D=\sum_{j=1}^{r}\nu_{j}D_{j}\sim nL$ whose reduced form
is SNC. We have,

$L^{(i)}=iL-\sum_{j=1}^{r}\left\lfloor\frac{i\nu_{j}}{n}\right\rfloor
D_{j}=\frac{1}{n}\left(iD-\sum_{j=1}^{r}n\left\lfloor\frac{i\nu_{j}}{n}\right\rfloor
D_{j}\right)=\frac{1}{n}\sum_{j=1}^{r}\\{i\nu_{j}\\}_{n}D_{j}.$

######  Proposition 4.1.

We have,

$\chi({\mathcal{O}}_{X_{n}})=n\chi({\mathcal{O}}_{Z})-\frac{1}{12}(R_{1}(n,D)+R_{2}(n,D)+R_{3}(n,D)),$

where

$R_{1}(n,D)=\frac{(n-1)^{2}}{2n}D^{[3]}+\frac{(n-1)(2n-1)}{2n}(D^{[1,2]}+D^{[2,1]})+\frac{3(n-1)}{2}D^{[1,1,1]},$
$R_{2}(n,D)=\frac{(1-n)}{2}c_{1}(Z)\left(\frac{(2n-1)}{n}D^{[2]}+3D^{[1,1]}\right)+\frac{(n-1)}{2}D_{red}(c_{1}^{2}(Z)+c_{2}(Z)),$
$\displaystyle R_{3}(n,D)$
$\displaystyle=6\left(\sum_{j<k}d(\nu_{j},\nu_{k},n)D_{j}D_{k}(D_{j}+D_{k}+K_{Z})\right.$
$\displaystyle\left.+\sum_{j<k<l}(d(\nu_{j},\nu_{k},n)+d(\nu_{j},\nu_{l},n)+d(\nu_{k},\nu_{l},n))D_{j}D_{k}D_{l}\right).$

###### Proof.

By Corollary 2.16, and Hirzebruch-Riemann-Roch theorem for $3$-folds we can
compute

$\displaystyle\chi({\mathcal{O}}_{X_{n}})$
$\displaystyle=\sum_{i=0}^{n-1}\chi({\mathcal{O}}_{Z}(-L^{(i)}))$
$\displaystyle=n\chi({\mathcal{O}}_{Z})-\frac{1}{12}\sum_{i=1}^{n-1}\left(L^{(i)}(L^{(i)}+K_{Z})(2L^{(i)}+K_{Z})+c_{2}(Z).L^{(i)}\right)$
$\displaystyle=n\chi({\mathcal{O}}_{Z})-\frac{1}{12}\sum_{i=1}^{n-1}\left(2(L^{(i)})^{3}+3(L^{(i)})^{2}K_{Z}+L^{(i)}K_{Z}^{2}+c_{2}(Z).L^{(i)}\right)$
$\displaystyle=n\chi({\mathcal{O}}_{Z})-\frac{1}{12}R(n,D),$

where $R(n,D)$ is a quantity depending only on $n$ and $D$. We have the
following identities: $\sum_{i=1}^{n-1}L^{(i)}=\frac{(n-1)}{2}D_{red}$, and

$\sum_{i=1}^{n-1}(L^{(i)})^{2}=\frac{(n-1)(2n-1)}{6n}D^{[2]}+\frac{(n-1)}{2}D^{[1,1]}+2\sum_{j<k}d(\nu_{j},\nu_{k},n)D_{j}D_{k}.$

The above is not difficult to deduce from the formulas in Lemma 2.24. To
illustrate, we compute $\sum_{i}(L^{(i)})^{3}$ as follows. The first step, we
compute explicitly:

$\displaystyle(L^{(i)})^{3}$
$\displaystyle=\frac{1}{n^{3}}\left(iD-\sum_{j=1}^{r}n\left\lfloor\frac{i\nu_{j}}{n}\right\rfloor
D_{j}\right)^{3}$
$\displaystyle=\frac{1}{n^{3}}\left(\sum_{j=1}^{r}\\{i\nu_{j}\\}_{n}D_{j}\right)^{3}$
$\displaystyle=\frac{1}{n^{3}}\left(\sum_{j=1}^{r}\\{i\nu_{j}\\}_{n}^{3}D_{j}^{3}+3\sum_{j<k}\\{i\nu_{j}\\}_{n}^{2}\\{i\nu_{k}\\}_{n}D_{j}^{2}D_{k}+\\{i\nu_{j}\\}_{n}\\{i\nu_{k}\\}_{n}^{2}D_{j}D_{k}^{2}\right.$
$\displaystyle\hskip
34.1433pt+\left.6\sum_{j<k<l}\\{i\nu_{j}\\}_{n}\\{i\nu_{k}\\}_{n}\\{i\nu_{l}\\}_{n}D_{j}D_{k}D_{l}\right)$

Applying directly the formulas in Lemma 2.24, we get.

$\displaystyle\sum_{i=1}^{n-1}(L^{(i)})^{3}$
$\displaystyle=\frac{(n-1)^{2}}{4n}D^{[3]}+\frac{(n-1)(2n-1)}{4n}(D^{[1,2]}+D^{[2,1]})+\frac{3(n-1)}{4}D^{[1,1,1]}$
$\displaystyle+3\sum_{j<k}d(\nu_{j},\nu_{k},n)(D_{j}^{2}D_{k}+D_{j}D_{k}^{2})$
$\displaystyle+3\sum_{j<k<l}(d(\nu_{j},\nu_{k},n)+d(\nu_{j},\nu_{l},n)+d(\nu_{k},\nu_{l},n))D_{j}D_{k}D_{l}.$

On the other hand,

$\sum_{i=1}^{n-1}(L^{(i)})^{2}K_{Z}=\frac{(1-n)}{2}c_{1}(Z)\left(\frac{(2n-1)}{3n}D^{[2]}+D^{[1,1]}\right)+2\sum_{j<k}d(\nu_{j},\nu_{k},n)D_{j}D_{k}K_{Z},$

$\sum_{i=1}^{n-1}L^{(i)}K_{Z}^{2}=\frac{(n-1)}{2}K_{Z}^{2}D_{red}=\frac{(n-1)}{2}D_{red}c_{1}^{2}(Z),$

$\sum_{i=1}^{n-1}L^{(i)}c_{2}(Z)=\frac{(n-1)}{2}D_{red}c_{2}(Z).$

From here it is not too difficult to note that we have
$R(n,D)=R_{1}(n,D)+R_{2}(n,D)+R_{3}(n,D)$, the quantities previously
mentioned.
∎

######  Theorem 4.2.

If $\\{D_{1},\ldots,D_{r}\\}$ is an asymptotic arrangement, then

$\frac{\chi({\mathcal{O}}_{X_{n}})}{n}\to\frac{\overline{c_{1}c_{2}}(Z,D)}{24},$

as $n\to\infty$ for prime numbers $n\gg 0$.

###### Proof.

First observe the following limits

$\lim_{n\to\infty}\frac{R_{1}(n,D)}{n}=\frac{1}{2}D^{[3]}+(D^{[1,2]}+D^{[2,1]})+\frac{3}{2}D^{[1,1,1]}=\frac{1}{2}D_{red}(D^{[2]}+D^{[1,1]}),$

$\lim_{n\to\infty}\frac{R_{2}(n,D)}{n}=-\frac{1}{2}c_{1}(Z)\left(2D^{[2]}+3D^{[1,1]}\right)+\frac{1}{2}D_{red}(c_{1}^{2}(Z)+c_{2}(Z)).$

From formulas in Corollary 2.7, we get the identity

$\lim_{n\to\infty}\frac{R_{1}(n,D)+R_{2}(n,D)}{n}=\frac{1}{2}(c_{1}c_{2}(Z)-\overline{c_{1}c_{2}}(Z,D)).$

Since the collection of divisors is an asymptotic arrangement, we use Theorem
2.27 to get $R_{3}(n,D)/n\approx 0$ for prime numbers $n\gg 0$. Thus, in this
case we have

$\frac{\chi({\mathcal{O}}_{X_{n}})}{n}\approx\frac{c_{1}c_{2}(Z)}{24}-\frac{1}{24}(c_{1}c_{2}(Z)-\overline{c_{1}c_{2}}(Z,D))=\frac{\overline{c_{1}c_{2}}(Z,D)}{24},$

for prime numbers $n\gg 0$.∎

######  Remark 4.3.

Observe that the formula for $(L^{(i)})^{3}$ can be extended to higher
dimensions. In the same way of Lemma 2.24, we can find formulas for
$(L^{(i)})^{e}$ depending only on the combinatorial aspects of $D$ and higher
dimensional Dedekind sums. Thus, for asymptoticity of
$\chi({\mathcal{O}}_{X_{n}})$ in any dimension, we need asymptoticity of
Dedekind sums, i.e., the higher dimensional analogs of Girstmair’s results
(Theorem 2.27). For dimension $d\geq 4$ this is an open problem.

###  4.2. Toric local resolutions

In this section, we study the $3$-fold singularity given by the normalization
of $t^{n}=x_{1}^{\nu_{1}}x_{2}^{\nu_{2}}x_{3}^{\nu_{3}}$, with $n\geq 2$ a
prime number and $0<\nu_{i}<n$. The aim is to achieve good local resolutions
of singularities in asymptotic terms with respect to $n$. Since this
singularity is toric, it can be resolved by subdivisions of its associated
cone obtaining a refinement fan. To assure asymptotic properties, we have to
pay attention to the combinatorial aspects of the refinement. Let $0<p,q<n$ be
integers such that $\nu_{1}+p\nu_{3}\equiv 0$ and $\nu_{1}+q\nu_{3}\equiv 0$
modulo $n$. By Section 2.3, this $3$-fold singularity is a toric variety
$Y_{p,q}:=\mbox{Spec}(\sigma^{\vee}\cap{\mathbb{Z}}^{3})$ defined by the cone
$\sigma=C(d_{1},d_{2},d_{3})\subset{\mathbb{R}}^{3}$, where
$d_{1}=ne_{1}-pe_{3},d_{2}=ne_{2}-qe_{3}$, and $d_{3}=e_{3}$ are the primitive
ray generators. A transversal section of this cone is sketched in Figure 2.

Figure 2. Transversal section of $\sigma$.

The fan defined by the cone $\sigma$ has 2-dimensional faces (walls) given by
$\tau_{jk}=C(d_{j},d_{k})$ for $j<k$. By Section 2.4.1, each wall $\tau_{jk}$
can be resolved by Hirzebruch-Jung sequences
$(m_{jk,\alpha},n_{jk,\alpha})_{\alpha=0}^{s_{jk}+1}$ in direction $d_{j}$ to
$d_{k}$ with initial data

$m_{jk,0}=n,\quad n_{jk,0}=0,\quad n_{jk,1}=1,\quad\forall j<k,$
$m_{13,1}=p^{\prime},\quad m_{23,1}=q^{\prime},\quad
m_{12,1}=\\{-p^{\prime}q\\}_{n},$

where $p^{\prime}$ and $q^{\prime}$ are the inverse modulo $n$ of $p$ and $q$.
Thus, there are integers $k_{jk,\alpha}\geq 2$, such that

$\frac{n}{m_{jk,1}}=[k_{jk,1},...,k_{jk,s_{jk}}].$

Then, the walls $\tau_{jk}$ can be resolved by subdividing them in a sequence
of steps by rays with generators $(e_{jk,\alpha})_{\alpha=1}^{s_{jk}}$ defined
recursively as

$e_{jk,\alpha}=\frac{m_{jk,\alpha}d_{j}+n_{jk,\alpha}d_{k}}{n},\quad
1\leq\alpha\leq s_{jk}.$

If we fix $j<k$, we denote by $e_{kj,\alpha}$ the exceptional divisors in
direction $d_{k}$ to $d_{j}$. We have the relation
$e_{jk,\alpha}=e_{kj,s+1-\alpha}$. In Figure 3 are illustrated the border
generators.

Figure 3. Lattice points that resolve each wall

In order to choose a good asymptotic local resolution of $\sigma$, imitating
the $2$-dimensional case, we can ask for a minimal local resolution, i.e.,
with nef canonical bundles. However, minimal varieties in higher dimensions
may have terminal singularities, minimal singular models are not necessarily
unique, and there are no efficient algorithms in the toric case. In this last,
at least there exists a kind of optimal method. Minimal resolutions can be
obtained by the canonical resolution of $\sigma$ which is obtained by the
canonical refinement of the cone [CLS11, Prop. 11.4.15]. However, the
canonical refinement appears not to have a regular pattern for any $p,q$. For
example, if $p=n-1$, and $0<q<n$ are free, the canonical and minimal
resolutions look very simple but release a lot of exceptional data. See Figure
4. The resolution to the case $p=q$ takes the same form, just rotate the
figure. And completely different from the above is the case $p+q=n$. See
Figure 5. Thus, we do not consider these cases in the following, i.e.,
$p,q\neq n-1$, $p=q$ or $\\{p+q\\}_{n}=0$. For more details see [TN23].

Figure 4. Case $p=n-1$. We have cones generated by 4 generators. These cones
define the canonical resolution of $\sigma$. The black dots, mark lattice
points to blowing up and get a minimal non-singular resolution. Figure 5. Case
$p+q=n$. The wall $\tau_{12}$ is of type $(1,1)/n$. Then the minimal non-
singular resolution depends only on the resolutions of the other walls.

To have a systematic way to construct resolutions, we choose the following
way: We construct a cyclic resolution, i.e., a refinement of $\sigma$ composed
by $3$-cones defining cyclic singularities. The advantage is that cyclic
singularities have a well-known algorithm to resolve them [Fuj74]. Also, as we
will see in the examples, we can construct some minimal resolutions. So, in
future work, we expect that, under suitable conditions, these resolutions have
good asymptotical behavior with respect to $n$.

####  4.2.1. Cyclic resolution.

From (4) we know that every $v\in P_{\sigma}\cap N$ can be written as

$v=\frac{v_{1}d_{1}+v_{2}d_{2}+v_{3}d_{3}}{n},\quad
v_{3}=\\{pv_{1}+qv_{2}\\}_{n}\quad 0\leq v_{i}<n.$

Fix a $v$, with $v_{1}+v_{2}+v_{3}\leq n$ and $v_{i}>0$.

Step 1: We refine by a star subdivision along $v$ obtaining a fan as
illustrate Figure 6.

Figure 6. Star subdivision along minimizer $v$

Step 2: Now we refine each wall by doing toric blow-ups following the
Hirzebruch-Jung algorithm. So we obtain a refinement $\sigma^{*}$ and denote
by $X$ the toric variety associated. This refinement gives us a birational
projective morphism $g\colon X\to Y_{p,q}$ [CLS11, 11.1.6]. This refinement is
sketched in Figure 7.

Figure 7. Cyclic local resolution

Denote each $3$-cone of $\sigma^{*}$ by

$\sigma_{jk,\alpha}=C(v,e_{jk,\alpha},e_{jk,\alpha+1}),\quad 0\leq\alpha\leq
s_{jk}.$

The $2$-cones of $\sigma^{*}$ are given in two types. The _exterior walls_
$\tau_{jk,\alpha}=C(e_{jk,\alpha},e_{jk,\alpha+1})$, and the _inner walls_
$\rho_{jk,\alpha}=C(v,e_{jk,\alpha})$. For any permutation
$(v_{j},v_{k},v_{l})$ with $j<k$, using determinants and properties of Section
2.4.1, we have

$\mbox{mult}(\sigma_{jk,\alpha})=v_{l},\quad\mbox{mult}(\rho_{jk,\alpha})=\mbox{gcd}(v_{j}n_{jk,\alpha}-v_{k}m_{jk,\alpha},v_{l}),\quad\mbox{mult}(\tau_{jk,\alpha})=1.$

######  Lemma 4.4.

Each cone $\sigma_{jk,\alpha}$ is a cyclic singularity of type

$\frac{(a_{jk,\alpha},b_{jk,\alpha},1)}{v_{l}},\quad
a_{jk,\alpha}=\\{m_{jk,\alpha+1}v_{k}-n_{jk,\alpha+1}v_{j}\\}_{v_{l}},\quad
b_{jk,\alpha}=\\{m_{jk,\alpha}v_{k}-n_{jk,\alpha}v_{j}\\}_{v_{l}},$

where $\\{\cdot\\}_{v_{l}}$ is the residue modulo $v_{l}$.

###### Proof.

Since $\tau_{jk,\alpha}$ is non-singular, then $\sigma_{jk,\alpha}$ is semi-
unimodular respect to $v$. By [Ash15, Prop. 1.2.3] if there is positive
integer $x,y$ such that

$\frac{xe_{jk,\alpha}+ye_{jk,\alpha+1}+v}{v_{l}}\in{\mathbb{Z}}^{3},$

then $x,y$ define the type of the cyclic singularity. Since $\gcd(n,v_{l})=1$,
we can solve the equations modulo $v_{l}$ and the result follows.
∎

Denote by $h\colon X\to Y_{p,q}\to{\mathbb{A}}^{3}$ the composition with the
natural projection to ${\mathbb{A}}^{3}$. Denote by $D_{j}$ the divisor in
${\mathbb{A}}^{3}$ defined by the coordinate $x_{j}$. Each ray on $\sigma^{*}$
generated by $d_{j},e_{jk,\alpha},$ or $v$ defines a toric divisor on $X$
given by

$\tilde{D}_{j}=V(C(d_{j})),\quad E_{jk,\alpha}=V(C(e_{jk,\alpha})),\quad
F=V(C(v)),$

where $V(\cdot)$ denotes the orbit closure of a cone [CLS11, p. 121]. At the
same time we fix notation for $j<k$ by $E_{kj,\alpha}:=E_{jk,s+1-\alpha}$. we
can compute

$h^{*}D_{j}=n\tilde{D}_{j}+\sum_{k}\sum_{\alpha=1}^{s_{jk}}m_{jk,\alpha}E_{jk,\alpha}+v_{j}F,$

$K_{X}=g^{*}K_{Y_{p,q}}+\sum_{j<k}\sum_{\alpha=1}^{s_{jk}}N_{jk,\alpha}E_{jk,\alpha}+\frac{v_{1}+v_{2}+v_{3}-n}{n}F,$
(6)

where

$N_{jk,\alpha}:=\frac{m_{jk,\alpha}+n_{jk,\alpha}}{n}-1.$

It is satisfied the relation
$k_{jk,\alpha}N_{jk,\alpha}-N_{jk,\alpha+1}=N_{jk,\alpha-1}-(k_{jk,\alpha}-2),$
which gives

$\sum_{\alpha=1}^{s_{jk}}N_{jk,\alpha}(k_{jk,\alpha}-2)=-(N_{jk,1}+N_{kj,1})-\sum_{\alpha=1}^{s_{jk}}(k_{jk,\alpha}-2).$

######  Proposition 4.5.

The ${\mathbb{Q}}$-divisor

$h^{*}\left(\frac{n-1}{n}(D_{1}+D_{2}+D_{3})\right)+\sum_{j<k}\sum_{\alpha}N_{jk,\alpha}E_{jk,\alpha}+\frac{v_{1}+v_{2}+v_{3}-n}{n}F,$

is an effective ${\mathbb{Z}}$-divisor.

###### Proof.

The local pullback $h^{*}(D_{1}+D_{2}+D_{3})$ equals

$n(\tilde{D_{1}}+\tilde{D_{2}}+\tilde{D_{3}})+\sum_{j<k,\alpha}(m_{jk,\alpha}+n_{jk,\alpha})E_{jk,\alpha}+(v_{1}+v_{2}+v_{3})F_{jkl,p}.$

Thus, $h^{*}\left(\frac{n-1}{n}(D_{j}+D_{k}+D_{l})\right)+\Delta$ equals to

$(n-1)(\tilde{D_{1}}+\tilde{D_{2}}+\tilde{D_{3}})+\sum_{j<k,\alpha}\left(m_{jk,\alpha}+n_{jk,\alpha}-1\right)E_{jk,\alpha}+(v_{1}+v_{2}+v_{3}-1)F,$

i.e., an effective ${\mathbb{Z}}$-divisor. ∎

Each inner wall defines a closed curve on $X$ given by

$C_{j}=V(C(d_{j},v)),\quad C_{jk,\alpha}=V(C(\rho_{jk,\alpha})).$

The refinement $\sigma^{*}$ is simplicial with cones of multiplicity one, and
the canonical divisor $K_{X}$ is a ${\mathbb{Q}}$-Cartier divisor. For any
pair $v_{j},v_{k}$, $j<k$, let $v_{l}$ be the another coordinate. The
following relation among lattices generators,

$e_{jk,\alpha-1}+(-k_{jk,\alpha})e_{jk,\alpha}+0\cdot v+e_{jk,\alpha+1}=0,$
$v_{j}e_{lj,1}+\left(\frac{v_{l}-m_{lj,1}v_{j}-m_{lk,1}v_{k}}{n}\right)d_{l}+(-1)v+v_{k}e_{lk,1}=0,$

describe the intersection theory on $X$ [Ful93, Ch. V]. We have,

$E_{jk,\alpha}C_{jk,\alpha\pm
1}=\frac{\mbox{mult}(\rho_{jk,\alpha})}{v_{l}},\quad
E_{jk,\alpha}C_{jk,\alpha}=-\frac{k_{jk,\alpha}\mbox{mult}(\rho_{jk,\alpha})}{v_{l}},\quad
FC_{jk,\alpha}=0,$
$\tilde{D}_{l}\,C_{l}=\frac{\gcd(v_{j},v_{k})(v_{l}-m_{lj,1}v_{j}-m_{lk,1}v_{k}))}{nv_{j}v_{k}},\quad
F\,C_{l}=-\frac{\gcd(v_{j},v_{k})}{v_{j}v_{k}},$
$K_{X}\,C_{l}=-\frac{\gcd(v_{j},v_{k})}{v_{j}v_{k}}\left(v_{j}+v_{k}-1+\frac{v_{l}-m_{lj,1}v_{j}-m_{lk,1}v_{k}}{n}\right),$
$K_{X}\,C_{jk,\alpha}=\frac{\mbox{mult}(\rho_{jk,\alpha})}{v_{l}}(k_{jk,\alpha}-2),\quad
1\leq\alpha\leq s_{jk}.$ (7)

######  Lemma 4.6.

We have

$F^{3}=\frac{n}{v_{1}v_{2}v_{3}}.$

###### Proof.

The divisor $v_{j}F$ is Cartier for any $j$, then from the pullback identities
above we have

$v_{1}v_{2}v_{3}F^{3}=\prod_{j=1}^{3}\left(h^{*}D_{j}-nD_{j}-\sum_{k}\sum_{\alpha}m_{jk,\alpha}E_{jk,\alpha}\right)=h^{*}D_{1}\,h^{*}D_{2}\,h^{*}D_{3}=n,$

where the last is by projection formula.∎

As a consequence, we can compute,

$\displaystyle K_{X}F^{2}$
$\displaystyle=-F^{2}\left(D_{1}+D_{2}+D_{3}+F\right)$
$\displaystyle=\frac{v_{1}+v_{2}+v_{3}-n}{v_{1}v_{2}v_{3}}.$

From the last one, we get

$\displaystyle K_{X}^{2}\,F$
$\displaystyle=-\sum_{l}\frac{K_{X}\,C_{l}}{\gcd(v_{j},v_{k})}-\sum_{j<k,\alpha}\frac{K_{X}\,C_{jk,\alpha}}{\mbox{mult}(\rho_{jk,\alpha})}-K_{X}F^{2}.$

######  Example 4.7.

_Case $\\{p+q\\}_{n}=1$:_ For $v_{1}=v_{2}=1$, we have
$v_{3}=\\{p+q\\}_{n}=1$. Thus, these coordinates define an interior lattice
point $v$, and it minimizes $v_{1}+v_{2}+v_{3}$. We have

$\mbox{mult}(\sigma_{jk,\alpha})=\mbox{mult}(\rho_{jk,\alpha})=\mbox{mult}(\tau_{jk,\alpha})=1,$

i.e., $X$ is non-singular. On the other hand,

$K_{X}C_{jk,\alpha}=(k_{jk,\alpha}-2),\quad K_{X}C_{l}=0,$

for all $j,k$ and $l$. Thus, $K_{X}$ is nef.

######  Example 4.8.

_Case $\\{p+q\\}_{n}=2$:_ In this case, again $v_{1}=v_{2}=1$ defines the
minimizer interior lattice point $v$. In this case, we have
$\sigma_{13,\alpha}$ and $\sigma_{23,\alpha}$ as non-singular cones. On the
other hand, each $\sigma_{12,\alpha}$ is cyclic singularity of order $2$.
Thus, they define canonical and terminal singularities. The first ones achieve
a terminal resolution with one blow-up. Moreover,

$K_{X}C_{3}=0,\quad K_{X}C_{j}\in\left\\{0,-\frac{1}{2}\right\\},\quad j=1,2$

so there are $p,q$ with canonical divisor $K_{X}$ nef. In the worst case,
i.e., $K_{X}\,C_{j}<0$ for $j=1,2$, we can do toric flips a to get a nef toric
variety given whose fan is sketched in Figure 8.

Figure 8. Flipped fan for $\\{p+q\\}_{n}=2$.

######  Example 4.9.

If we drag the lattice point $v$ to one of the generators of the cone
$\sigma$, we get a degenerated fan as in Figure 9. In this case, the
singularities are of order $n$, and as an advantage, we do not have a divisor
$F$.

Figure 9. Degenerated cyclic resolution

As we see, having $v_{j}\leq 2$ gives us good singularities to work. Indeed,
if the $v_{j}^{\prime}s$ are small enough, the singularities are also good in
asymptotic terms. The following arithmetic lemma will be useful in Section
4.3.

######  Lemma 4.10.

There exists $v\in P_{\sigma}\cap N$ such that $v_{1}+v_{2}+v_{3}=n$, so
$K_{X}$ has multiplicity zero at $F$. Moreover, for $n\gg 0$ we can choose $v$
such that the slopes $v_{j}/v_{k}\leq 3$.

###### Proof.

By (4) we have $v_{3}=\\{v_{1}p+v_{2}q\\}_{n}$. So $v_{1}+v_{2}+v_{3}=n$,
implies that $(v_{1},v_{2})$ are solutions $(x,y)$ of the Diophantine equation

$y\equiv cx\mod n,\quad c=\\{-(p+1)(q+1)^{\prime}\\}_{n}.$

Moreover, for any of those solutions with $x+y<n$, we have $v_{3}=n-x-y$. A
degenerate case is $p+q=n-2$, equivalently $c=1$, thus the solution to the
equation is the diagonal. Thus, we can choose $x=y=\lfloor\frac{n}{3}\rfloor$,
and the result follows for this case. Now, we assume that $c<n/2$, otherwise
we do $(x,y)\mapsto(-x,y)$. The integer points in the square $[1,n-1]^{2}$
solving the equation distribute in ${\mathbb{R}}^{2}$ along the lines
$L_{\beta}:y=cx-\beta n$ for $0\leq\beta\leq c-1$. Thus, over each $L_{\beta}$
the integer solutions over the line are defined by those integers in the
interval

$I_{\beta}=\left(\left\lfloor\frac{\beta
n}{c}\right\rfloor,\left\lfloor\frac{(\beta+1)n}{c}\right\rfloor\right].$

For each $1\leq k\leq\lfloor\frac{n}{c}\rfloor$, we have

$y_{k}=y\left(\left\lfloor\frac{\beta n}{c}\right\rfloor+k\right)=ck-r,$

where $0\leq r<c$ is the residue of $\beta n$ modulo $c$. So, $c(k-1)\leq
y_{k}\leq ck$. Let us choose $\beta=\lfloor\frac{c-1}{3}\rfloor,$ and
$k=\lfloor\frac{n}{3c}\rfloor$. In particular $\lfloor\frac{n}{3}\rfloor\in
I_{\beta}$. So, as $n\gg 0$ we have $y_{k}\approx\frac{n}{3}$. By
construction, we have $x_{k}=\left\lfloor\frac{\beta n}{c}\right\rfloor+k$.
Let $0\leq r^{*}\leq 2$ the residue of $c-1$ modulo $3$, then
$x_{k}\approx\frac{n(c-r^{*})}{3c}$. Since
$\frac{1}{2}\leq\frac{c-r^{*}}{c}\leq 1$, the result follows choosing
$v_{1}=x_{k}$ and $v_{2}=y_{k}$ as $n\gg 0$. ∎

###  4.3. Global resolution

Let $\\{D_{1},...,D_{r}\\}$ be an asymptotic arrangement on $Z$. Thus, for
prime numbers $n\gg 0$ we have multiplicities $0<\nu_{j}<n$ depending on $n$,
with its respective $q_{jk}\in O_{n}$. We have $n$-th root covers $f_{n}\colon
Y_{n}\to Z$ branched at each $D=\sum_{j}\nu_{j}D_{j}$. Let us fix a $n\gg 0$,
so we drop the subscript $n$ of the morphisms, i.e., we are fixing a $f\colon
Y_{n}\to Z$.

For $j<k$, the singularities of $Y_{n}$ over curves in $D_{jk}:=D_{j}\cap
D_{k}$ are locally analytically isomorphic to $C_{q_{jk},n}\times{\mathbb{C}}$
where $C_{q_{jk},n}$ is the surface cyclic quotient singularity of type
$\frac{1}{n}(q_{jk},1)$. For a triple $j<k<l$, the singularity of $Y_{n}$ over
a point in $D_{jkl}:=D_{j}\cap D_{k}\cap D_{l}$ is locally analytically
isomorphic to the normalization of $\mbox{Spec}(\sigma_{jkl}^{\vee}\cap M)$
where $\sigma_{jkl}$ is a cone with walls of types
$\frac{1}{n}(q_{jk},1),\frac{1}{n}(q_{jl},1),\frac{1}{n}(q_{kl},1)$ as we see
in Section 4.2. We will get the _cyclic resolution_ $X_{n}\to Y_{n}$ via
weighted blow-ups in two steps.

Step 1: Since singularities over $D_{jkl}$ are isolated, for each
$\sigma_{jkl}$, we do a weighted blow-up at a convenient interior lattice
point $v^{jkl}$. So, this refinement locally gives a projective morphism which
is a blow-up over an isolated point [CLS11, 11.1.6]. In this case, we get a
projective birational morphism $h^{\prime}:X_{n}^{\prime}\to Y_{n}$. We have
exceptional divisors $F_{jkl}$ whose components are over the points of
$D_{jkl}$ and they are isomorphic to weighted fake projective planes [Buc08].
For future computations, we fix the notation of $v^{jkl}$ and $F_{jkl}$
independent of the order of the triple $j,k,l$. For example,
$v^{jkl}=v^{kjl}$.

Step 2: Since the centers of the above blow-ups are points, the singularity
type over intersections $D_{jk}$ was not affected. For curves in $D_{jk}$,
locally by SNC property, we can assume that they are supported on a local
equation $xy=0$ for local coordinates $x,y$. Then over such curves, the
singularities on $X^{\prime}_{n}$ are locally analytically isomorphic to
$C_{q_{jk},n}\times{\mathbb{C}}$, thus we use the Hirzebruch-Jung algorithm
which is a weighted blow-up to resolve $C_{q_{jk},n}$. The Hirzebruch-Jung
resolution can be realized by a single blow-up
$Bl_{{\mathfrak{m}}}(C_{q_{jk,n}})\to C_{q_{jk,}}$ where ${\mathfrak{m}}$ is a
maximal ideal determined explicitly in coordinates $x,y$ as we see at the end
of Section 2.4.1 (also see [KM92, 10.5]). In terms of local resolutions, we
need to follow an order compatible with the resolution, i.e., if we locally
blow-up a curve in $D_{jk}$ then this operation must be reflected on the other
local toric pictures following the centers to blowing-up. See Figure 10. Thus,
this construction extends and we have resolved the curves $D_{jk}$.
Consequently, we get a projective morphism denoted by $g\colon X_{n}\to
X^{\prime}_{n}\to Y_{n}$, and denote by $h\colon X_{n}\to Z$ the composition.
Since $X_{n}$ was constructed by a sequence of weighted blow-ups with cyclic
singularities, then, $X_{n}$ is an embedded ${\mathbb{Q}}$-resolution of
$Y_{n}$ [ABMMOG12, 2.1]. As we see in 2.17, the varieties $X_{n}$ are
irreducible. We summarize this in the following.

######  Proposition 4.11 (and Definition).

There exists a _cyclic partial resolution_ $g:X_{n}\to Y_{n}$, i.e. a
projective, surjective, birational morphism such that $X_{n}$ is irreducible
and, it has at most isolated cyclic quotient singularities of order lower than
$n$.

∎

Figure 10. Assume $r=4$ with $D_{jkl}=1$, then on $Y$ the singularities over
each $D_{jkl}$ can be sketched as in the figure. So the resolution process is
in the following order: First the internals blow-ups, and then the walls in
the following order $D_{12},D_{13},D_{14},D_{23},D_{24},D_{34}$.

In the rest, we will abuse notation using $D_{jk}$ and $D_{jkl}$ for both,
set-theoretic intersections $D_{j}\cap D_{k}$ and $D_{j}\cap D_{k}\cap D_{l}$,
and for intersections of cycles $D_{j}D_{k}$ and $D_{j}D_{k}D_{l}$. Over each
$D_{jk}$, we get exceptional divisors $E_{jk,\alpha}$, $0\leq\alpha\leq
s_{jk}$, where $s_{jk}=\ell(n,q_{jk})$ and whose components are over those of
$D_{jk}$. From the local computations of the section above, we have

$h^{*}D_{j}=n\tilde{D}_{j}+\sum_{D_{k}D_{j}\neq\emptyset}\sum_{\alpha=1}^{s_{jk}}m_{jk,\alpha}E_{jk,\alpha}+\sum_{D_{kl}D_{j}\neq\emptyset}{F_{kl,j}},$
(8)

where $F_{kl,j}$ is a divisor whose components are the exceptional divisors
over points in $D_{jkl}$.

Explicitly, for any triple of positive integers numbers $j,k,l$, let
$\rho_{kl}(j)\in\\{1,2,3\\}$ the position of $j$ if we order the triple. For
example $\rho_{23}(1)=1$, $\rho_{57}(6)=2$ or $\rho_{54}(8)=3$. Thus, if
$F_{jkl,p}$ is the exceptional divisor over a point $p\in D_{jkl}$, then

$F_{jkl}=\sum_{p\in D_{jkl}}F_{jkl,p}$ $F_{kl,j}=\sum_{p\in
D_{jkl}}\rho_{kl}(j)F_{jkl,p}.$ (9)

In terms of intersection theory, we have

$F_{jkl}^{3}=\frac{n}{v^{jkl}_{1}v^{jkl}_{2}v^{jkl}_{3}}D_{jkl},\quad\tilde{D_{j}}\tilde{D_{k}}=0.$

From Theorem 2.13 we have

$K_{X_{n}}\sim_{{\mathbb{Q}}}h^{*}K+\Delta,$ (10)
$K:=\left(K_{Z}+\frac{n-1}{n}\sum_{j=1}^{r}D_{j}\right),\quad\Delta=\sum_{j<k}E_{jk}+\sum_{j<k<l}V_{jkl}F_{jkl},$
$E_{jk}=\sum_{\alpha=1}^{s_{jk}}N_{jk,\alpha}E_{jk,\alpha}\quad
N_{jk,\alpha}=\frac{m_{jk,\alpha}+n_{jk,\alpha}-n}{n},\quad
V_{jkl}=\frac{v_{1}^{jkl}+v_{2}^{jkl}+v_{3}^{jkl}-n}{n}.$

Now we describe how the intersection theory on $X_{n}$ behaves under pullbacks
of divisors of $Z$. In what follows, we set $E_{jk,0}=\tilde{D}_{j}$ and
$E_{jk,s_{jk}+1}=\tilde{D}_{k}$.

######  Proposition 4.12.

Let $G,G^{\prime}$ any divisors on $Z$, then $h^{*}GF_{kl,j}=0$ as $2$-cycle,
for any $1\leq\alpha\leq s_{jk}$,

$h^{*}Gh^{*}G^{\prime}E_{jk,\alpha}=0,$
$h^{*}GE_{jk,\alpha}^{2}=-k_{jk,\alpha}GD_{jk},\quad
h^{*}GE_{jk,\alpha}E_{jk,\alpha\pm 1}=GD_{jk}.$

###### Proof.

We use the projection formula repeatedly. The first one is given by the fact
that $h_{*}F_{jk,l}$ has codimension $3$. Now for a $1\leq\alpha\leq s_{jk}$
we have $h_{*}E_{jk,\alpha}$ supported in codimension $2$, thus

$h^{*}Gh^{*}G^{\prime}E_{jk,\alpha}=GG^{\prime}h_{*}E_{jk,\alpha}=0.$

Finally, for any $\alpha$ we have

$h^{*}D_{j}h^{*}GE_{jk,\alpha}=0=h^{*}G(m_{jk,\alpha-1}E_{jk,\alpha-1}E_{jk,\alpha}+m_{jk,\alpha}E_{jk,\alpha}^{2}+m_{jk,\alpha+1}E_{jk,\alpha+1}E_{jk,\alpha}),$
$h^{*}D_{k}h^{*}GE_{jk,\alpha}=0=h^{*}G(n_{jk,\alpha-1}E_{jk,\alpha-1}E_{jk,\alpha}+n_{jk,\alpha}E_{jk,\alpha}^{2}+n_{jk,\alpha+1}E_{jk,\alpha+1}E_{jk,\alpha}).$

The recursive relations with $k_{jk,\alpha}$ give

$\left[\begin{array}[]{cc}m_{jk,\alpha}&m_{jk,\alpha+1}\\\
n_{jk,\alpha}&n_{jk,\alpha+1}\\\
\end{array}\right]\left[\begin{array}[]{c}h^{*}G(E_{jk,\alpha}^{2}+k_{jk,\alpha}E_{jk,\alpha-1}E_{jk,\alpha})\\\
h^{*}G(E_{jk,\alpha+1}E_{jk,\alpha}-E_{jk,\alpha-1}E_{jk,\alpha})\end{array}\right]=\left[\begin{array}[]{c}0\\\
0\end{array}\right].$

From Lemma 2.23 we have the determinant
$m_{jk,\alpha}n_{jk,\alpha+1}-m_{jk,\alpha+1}n_{jk,\alpha}=n$, thus

$h^{*}G(E_{jk,\alpha}^{2}+k_{jk,\alpha}E_{jk,\alpha-1}E_{jk,\alpha})=0$
$h^{*}G(E_{jk,\alpha+1}E_{jk,\alpha}-E_{jk,\alpha-1}E_{jk,\alpha})=0.$

In particular, we have

$h^{*}GE_{jk,\alpha+1}E_{jk,\alpha}=h^{*}G\tilde{D}_{j}E_{jk,1}=GD_{jk},$

and the result follows. ∎

######  Corollary 4.13.

For any divisor $G$ on $Z$ we have

$h^{*}GE_{jk}K_{X}=-D_{jk}G\left((N_{jk,1}+N_{kj,1})+\sum_{\alpha=1}^{s_{jk}}(k_{jk,\alpha}-2)\right).$

###### Proof.

Since $h^{*}G\,F$ vanishes at top-dimensional intersections, and
$E_{jk,\alpha_{1}}E_{jk,\alpha_{2}}=0$ for $|\alpha_{1}-\alpha_{2}|>1$, we
have

$\displaystyle h^{*}GE_{jk}K_{X}$
$\displaystyle=h^{*}G(E_{jk})^{2}=h^{*}G\sum_{\alpha=1}^{s_{jk}}N_{jk,\alpha}^{2}E_{jk,\alpha}^{2}+2N_{jk,\alpha}N_{jk,\alpha+1}E_{jk,\alpha}E_{jk,\alpha+1}$
$\displaystyle=D_{jk}G\sum_{\alpha=1}^{s_{jk}}-k_{jk,\alpha}N_{jk,\alpha}^{2}+2N_{jk,\alpha}N_{jk,\alpha+1}$
$\displaystyle=D_{jk}G\sum_{\alpha=1}^{s_{jk}}N_{jk,\alpha}N_{jk,\alpha+1}-N_{jk,\alpha}N_{jk,\alpha-1}+N_{jk,\alpha}(k_{jk,\alpha}-2)$
$\displaystyle=D_{jk}G\sum_{\alpha=1}^{s_{jk}}N_{jk,\alpha}(k_{jk,\alpha}-2)$
$\displaystyle=-D_{jk}G\left((N_{jk,1}+N_{kj,1})+\sum_{\alpha=1}^{s_{jk}}(k_{jk,\alpha}-2)\right).$

where the last identity is by telescoping sum argument. ∎

######  Corollary 4.14.

If $C$ is a curve on $X_{n}$ contained in $\tilde{H}_{j}$ and disjoint to any
exceptional divisor $F_{jkl}$, then

$\tilde{D}_{j}C=\frac{h_{*}C}{n}\left(D_{j}-\sum_{j\neq
k}m_{jk,1}D_{k}\right).$

###### Proof.

From (8) we have

$h^{*}D_{j}C=D_{j}h_{*}C=n\tilde{D}_{j}+\sum_{k}m_{jk,1}E_{jk,1}C,$
$h^{*}D_{k}C=D_{k}h_{*}C=E_{jk,1}C,\quad\forall j\neq k,$

and the result follows. ∎

###  4.4. Asymptoticity of $K_{X_{n}}^{3}$.

For simplicity, let us denote $K_{X}=K_{X_{n}}$. Let us introduce the
following notation

$|D|_{jk}:=|D_{j}|_{k}+|D_{k}|_{j},$

where $|D_{j}|_{k}$ satisfy

$\sum_{D_{ll^{\prime}}D_{j}\neq
0}F_{ll^{\prime},j}E_{jk,\alpha}K_{X}=|D_{j}|_{k}(k_{jk,\alpha}-2).$

######  Lemma 4.15.

We have

$|D_{j}|_{k}=\sum_{l}\dfrac{v^{jkl}_{\rho_{kl}(j)}}{{v^{jkl}_{\rho_{kj}(l)}}}D_{jkl}.$

###### Proof.

Using equations from (7), observe that $|D_{j}|_{k}$ depends on slopes of
weights $v^{jkl}_{1},v^{jkl}_{2},v^{jkl}_{3}$ of the lattice points $v^{jkl}$.
Using (9), we get the result.
∎

We need this to compute the intersection of $K_{X}$ with the external walls of
the local toric resolution. Recursively we denote,

$x_{jk,\alpha}=K_{X}E_{jk,\alpha-1}E_{jk,\alpha},\quad 1\leq\alpha\leq
s_{jk}+1$ $y_{jk,\alpha}=K_{X}E_{jk,\alpha}^{2},\quad 1\leq\alpha\leq s_{jk}.$

Thus, we can write

$h^{*}D_{j}E_{jk,\alpha}K_{X}=m_{jk,\alpha-1}x_{jk,\alpha}+m_{jk,\alpha}y_{jk,\alpha}+m_{jk,\alpha+1}x_{jk,\alpha+1}+|D_{j}|_{k}(k_{jk,\alpha}-2)$
(11)

Using the ${\mathbb{Q}}$-numerical equivalence of $K_{X}$ in 2.13, we compute

$x_{jk,1}=h^{*}D_{k}\tilde{D_{j}}K_{X}=D_{jk}\left(K+\sum_{l\neq
j}N_{jl,1}D_{l}\right),$
$x_{jk,s+1}=h^{*}D_{j}\tilde{D_{k}}K_{X}=D_{jk}\left(K+\sum_{l\neq
k}N_{kl,1}D_{l}\right).$

######  Proposition 4.16.

We have

$x_{jk,\alpha}=x_{jk,1}+\frac{1}{n}(m_{jk,\alpha}^{*}(D_{jk}D_{k}-|D_{k}|_{j})-n_{jk,\alpha}^{*}(D_{jk}D_{j}-|D_{j}|_{k})),$
$y_{jk,\alpha}=-k_{jk,\alpha}x_{jk,\alpha}+\frac{(k_{jk,\alpha}-2)}{n}(n_{jk,\alpha+1}(D_{jk}D_{j}-|D_{j}|_{k})-m_{jk,\alpha+1}(D_{jk}D_{k}-|D_{k}|_{j})).$

Where $m_{jk,\alpha}^{*}=m_{jk,\alpha}-m_{jk,\alpha-1}-m_{jk,1}+m_{jk,0}$ and
analogous for $n_{jk,\alpha}^{*}$.

###### Proof.

Using the recursion given by the $k_{jk,\alpha}^{\prime}s$, and formulas for
$h^{*}D_{j}\,E_{jk,\alpha}K_{X}$ and $h^{*}D_{k}\,E_{jk,\alpha}K_{X}$ of (11),
we have

$\left[\begin{array}[]{c}(D_{j}^{2}D_{k}-|D_{j}|_{k})(k_{jk,\alpha}-2)\\\
(D_{j}D_{k}^{2}-|D_{k}|_{j})(k_{jk,\alpha}-2)\end{array}\right]=\left[\begin{array}[]{cc}m_{jk,\alpha}&m_{jk,\alpha+1}\\\
n_{jk,\alpha}&n_{jk,\alpha+1}\end{array}\right]\left[\begin{array}[]{c}k_{jk,\alpha}x_{jk,\alpha}+y_{jk,\alpha}\\\
x_{jk,\alpha+1}-x_{jk,\alpha}\end{array}\right]$

The determinant $m_{jk,\alpha}n_{jk,\alpha+1}-m_{jk,\alpha+1}n_{jk,\alpha}=n$,
implies second relation for $y_{jk,\alpha}$, and

$x_{jk,\alpha+1}=x_{jk,\alpha}+\frac{(k_{jk,\alpha}-2)}{n}(m_{jk,\alpha}(D_{jk}D_{k}-|D_{k}|_{j})-n_{jk,\alpha}(D_{jk}D_{j}-|D_{j}|_{k}))$

The recurrence for $x_{jk,\alpha}$ with a telescopic sum argument give the
result.
∎

######  Proposition 4.17.

We have,

$\displaystyle K_{X}^{3}=$ $\displaystyle
nK^{3}-2\sum_{j<k}\left(D_{jk}K+\sum_{j<k<l}V_{jkl}D_{jkl}\right)(N_{jk,1}+N_{kj,1}+(k-2)_{jk})$
$\displaystyle+\sum_{j<k<l}\frac{nV_{jkl}^{3}}{v_{1}^{jkl}v_{2}^{jkl}v_{3}^{jkl}}D_{jkl}+\sum_{j<k}\sum_{\alpha=1}^{s_{jk}}\frac{D_{jk}(D_{j}+D_{k})-|D|_{jk}}{n}N_{jk,\alpha}(k_{jk,\alpha}-2)$
$\displaystyle-\sum_{j<k}\sum_{\alpha=1}^{s_{jk}}N_{jk,\alpha}(x_{jk,\alpha}+y_{jk,\alpha}+x_{jk,\alpha+1}).$

###### Proof.

We will compute $K_{X}^{3}$ using the numerical equivalence of (10). Squaring
we get

$\displaystyle
K_{X}^{2}\sim_{{\mathbb{Q}}}h^{*}K^{2}+\left(\sum_{j<k}E_{jk}\right)^{2}+\sum_{j<k<l}V_{jkl}^{2}F_{jkl}^{2}+2\sum_{j<k}E_{jk}\left(h^{*}K+\sum_{j<k<l}V_{jkl}F_{jkl}\right).$

We have explicitly

$(h^{*}K)^{2}K_{X}=(h^{*}K)^{3}=nK^{3}.$
$E_{jk}F_{jkl}K_{X}=D_{jkl}\sum_{\alpha=1}^{s_{jk}}N_{jk,\alpha}(k_{jk,\alpha}-2)=-D_{jkl}\left((N_{jk,1}+N_{kj,1})+\sum_{\alpha=1}^{s_{jk}}(k_{jk,\alpha}-2)\right).$

In the rest, we will denote

$(k-2)_{jk}:=\sum_{\alpha=1}^{s_{jk}}(k_{jk,\alpha}-2).$

Using Corollary 4.13 for $G=K$, we get

$\displaystyle K_{X}^{3}=$ $\displaystyle
nK^{3}-2\sum_{j<k}\left(D_{jk}K+\sum_{j<k<l}V_{jkl}D_{jkl}\right)(N_{jk,1}+N_{kj,1}+(k-2)_{jk})$
$\displaystyle+\sum_{j<k<l}\frac{nV_{jkl}^{3}}{v_{1}^{jkl}v_{2}^{jkl}v_{3}^{jkl}}D_{jkl}+K_{X}\left(\sum_{j<k}E_{jk}\right)^{2}.$

Just rest to compute

$K_{X}\left(\sum_{j<k}E_{jk}\right)^{2}=\sum_{j<k}\sum_{\alpha=1}^{s_{jk}}N_{jk,\alpha}(N_{jk,\alpha-1}x_{jk,\alpha}+N_{jk,\alpha}y_{jk,\alpha}+N_{jk,\alpha+1}x_{jk,\alpha+1}).$

From Corollary 4.13, we have

$\displaystyle\frac{D_{jk}(D_{j}+D_{k})(k_{jk,\alpha}-2)}{n}$
$\displaystyle=\frac{h^{*}(D_{j}+D_{k})E_{jk,\alpha}K_{X}}{n}$
$\displaystyle=N_{jk,\alpha-1}x_{jk,\alpha}+N_{jk,\alpha}y_{jk,\alpha}+N_{jk,\alpha+1}x_{jk,\alpha+1}$
$\displaystyle+(x_{jk,\alpha}+y_{jk,\alpha}+x_{jk,\alpha+1})+\frac{(k_{jk,\alpha}-2)}{n}|D|_{jk},$

So, we have explicitly

$\displaystyle K_{X}\left(\sum_{j<k}E_{jk}\right)^{2}$
$\displaystyle=\sum_{j<k}\sum_{\alpha=1}^{s_{jk}}\frac{D_{jk}(D_{j}+D_{k})-|D|_{jk}}{n}N_{jk,\alpha}(k_{jk,\alpha}-2)$
$\displaystyle-\sum_{j<k}\sum_{\alpha=1}^{s_{jk}}N_{jk,\alpha}(x_{jk,\alpha}+y_{jk,\alpha}+x_{jk,\alpha+1}).$

∎

######  Theorem 4.18.

If $\\{D_{1},\ldots,D_{r}\\}$ is an asymptotic arrangement, then

$\frac{K_{X_{n}}^{3}}{n}\approx-\bar{c}_{1}^{3}(Z,D),$

for prime numbers $n\gg 0$.

###### Proof.

The first term of the sum contains $\sum_{\alpha}N_{jk,\alpha}(k_{jk}-2)$,
which is asymptotic respect to $n$ by previous discussion (Section 2.5). Thus,
we just have to prove asymptoticity for

$\sum_{j<k}\sum_{\alpha=1}^{s_{jk}}N_{jk,\alpha}(x_{jk,\alpha}+y_{jk,\alpha}+x_{jk,\alpha+1})$
(12)

Proceeding as above, it is not difficult to show the following identity,

$\displaystyle\sum_{\alpha=1}^{s_{jk}}\frac{D_{jk}(D_{j}+D_{k})-|D|_{jk}}{n}(k_{jk,\alpha}-2)$
$\displaystyle=\sum_{\alpha=1}^{s_{jk}}(N_{jk,\alpha}+1)(x_{jk,\alpha}+y_{jk,\alpha}+x_{jk,\alpha+1})$
$\displaystyle-N_{jk,1}x_{jk,1}-N_{jk,s}x_{jk,s+1}.$

So, the asymptoticity of (12) depends only on the asymptoticity of

$\sum_{j<k}\sum_{\alpha=1}^{s_{jk}}(x_{jk,\alpha}+y_{jk,\alpha}+x_{jk,\alpha+1}).$
(13)

By 4.16, $x_{jk,\alpha}+y_{jk,\alpha}+x_{jk,\alpha+1}$ equals to

$\frac{k_{jk,\alpha}-2}{n}(m_{jk,\alpha}^{**}(D_{jk}D_{k}-|D_{k}|_{j})-n_{jk,\alpha}^{**}(D_{jk}D_{j}-|D_{j}|_{k})-nx_{jk,1}),$

where $m_{jk,\alpha}^{**}=m_{jk,\alpha-1}-m_{jk,\alpha+1}-m_{jk,0}+m_{jk,1}$,
and analogous for $n_{jk,\alpha}^{**}$. Observe that these terms are bounded
by $Cn$ for some constant $C>0$. On the other hand, the terms
$(D_{jk}D_{j}-|D_{j}|_{k})$ and $(D_{jk}D_{k}-|D_{k}|_{j})$ asymptotically
depend only on the slopes of coordinates of the chosen lattice points
$v^{jkl}$ on each intersection $D_{jkl}$. By Lemma 4.10, we can choose lattice
points with slopes asymptotically bounded by $3$ as $n$ grows, with $K_{X}$
having $V_{jkl}=0$ for all $j<k<l$. So, we have

$|D|_{jk}\leq 6\sum_{l}D_{jkl}.$

Thus, as $n$ grows, all the terms in $K_{X}^{3}/n$ vanish except
$nK^{3}\approx-n\bar{c}_{1}^{3}(Z,D)$. ∎

###  4.5. Asymptoticity of $e(X_{n})$.

The topological characteristic can be computed from the topology of $(Z,D)$
and the exceptional divisors $E_{jk,\alpha},F_{jkl}$. The divisors
$F_{jkl}=\sum_{p\in D_{jkl}}F_{jkl,p}$ where $F_{jkl,p}$ is the corresponding
exceptional divisor over a $p\in D_{jkl}$. Thus,
$e(F_{jkl})=D_{jkl}e(F_{jkl,p})$. In the toric picture (Section 4.2) of
$X_{n}$ over $p$, let $v=v^{jkl}$ the ray generator defining $F_{jkl,p}$.

######  Lemma 4.19.

We have $e(F_{jkl,p})=s_{jk}+s_{jl}+s_{kl}+3$.

###### Proof.

It is well-known that $F_{jkl,p}$ is the toric variety associated to the star-
fan $\mbox{Star}(C(v))\subset(N_{v})_{{\mathbb{R}}}$, i.e., the induced fan by
the lattice quotient $N_{v}=N/vN$. In this case, the $2$-cones of
$\mbox{Star}(C(v))$ are the induced by each
$C(e_{jk,\alpha},e_{jk,\alpha+1})$. Since $e(F_{jkl,p})$ is the sum of its
top-dimensional cones [CLS11, Thm. 12.3.9], we have the result.
∎

The components of divisors $E_{jk,\alpha}$ are determined locally as exterior
divisors of the toric picture of $X_{n}$. They intersect $F$ at the rational
curves $C_{jk,\alpha}$. Locally each component of $E_{jk,\alpha}$ is
isomorphic to ${\mathbb{A}}^{1}\times{\mathbb{P}}^{1}_{w}$ over $D_{jk}$,
where ${\mathbb{P}}^{1}_{w}$ is a weighted projective line. This follows for
the star-fan construction.

######  Lemma 4.20.

If $E_{jk,\alpha}=\sum_{C\in D_{jk}}E_{jk,\alpha,C}$ is the decomposition in
componenets, then we compute

$e(E_{jk,\alpha})=4\sum_{C\in D_{jk}}(1-p_{g}(C)).$

###### Proof.

We have $e(E_{jk,\alpha})=\sum_{C}e(E_{jk,\alpha,C})$. Since $E_{jk,\alpha,C}$
is a fibration over $C$ with fiber ${\mathbb{P}}^{1}_{w}$. Since the
topological characteristics of weighted and non-weighted projective spaces are
the same, we have that
$e(E_{jk,\alpha,C})=e({\mathbb{P}}^{1}_{w})e(C)=4(1-p_{g}(C))$. ∎

######  Lemma 4.21.

If $X$ is a complex algebraic variety, and $A\subset X$ is a subvariety such
that $X\setminus A$ is smooth, then $e(X)=e(X\setminus A)+e(A)$.

###### Proof.

See [Ful93, p. 141].∎

######  Remark 4.22.

The above lemma implies the _exclusion-inclusion principle_ , i.e., for
subvarieties $V,W\hookrightarrow X$ we have $e(V\cup W)+e(V\cap W)=e(V)+e(W)$.

######  Proposition 4.23.

We have,

$\displaystyle e(X_{n})$ $\displaystyle=n(e(Y)-e(D))+e(D)-e(\mbox{Sing}(D))$
$\displaystyle+\sum_{j<k}\sum_{C\in
D_{jk}}[s_{jk}(3-4p_{g}(C))-1]-\sum_{j<k<l}(s_{jk}+s_{jl}+s_{kl}-3)D_{jkl}$

###### Proof.

Denote by $R$ the ramification divisor of $h\colon X_{n}\to Z$. This is a
$n:1$ morphism which is an isomorphism outside $R$ we have

$e(X\setminus R)=ne(Y\setminus D)=n(e(Y)-e(D)).$

On the other hand, $R=\bigcup_{j}\tilde{D}_{j}\cup\mbox{Exc}(h)$, where
$\mbox{Exc}(h)$ is the exceptional data of $h$. Topologically is given by

$\mbox{Exc}(h)=\bigcup_{j<k<l}F_{jkl}\cup\bigcup_{j<k}\bigcup_{\alpha}E_{kj,\alpha}.$

By the exclusion-inclusion observe that

$e\left(\bigcup_{j}\tilde{D}_{j}\right)-e\left(\bigcup_{j}\tilde{D}_{j}\cap\mbox{Exc}(h)\right)=e(D)-e(\mbox{Sing}(D)).$

So, we get

$e(R)=e(D)-e(\mbox{Sing}(D))+e(\mbox{Exc}(h)).$

On the other hand, the components of $E_{jk,\alpha}E_{jk,\alpha}$ are curves
over $D_{jk}$ isomorphic to their respective components. Also, each component
of $E_{jk,\alpha}F_{jkl}$ is a rational curve over its corresponding point in
$D_{jkl}$. Thus, we have identities,

$e(E_{jk,\alpha}E_{jk,\alpha+1})=e(D_{jk})$
$e(E_{jk,\alpha}F_{jkl})=2D_{jkl}.$

Using repeatedly the exclusion-inclusion principle we we compute

$e(\mbox{Exp}(h))=\sum_{j<k}\sum_{C\in
D_{jk}}[s_{jk}(3-4p_{g}(C))-1]-\sum_{j<k<l}(s_{jk}+s_{jl}+s_{kl}-3)D_{jkl}.$

∎

######  Theorem 4.24.

If $\\{D_{1},\ldots,D_{r}\\}$ is an asymptotic arrangement, then

$\frac{e(X_{n})}{n}\approx\bar{c}_{3}(Z,D),$

for prime numbers $n\gg 0$.

###### Proof.

As a corollary of the previous identity for $e(X_{n})$, and by the previous
discussion in Section 2.5, the lengths $s_{jk}/n$ are asymptotically zero as
$n$ grows. Thus, $e(X_{n})\approx n(e(Z)-e(D))$ as $n$ grows. ∎

##  5\. Applications to the geography of 3-folds

###  5.1. The case of $3$ hyperplanes.

Consider $Z\hookrightarrow{\mathbb{P}}^{4}$ of degree $d=\deg(Z)$. In this
case, we have explicitly

$K_{Z}=(d-5)H|_{Z},\quad c_{2}(Z)=(10+d(d-5))H^{2}|_{Z},$
$c_{3}(Z)=-d\left(d^{2}(d-5)+10d-10\right),$

for a generic hyperplane section $H$. Take 3 hyperplane sections
$\\{H_{1},H_{2},H_{3}\\}$ in general position, and asymptotic partitions
$\nu_{1}+\nu_{2}+\nu_{3}=n$. Along $D=\sum_{j}\nu_{j}H_{j}\sim nH$ consider
the respective $n$-th root cover $Y_{n}\to Z$. Its singularities are over $d$
points in $H_{1}H_{2}H_{3}$. As we see in Example 4.7, these singularities
admit a locally nef non-singular resolution. Unfortunately, in this resolution
the lattice point $v$ does not satisfy the condition of Lemma 4.10, i.e., the
volume $K_{X}^{3}$ will not be completely asymptotic to the logarithmic Chern
number $\bar{c}_{1}^{3}(Z,D)$. However, since the chosen $v$ satisfy
$v_{j}=1$. So, following the methods of Theorem 4.18 to compute $K_{X}^{3}$,
we get

$x_{jk,\alpha}=E_{jk,\alpha}E_{jk,\alpha+1}K_{X}=d(d-3),\quad 1\leq\alpha\leq
s_{jk}.$

Now we compute

$\displaystyle K_{X}^{3}$
$\displaystyle=d(d-3)\left(nd^{2}-3nd+3n-9d+18-3\sum_{j<k}(k-2)_{jk}\right),$

since

$\sum_{j<k}(N_{jk,1}+N_{kj,1})=-3\frac{n-3}{n},$ $\displaystyle
K_{X}\,\left(\sum_{j<k}E_{jk}\right)^{2}=-d(d-3)\sum_{j<k}(N_{jk,1}+N_{jk,s}+(k-2)_{jk}).$

In particular for prime numbers $n\gg 0$,

$\frac{K_{X}^{3}}{n}\approx(K_{Z}+H_{1}+H_{2}+H_{3})^{3}-d=d(d-2)^{3}-d.$

On the other hand, from Section 4.1 we have

$\chi({\mathcal{O}}_{X})=n\chi(Z,{\mathcal{O}}_{Z})-\frac{1}{12}(R_{1}(n)+R_{2}(n)+R_{3}(n)),$

where

$\chi(Z,{\mathcal{O}}_{Z})=-\frac{d(d-5)(10+d(d-5))}{24}$
$R_{1}(n)=\frac{9d(n-1)(2n-1)}{2n},$
$R_{2}(n)=\frac{3d(d-5)(n-1)(5n-1)}{2n}+\frac{3d((d-5)^{2}+d(d-5)+10)(n-1)}{2}$
$R_{3}(n)=6d(d-2)(d(\nu_{1},\nu_{2},n)+d(\nu_{1},\nu_{3},n)+d(\nu_{2},\nu_{3},n)).$

Since, the partition is asymptotic, for $n\gg 0$ we have

$\frac{\chi({\mathcal{O}}_{X})}{n}\approx-\frac{d(d-2)(d-1)^{2}}{24}.$

For $n\gg 0$, the topological characteristic behaves as

$\frac{e(X)}{n}\approx c_{3}({\mathbb{P}}^{3},D_{red})=-d(d-5)(d^{2}+2d+6).$

Following the proof of Theorem 5.2, we get $K_{X}$ nef for $n\gg 0$. As a
consequence of the above computations, we have.

######  Theorem 5.1.

For $d\geq 5$ and $n\gg 0$ there are minimal non-singular $3$-folds $X$ of
general type having degree $n$ over $Z$ with slopes

$\frac{c_{1}^{3}}{c_{1}c_{2}}\approx\frac{(d-2)^{3}-1}{(d-2)(d-1)^{2}},\quad\frac{c_{3}}{c_{1}c_{2}}\approx\frac{(d-5)(d^{2}+2d+6)}{(d-2)(d-1)^{2}}.$

In particular, as the degree of $Z$ grows, the slopes have limit point
$(1,1)$.

∎

###  5.2. Hyperplane sections arrangements.

The above partial resolution can be seen as a resolution of pairs

$h\colon(X_{n},\tilde{D}_{red})\to(Z,D_{red}),$

where $\tilde{D}$ is the inverse direct image of $D$. The reduced divisor of
$D^{\prime}$ is an SNC divisor. Indeed, in terms of $\log$-resolutions [KM92,
p. 5], we can see that our partial resolution has good behavior in logarithmic
terms, i.e., they preserves the $\log$-structure of the variety $n$-th root
cover $Y_{n}$. The following result illustrates these ideas.

######  Theorem 5.2.

Let $Z$ be a minimal non-singular projective $3$-fold, and let $\\{H_{1}$,
$\ldots,$ $H_{r}\\}$ be a collection of hyperplane sections in general
position. Then, for prime numbers $n\gg 0$ there are $\log$-morphisms
$(X_{n},\tilde{D}_{red})\to(Z,D_{red})$ of degree $n$ such that:

  1. (1)

$X_{n}$ is of $\log$-general type, i.e., $K_{X_{n}}+\tilde{D}_{red}$ is big
and nef,

  2. (2)

$X_{n}$ has cyclic quotient singularities, and so $\log$-terminal of order
lower than $n$, and

  3. (3)

the slopes $(-K^{3}/24\chi,e/24\chi)$ of $X_{n}$ are arbitrarily near to
$(2,1/3)$.

###### Proof.

We take $D=\sum_{j=1}^{r}\nu_{j}H_{j}$, where $H_{j}$ are hyperplane sections
on $Z$ and $\sum_{j=1}^{r}\nu_{j}=n$ an asymptotic partition. Recall that
$H_{j}H_{k}H_{l}=H_{j}^{2}H_{k}=\deg(Z)$ for any $j<k<l$. Take $h\colon
X_{n}\to Y_{n}\to Z$ the asymptotic cyclic resolution constructed in Theorem
4.18. Again for simplicity let us denote $K_{X}=K_{X_{n}}$. From, the explicit
description given in 4.5, we have

$K_{X}+D^{\prime}_{red}=K_{X}+\sum_{j}\tilde{D}_{j}+\sum_{j<k,\alpha}E_{jk,\alpha}+\sum_{j<k<l}F_{jkl}=h^{*}(K_{Z}+D_{red}).$

First observe that for any curve $C$ outside the exceptional data of $h$, we
have $(K_{X}+D^{\prime}_{red})C\geq 0$, by projection formula and since
$K_{Z}+D_{red}$ is ample. For every closed curve $C=C_{jk,\alpha}$ of
$C=C_{l}$ of the local toric picture (Section 4.2) of the resolution, we have
$(K_{X}+D^{\prime}_{red})\,C=0$. For the remainder curves, we just need to
concern about the positivity of its intersection with $K_{X}$. Since $K_{Z}$
is a nef divisor, by Corollary 2.14 we must have any $K_{X}$-negative curve
contained in the support of $h^{*}(D)$. Thus, the rest of rational curves in
$\mbox{Supp}(h^{*}D)$ are of the following types:

  1. (1)

Curves defined by the closure of a wall $E_{jk,\alpha-1}E_{jk,\alpha}$ for
$1\leq\alpha\leq s_{jk}.$

  2. (2)

A curve contained in $E_{jk,\alpha}$ but not in $E_{jk,\alpha\pm 1}$ for
$1\leq\alpha\leq s_{jk}$.

  3. (3)

A curve contained in $\tilde{H}_{j}$.

If $C$ is of type (1), from 4.12 we have,

$(K_{X}+D^{\prime}_{red})E_{jk,\alpha}E_{jk,\alpha+1}=h^{*}(K_{Z}+D)E_{jk,\alpha}E_{jk,\alpha+1}=(K_{Z}+D_{red})H_{jk}>0,$

for any $\alpha$. If $C$ is of type (2), then $C$ must be a fiber of the ruled
surface $E_{jk,\alpha}$,i.e., is in the class of $C_{jk,\alpha}$. But, by (6)
we have $K_{X}C>0$. Finally, if $C$ is of type (3), we assume that it does not
intersect interior divisors $F_{jkl}$. If does it, then by the toric local
description $C$ must be of the form $E_{jk,1}\tilde{H}_{j}$ for some $k$.
Again by projection formula, we have
$(K_{X}+D^{\prime}_{red})C=(K_{Z}+D)h_{*}C\geq 0$. Then,
$K_{X}+D^{\prime}_{red}$ is a nef divisor, and moreover
$(K_{X}+D^{\prime}_{red})^{3}=(K_{Z}+D)^{3}>0$.Thus, by [Laz04, Th. 2.2.16.],
the divisor $K_{X}+D^{\prime}_{red}$ is big. Now, from Theorem 4.18 we know
that for $n\gg 0,$

$\displaystyle\frac{K_{X}^{3}}{n}$ $\displaystyle\approx-
c_{1}^{3}(Z,D)=\left(K_{Z}+rH\right)^{3}$
$\displaystyle=K_{Z}^{3}+r^{3}\deg(Z)+3rK_{Z}H^{2}+3K_{Z}^{2}H,$

where $H$ is a generic hyperplane section on $Z$. Thus, if we choose $r$
depending on $n$ with $r(n)/n\to 0$ as $n$ grows, then the numbers $|D|_{jk}$
goes to zero respect with $n$. Then, we have $K_{X}^{3}>0$. Moreover, from
Example 2.9 we have $(-K^{3}/24\chi,e/24\chi)(X)$ arbitrarily near to
$(2,1/3)$. ∎

##  6\. Discussion

In this section, we will briefly discuss some possible future paths in order
to extend this work.

###  6.1. Asymptoticity through minimal models

One of the main horizons of this research is to achieve the asymptoticity of
invariants through minimal models. This means that the invariants of $X_{n}$,
with respect to $n$, could be asymptotically equal to the respective
invariants of its minimal model. Thus, we will be in a very nice position to
do geography, i.e., the study of arrangements of hypersurfaces is identified
through the slopes of Chern numbers with a "region" of minimal projective
varieties. As we see in Theorem 5.2 and Theorem 5.1, if the basis pair $(Z,D)$
has $Z$ minimal of general type and $D$ composed by ample divisors, then our
constructions preserve important features in terms of minimal models. However,
this in general is not something easy to work on. For the future of this work,
$3$ aspects are important.

  1. (1)

Asymptotic study of (partial) desingularization of cyclic quotient
singularities of dimension $\geq 3$.

  2. (2)

Hirzebruch-Riemann-Roch for singular varieties with terminal and $\log$ \-
terminal singularities with their asymptotic analogs. In particular, this
requires to establish what invariants will be the correct version of the Chern
numbers. As we did in the case of dimension $3$.

  3. (3)

The behavior of the invariants after applying the MMP to our constructed
varieties.

In the next section, we discuss (1). If we achieve our goal we will be, able
to construct _good partial resolutions_ $X_{n}\to Y_{n}$, i.e., the Chern
numbers, with respect to $n$, are asymptotically equal to the logarithmic
Chern numbers of the basis $(Z,D)$. We expect that we can improve the
singularities to the terminal ones, so we will be able to run the MMP, i.e. we
want to construct a _terminal good partial resolution_. For (2), we have
results of [Rei87] and [BS05] which are a kind of starting point for future
work. These contain versions of the Hirzebruch-Riemann-Roch theorem for
varieties with canonical and cyclic quotient singularities. For (3), we think
that the answer could be hidden in all the massive previous work done around
the minimal model program [BCHM10], [KM92]. We expect, that the involved
invariants do not suffer dramatic changes after flipping or contractions
operations as occur in the case of surfaces. Then, asymptotically with respect
to $n$, the invariants remain unchanged. We state the above discussion as
conjecture.

######  Conjecture 6.1.

Let $X_{n}\to Y_{n}\to(Z,D)$ be a terminal good partial resolution of
singularities of the $n$-th root cover construction. Assume that $K_{Y_{n}}$
is nef, and let $X^{\prime}_{n}$ a minimal model of $X_{n}$. Then, for any
partition $i_{1}+\ldots+i_{m}=d$ we have

$\frac{c_{i_{1}}\ldots
c_{i_{m}}(X^{\prime}_{n})}{n}\approx\bar{c}_{i_{1}}\ldots\bar{c}_{i_{m}}(Z,D),$

for prime numbers $n\gg 0$.

###  6.2. What about the length of resolution of $3$-fold c.q.s

Cyclic quotient singularities of dimension $3$ can be desingularized using a
generalization of the Hirzebruch-Jung algorithm, which is the Fujiki-Oka
algorithm. See [Ash19] for a modern treatment. After the local cyclic toric
resolution of Section 4.2, instinctively we want to desingularize each one
with the Fujiki-Oka process. However, since we want asymptoticy of invariants
in our resolutions, we ask for the _topological length_ and the _intersection
number behavior_ of such an algorithm. For the first, we mean the amount of
new topological data, i.e., how the Betti numbers grow for the chosen
resolutions. For last, we mean how the new curves and divisors on the
exceptional data affect the volume $K_{X}^{3}$. As we discussed in Section
2.5, the algorithm in dimension $2$ has both aspects behaving as
$\sim\sqrt{n}$ for a suitable class of integer numbers.

Let us assume that we choose a partial resolution for the local cyclic
resolution, so the amount of new topological data will behave approximately as

$\sim 3\sqrt{n}\sum_{j<k,\alpha}\ell(v_{l},a_{jk,\alpha},b_{jk,\alpha}),$

where $\ell(v_{l},a_{jk,\alpha},b_{jk,\alpha})$ is a length number depending
on each cyclic singularities given in Lemma 4.4. Thus, asymptotically respect
with $n$, we require that $\ell(v_{l},a_{jk,\alpha},b_{jk,\alpha})\sim
n^{1/c}$ for $c<1/2$. In particular, Fujiki-Oka algorithm for a cyclic
quotient singularity of type $\frac{1}{n}(a,b,1)$ contains the processes for
those of dimension two $\frac{1}{n}(a,1)$ and $\frac{1}{n}(b,1)$. Thus, in the
best case, we will have
$\ell(v_{l},a_{jk,\alpha},b_{jk,\alpha})\sim\sqrt{v_{l}}$. To assure
asymptoticity in Theorem 4.18 we must have $v_{l}\sim n/3$, thus after resolve
we lose the asymptoticity on the topological side. On the other hand, if we
admit all $v_{l}^{\prime}s$ small as we see in Theorem 5.1, then after resolve
we lose the asymptoticity of the volume. These observations lead us to the
following problem: the existence of a well-behaved algorithmic _terminal
resolution_ for cyclic quotient singularities, i.e., having only terminal
singularities.

The terminalization of a toric singularity is a well-known process [CLS11,
Sec. 11.4]. Indeed, assume that our toric singularity has associated cone
$\sigma\subset{\mathbb{R}}^{d}$. First, we have to compute the convex hull of
$\sigma\cap{\mathbb{Z}}^{d}-\\{0\\}$. This will give us a refinement of
$\sigma$, which is a canonical resolution, i.e. having at most canonical
singularities with ample canonical bundle. Finally, each canonical toric
singularity defined by a cone can be terminalizated by blowing-up each lattice
point on the plane generated by the primitive generators of the cone. However,
we do not know the growing behavior of this algorithm. In fact, it is known
that the best convex-hull algorithm behaves as $\sim n\log n$ when the number
of lattice points is $n$ [Gre90]. This not seems like a good algorithm to
choose.

######  Question 6.2.

How can we construct a terminal algorithm for cyclic quotient singularities
with the desired asymptotic properties? Is it possible?

As we see in Figure 4 and Figure 5, to achieve a well-behaved resolution it is
probable that we will have to impose different conditions on the integer $a$
and $b$. In Theorem 5.2 we see that there are $3$-folds with cyclic quotient
singularities accumulating in the well-known point of the map $(2,1/3)$. We
are curious if after applying the process proposed in Section 6.1, the minimal
$3$-folds expected will preserve the accumulating point or they move out.
Finally, the principal motivation for all this work is the connection between
arrangements of hypersurfaces and the geography of invariants of minimal
varieties. For us will be interesting to explore the geography through
arbitrary arrangements of planes on ${\mathbb{P}}^{3}$.

######  Question 6.3.

What is the region covered by minimal models of $n$-th root cover $Y_{n}$
along arrangements of planes in ${\mathbb{P}}^{3}$?

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