# The Marshall-Olkin Flexible Weibull Extension Distribution

Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura
35516, Egypt

###### Abstract

This paper introduces a new generalization of the flexible Weibull
distribution with three parameters this model called the Marshall-Olkin
flexible Weibull extension (MO-FWE) distribution which exhibits bathtub-shaped
hazard rate. We studied it’s statistical properties include, quantile function
skewness and kurtosis, the mode, $rth$ moments and moment generating function
and order statistics. We used the method of maximum likelihood for estimating
the model parameters and the observed Fisher’s information matrix is derived.
We illustrate the usefulness of the proposed model by applications to real
data.

Keywords: Weibull distribution; flexible Weibull extension distribution;
Marshall-Olkin flexible Weibull; maximum likelihood estimation.

##  1 Introduction

The Weibull distribution (WD) introduced by Weibull [20], is a popular
distribution for modeling lifetime data where the hazard rate function is
monotone. Recently appeared new classes of distributions were based on
modifications of the Weibull distribution (WD) to provide a good fit to data
set with bathtub hazard failure rate Xie and Lai [18]. Among of these,
modified Weibull distribution (MWD), Lai et al. [8] and Sarhan and Zaindin
[14], moreover the beta-Weibull distribution (BWD) has been derived by Famoye
et al. [6], beta modified Weibull distribution (BMWD), Silva et al. [17] and
Nadarajah et al. [12]. Recently, there are many generalization of the WD like
a Kumaraswamy Weibull distribution (KWD), Cordeiro et al. [5], generalized
modified Weibull distribution (GMWD), Carrasco et al. [4] and exponentiated
modified Weibull extension distribution (EMWED), Sarhan and Apaloo [15].

The flexible Weibull distribution (FWED), Bebbington et al. [3] has a wide
range of applications including life testing experiments, reliability
analysis, applied statistics and clinical studies. The origin and other
aspects of this distribution can be found in [3].

A random variable $X$ is said to have the flexible Weibull Extension
distribution with parameters $\alpha,\beta>0$ if it’s probability density
function (pdf) is given by

$g(x)=\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}\exp\left\\{-e^{\alpha x-\frac{\beta}{x}}\right\\},\quad
x>0,$ (1.1)

while the cumulative distribution function (cdf) is given by

$G(x)=1-\exp\left\\{-e^{\alpha x-\frac{\beta}{x}}\right\\},\quad x>0.$ (1.2)

The survival function is given by the equation

$S(x)=\exp\left\\{-e^{\alpha x-\frac{\beta}{x}}\right\\},\quad x>0,$ (1.3)

the hazard rate function is

$h(x)=\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha x-\frac{\beta}{x}},$
(1.4)

and the reversed hazard rate function is

$r(x)=\frac{\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}\exp\left\\{-e^{\alpha
x-\frac{\beta}{x}}\right\\}}{1-\exp\left\\{-e^{\alpha
x-\frac{\beta}{x}}\right\\}}.$ (1.5)

In this paper we present a new generalization of the flexible Weibull
extension distribution called the Marshall-Olkin flexible Weibull extension
distribution (MO-FWED). By using the Marshall and Olkin’s [11] method for
adding a new parameter to an existing distribution, this new generalized
referred to as the Marshall-Olkin flexible Weibull extension distribution.

Marshall and Olkin [11] proposed a new family of distributions called the
Marshall-Olkin extended (MOE) family by adding a new parameter to the baseline
distribution. They defined a new survival function $S_{MO}(x)$ by introducing
the additional shape parameter $\theta$ such that $\theta>0$ and
$\bar{\theta}=1-\theta$. Marshall and Olkin called the parameter $\theta$ the
tilt parameter and they interpreted $\theta$ in terms of the behavior of the
hazard rate function of $S_{MO}(x)$. Their ratio is increasing in $x$ for
$\theta\geq 1$ and decreasing in $x$ for $0<\theta<1$. They consider for any
arbitrary continuous distribution called baseline distribution having
cumulative distribution function $G(x,\varphi)$ with the related probability
density function pdf $g(x,\varphi)$, then the cumulative distribution function
of the Marshall Olkin (MO) family of distribution is given by

$F_{MO}(x)=\frac{G(x,\varphi)}{1-\bar{\theta}S(x,\varphi)},\quad-\infty<x<\infty,$
(1.6)

where $\theta>0$ and $\bar{\theta}=1-\theta$.
The probability density function corresponding to Eq.(1.6) becomes

$f_{MO}(x)=\frac{\theta
g(x,\varphi)}{\left[1-\bar{\theta}S(x,\varphi)\right]^{2}},\quad-\infty<x<\infty.$
(1.7)

The survival function, hazard rate function, reversed hazard rate function and
cumulative hazard rate function of the Marshall-Olkin (MO) family of a
probability distribution are given by

$\displaystyle S_{MO}(x)$ $\displaystyle=$ $\displaystyle\frac{\theta
S(x,\varphi)}{1-\bar{\theta}S(x,\varphi)},$ (1.8a) $\displaystyle h_{MO}(x)$
$\displaystyle=$
$\displaystyle\frac{h(x,\varphi)}{1-\bar{\theta}S(x,\varphi)},$ (1.8b)
$\displaystyle r_{MO}(x)$ $\displaystyle=$ $\displaystyle\frac{\theta
r(x,\varphi)}{1-\bar{\theta}S(x,\varphi)},$ (1.8c) $\displaystyle H_{MO}(x)$
$\displaystyle=$
$\displaystyle-\log\left(S_{MO}(x)\right)=-\log\left(\frac{\theta
S(x,\varphi)}{1-\bar{\theta}S(x,\varphi)}\right),$ (1.8d)

respectively, where $\theta>0$, $\bar{\theta}=1-\theta$.

This paper is organized as follows, we define the cumulative distribution,
probability density and hazard functions of the Marshall-Olkin flexible
Weibull extension distribution (MO-FWED) in Section 2. In Sections 3 and 4, we
introduced the statistical properties include, quantile function, the mode,
skewness and kurtosis, $rth$ moments and moment generating function. The
distribution of the order statistics is expressed in Section 5. The maximum
likelihood estimation of the parameters is determined in Section 6. Real data
sets are analyzed in Section 7 and the results are compared with existing
distributions. The conclusions are introduced in Section 8.

##  2 Marshall-Olkin Flexible Weibull Extension Distribution

In this section, we studied the three parameters Marshall-Olkin flexible
Weibull extension distribution. Substituting from Eqs. (1.2) and (1.3) into
Eq. (1.6), the cumulative distribution function of the Marshall-Olkin flexible
Weibull extension distribution (MO-FWE) is given by

$F(x;\alpha,\beta,\theta)=\frac{1-e^{-e^{\alpha
x-\frac{\beta}{x}}}}{1-(1-\theta)e^{-e^{\alpha
x-\frac{\beta}{x}}}},\;x>0,\;\alpha,\beta,\theta>0.$ (2.1)

Substituting from Eqs. (1.1) and (1.3) in Eq. (1.7), the pdf corresponding to
Eq. (2.1) is given by

$f(x;\alpha,\beta,\theta)=\frac{\theta\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{-e^{\alpha
x-\frac{\beta}{x}}}}{\left[1-(1-\theta)e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]^{2}},\;x>0,\;\alpha,\beta,\theta>0.$ (2.2)

The survival function, hazard rate function, reversed-hazard rate function and
cumulative hazard rate function of $X\sim$ MO-FWED($\alpha,\beta,\theta$) are
given by

$\displaystyle S(x;\alpha,\beta,\theta)$ $\displaystyle=$
$\displaystyle\frac{\theta e^{-e^{\alpha
x-\frac{\beta}{x}}}}{1-(1-\theta)e^{-e^{\alpha x-\frac{\beta}{x}}}},$ (2.3a)
$\displaystyle h(x;\alpha,\beta,\theta)$ $\displaystyle=$
$\displaystyle\frac{\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}}{1-(1-\theta)e^{-e^{\alpha x-\frac{\beta}{x}}}},$ (2.3b)
$\displaystyle r(x;\alpha,\beta,\theta)$ $\displaystyle=$
$\displaystyle\frac{\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{-e^{\alpha x-\frac{\beta}{x}}}}{\left(1-e^{-e^{\alpha
x-\frac{\beta}{x}}}\right)\left[1-(1-\theta)e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]},$ (2.3c) $\displaystyle H(x;\alpha,\beta,\theta)$
$\displaystyle=$ $\displaystyle-\log\left(\frac{\theta e^{-e^{\alpha
x-\frac{\beta}{x}}}}{1-(1-\theta)e^{-e^{\alpha x-\frac{\beta}{x}}}}\right)$
(2.3d)

respectively, $x>0$ and $\alpha,\beta,\theta>0$.

Figures (1–6) display the cdf, pdf, survival function, hazard rate function,
reversed hazard rate function and cumulative hazard rate function of the MO-
FWED($\alpha$, $\beta$, $\theta$) for some parameter values.

Figure 1: The cdf of MO-FWED for different values of parameters.

Figure 2: The pdf of MO-FWED for different values of parameters.

Figure 3: The survival function of MO-FWED for different values of parameters.

Figure 4: The hazard rate function of MO-FWED for different values of
parameters.

Figure 5: The reversed hazard rate function of MO-FWED for different values of
parameters.

Figure 6: The cumulative hazard rate function of MO-FWED for different values
of parameters.

##  3 Statistical Properties

In this section, we study the statistical properties for the MO-FWE
distribution, specially quantile and simulation, median, skewness, kurtosis
and moments.

###  3.1 Quantile and simulation

The quantile $x_{q}$ of the MO-FWE($\alpha,\beta,\theta$) random variable is
given by

$\displaystyle F(x_{q};\alpha,\beta,\theta)$ $\displaystyle=$ $\displaystyle
q,\quad 0<q<1.$ (3.1)

Using the cumulative distribution function of the MO-FWE distribution, from
(2.1) in Eq. (3.1), we have

$\alpha x_{q}^{2}-k(q)x_{q}-\beta=0,$ (3.2)

where

$k(q)=\ln\left[-\ln\left(\frac{1-q}{1-(1-\theta)q}\right)\right].$ (3.3)

So, the simulation of the MO-FWE random variable is straightforward. Let $U$
be a uniform variate on the unit interval $(0,1)$, thus, by means of the
inverse transformation method, we consider the random variable $X$ given by

$X=\frac{k(u)\pm\sqrt{k(u)^{2}+4\alpha\beta}}{2\alpha}.$ (3.4)

Since the median is $50\%$ quantile then by setting $q=0.5$ in Eq. (3.2), the
median $M$ of the MO-FWED can be obtained the median .

###  3.2 The Mode of MO-FWE

In this subsection, we will derive the mode of the MO-
FWED$(\alpha,\beta,\theta)$ by deriving its pdf with respect to $x$ and equal
it to zero thus the mode of the MO-FWED$(\alpha,\beta,\theta)$ can be obtained
as a nonnegative solution of the following nonlinear equation

$\left[1-(1-\theta)e^{-e^{\alpha x-\frac{\beta}{x}}}\right]\left[-2\beta
x+\left(\alpha x^{2}+\beta\right)^{2}\right]-\left(\alpha
x^{2}+\beta\right)^{2}\left[1+(1-\theta)e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]e^{\alpha x-\frac{\beta}{x}}=0.$ (3.5)

From Figure 2, the pdf for MO-FWED has only one peak, It is a unimodal
distribution, so the above equation has only one solution. It is not possible
to get an explicit solution of Eq.3.5 in the general case. Numerical methods
should be used such as bisection or fixed-point method to solve it.

###  3.3 The Skewness and Kurtosis

The analysis of the variability Skewness and Kurtosis on the shape parameters
$\alpha,\beta,\theta$ can be investigated based on quantile measures. The
short comings of the classical Kurtosis measure are well-known. The Bowely’s
skewness based on quartiles is given by, Kenney and Keeping [7],

$S_{k}=\frac{q_{(0.75)}-2q_{(0.5)}+q_{(0.25)}}{q_{(0.75)}-q_{(0.25)}},$ (3.6)

and the Moors Kurtosis based on quantiles, Moors [10],

$K_{u}=\frac{q_{(0.875)}-q_{(0.625)}-q_{(0.375)}+q_{(0.125)}}{q_{(0.75)}-q_{(0.25)}},$
(3.7)

where $q_{(.)}$ represents quantile function.

###  3.4 The Moments

Now in this subsection, we derive the $rth$ moment for MO-FWED. Moments are
important in any statistical analysis, especially in applications. It can be
used to study the most important features and characteristics of a
distribution (e.g. tendency, dispersion, skewness and kurtosis).

######  Theorem 3.1.

If $X$ has MO-FWED $(\alpha,\beta,\theta)$, then the $r$th moments of random
variable $X$, is given by the following

$\mu_{r}^{{}^{\prime}}=\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\sum_{i=0}^{\infty}\frac{(-1)^{i+j}(k+1)^{j+1}\beta^{i}\theta(1-\theta)^{k}}{i!j!(j+1)^{r-2i-1}\alpha^{r-i-1}}\left[\frac{\Gamma(r-i+1)}{\alpha(j+1)^{2}}+\beta\Gamma(r-i-1)\right].$
(3.8)

###### Proof.

The $r$th moment of the random variable $X$ with probability density function
$f(x)$ is given by

$\mu_{r}^{{}^{\prime}}=\int_{0}^{\infty}x^{r}f(x;\alpha,\beta,\theta)dx.$
(3.9)

Substituting from Eq. (2.2) into Eq. (3.9) we get

$\displaystyle\mu_{r}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}x^{r}\theta\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{-e^{\alpha
x-\frac{\beta}{x}}}\left[1-(1-\theta)e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]^{-2}dx.$

Since $0<(1-\theta)e^{-e^{\alpha x-\frac{\beta}{x}}}<1$ for $x>0$ we can use
the binomial series expansion of
$\left[1-(1-\theta)e^{-e^{\alpha x-\frac{\beta}{x}}}\right]^{-2}$ yields

$\left[1-(1-\theta)e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]^{-2}=\sum_{k=0}^{\infty}(k+1)(1-\theta)^{k}e^{-ke^{\alpha
x-\frac{\beta}{x}}},$

then we get

$\mu_{r}^{{}^{\prime}}=\sum_{k=0}^{\infty}(k+1)\theta(1-\theta)^{k}\int_{0}^{\infty}x^{r}\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{-(k+1)e^{\alpha x-\frac{\beta}{x}}}dx,$

using series expansion of $e^{-(k+1)e^{\alpha x-\frac{\beta}{x}}}$,

$e^{-(k+1)e^{\alpha
x-\frac{\beta}{x}}}=\sum_{j=0}^{\infty}\frac{(-1)^{j}(k+1)^{j}}{j!}e^{j(\alpha
x-\frac{\beta}{x})},$

we obtain

$\mu_{r}^{{}^{\prime}}=\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\frac{(-1)^{j}(k+1)^{j+1}\theta(1-\theta)^{k}}{j!}\int_{0}^{\infty}x^{r}\left(\alpha+\beta
x^{-2}\right)e^{(j+1)\alpha x}e^{-(j+1)\frac{\beta}{x}}dx,$

using series expansion of $e^{-(j+1)\frac{\beta}{x}}$,

$e^{-(j+1)\frac{\beta}{x}}=\sum_{i=0}^{\infty}\frac{(-1)^{i}(j+1)^{i}\beta^{i}}{i!}x^{-i},$

we obtain

$\displaystyle\mu_{r}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\sum_{i=0}^{\infty}\frac{(-1)^{i+j}(k+1)^{j+1}(j+1)^{i}\beta^{i}\theta(1-\theta)^{k}}{i!j!}\int_{0}^{\infty}x^{r-i}\left(\alpha+\beta
x^{-2}\right)e^{(j+1)\alpha x}dx,$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\sum_{i=0}^{\infty}\frac{(-1)^{i+j}(k+1)^{j+1}(j+1)^{i}\beta^{i}\theta(1-\theta)^{k}}{i!j!}\times$
$\displaystyle\hskip 71.13188pt\left[\int_{0}^{\infty}\alpha
x^{r-i}e^{(j+1)\alpha x}dx+\int_{0}^{\infty}\beta x^{r-i-2}e^{(j+1)\alpha
x}dx\right],$

by using the definition of gamma function ( Zwillinger [19]), in the form,

$\Gamma(z)=x^{z}\int_{0}^{\infty}t^{z-1}e^{xt}dt,\quad z,x>0.$

Finally, we obtain the $r$th moment of MO-FWE distribution in the form

$\displaystyle\mu_{r}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\sum_{i=0}^{\infty}\frac{(-1)^{i+j}(k+1)^{j+1}(j+1)^{i}\beta^{i}\theta(1-\theta)^{k}}{i!j!}\times$
$\displaystyle\hskip
71.13188pt\left[\frac{\Gamma(r-i+1)}{\alpha^{r-i}(j+1)^{r-i+1}}+\frac{\beta\Gamma(r-i-1)}{\alpha^{r-i-1}(j+1)^{r-i-1}}\right]$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\sum_{i=0}^{\infty}\frac{(-1)^{i+j}(k+1)^{j+1}\beta^{i}\theta(1-\theta)^{k}}{i!j!(j+1)^{r-2i-1}\alpha^{r-i-1}}\left[\frac{\Gamma(r-i+1)}{\alpha(j+1)^{2}}+\beta\Gamma(r-i-1)\right].$

This completes the proof.

∎

##  4 The Moment Generating Function

The moment generating function (mgf), $M_{X}(t)$, of a random variable $X$
provides the basis of an alternative route to analytic results compared with
working directly with the pdf and cdf of $X$.

######  Theorem 4.1.

The moment generating function (mgf) of MO-FWE distribution is given by

$M_{X}(t)=\sum_{r=0}^{\infty}\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\sum_{i=0}^{\infty}\frac{(-1)^{i+j}(k+1)^{j+1}\beta^{i}\theta(1-\theta)^{k}t^{r}}{r!i!j!(j+1)^{r-2i-1}\alpha^{r-i-1}}\left[\frac{\Gamma(r-i+1)}{\alpha(j+1)^{2}}+\beta\Gamma(r-i-1)\right].$
(4.1)

###### Proof.

The moment generating function of the random variable $X$ with probability
density function $f(x)$ is given by

$M_{X}(t)=\int_{0}^{\infty}e^{tx}f(x)dx,$ (4.2)

using series expansion of $e^{tx}$, we obtain

$M_{X}(t)=\sum_{r=0}^{\infty}\frac{t^{r}}{r!}\int_{0}^{\infty}x^{r}f(x)dx=\sum_{r=0}^{\infty}\frac{t^{r}}{r!}\mu_{r}^{{}^{\prime}}.$
(4.3)

Substituting from Eq. (3.8) into Eq. (4.3), we obtain the moment generating
function of MO-FWED in the following form

$M_{X}(t)=\sum_{r=0}^{\infty}\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\sum_{i=0}^{\infty}\frac{(-1)^{i+j}(k+1)^{j+1}\beta^{i}\theta(1-\theta)^{k}t^{r}}{r!i!j!(j+1)^{r-2i-1}\alpha^{r-i-1}}\left[\frac{\Gamma(r-i+1)}{\alpha(j+1)^{2}}+\beta\Gamma(r-i-1)\right].$

This completes the proof. ∎

##  5 Order Statistics

In this section, we derive closed form expressions for the probability density
function of the $r$th order statistic of the MO-FWED. Let
$X_{1:n},X_{2:n},\cdots,X_{n:n}$ denote the order statistics obtained from a
random sample $X_{1}$, $X_{2}$, $\cdots$, $X_{n}$ which taken from a
continuous population with cumulative distribution function $F(x;\varphi)$ and
probability density function $f(x;\varphi)$, then the probability density
function of $X_{r:n}$ is given by

$f_{r:n}(x;\varphi)=\frac{1}{B(r,n-r+1)}\left[F(x;\varphi)\right]^{r-1}\left[1-F(x;\varphi)\right]^{n-r}f(x;\varphi),$
(5.1)

where $f(x;\varphi)$, $F(x;\varphi)$ are the pdf and cdf of MO-FWED
$(\alpha,\beta,\theta)$ given by Eqs. (2.2) and (2.1) respectively,
$\varphi=(\alpha,\beta,\theta)$ and $B(.,.)$ is the beta function, also we
define first order statistics $X_{1:n}=\min(X_{1},X_{2},\cdots,X_{n})$, and
the last order statistics as $X_{n:n}=\max(X_{1},X_{2},\cdots,X_{n})$. Since
$0<F(x;\varphi)<1$ for $x>0$, we can use the binomial expansion of
$[1-F(x;\varphi)]^{n-r}$ given as follows.

$\left[1-F(x;\varphi)\right]^{n-r}=\sum_{i=0}^{n-r}\binom{n-r}{i}(-1)^{i}[F(x;\varphi)]^{i}.$
(5.2)

Substituting from Eq. (5.2) into Eq. (5.1), we obtain

$\displaystyle f_{r:n}(x;\varphi)$ $\displaystyle=$
$\displaystyle\frac{1}{B(r,n-r+1)}f(x;\varphi)\sum_{i=0}^{n-r}\binom{n-r}{i}(-1)^{i}\left[F(x;\varphi)\right]^{i+r-1}$
(5.3) $\displaystyle=$
$\displaystyle\sum_{i=0}^{n-r}\frac{(-1)^{i}n!}{i!(r-1)!(n-r-i)!}\left[F(x,\varphi)\right]^{i+r-1}f(x;\varphi).$

Substituting from Eqs. (2.1) and (3.2) into Eq. (5.3), we obtain pdf of $rth$
order statistics for MO-FWED($\alpha,\beta,\theta$).

Relation (5.3), shows that $f_{r:n}(x;\varphi)$ is the weighted average of the
Marshall Olkin flexible Weibull extension MO-FWED withe different shape
parameters.

##  6 Parameters Estimation

In this section, point and interval estimation of the unknown parameters of
the MO-FWED are derived by using the method of maximum likelihood based on a
complete sample.

###  6.1 Maximum likelihood estimation

Let $x_{1},x_{2},\cdots,x_{n}$ denote a random sample of complete data from
the MO-FWED. The likelihood function is given as

$L=\prod_{i=1}^{n}f(x_{i};\alpha,\beta,\theta),$ (6.1)

substituting from Eq. (2.2) into Eq. (6.1), we have

$L=\prod_{i=1}^{n}\theta\left(\alpha+\frac{\beta}{x_{i}^{2}}\right)e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}e^{-e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\left[1-(1-\theta)e^{-e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\right]^{-2}.$

The log-likelihood function is

$\mathcal{L}=n\ln(\theta)+\sum_{i=1}^{n}\ln\left(\alpha+\frac{\beta}{x_{i}^{2}}\right)+\sum_{i=1}^{n}\left(\alpha
x_{i}-\frac{\beta}{x_{i}}\right)-\sum_{i=1}^{n}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}-2\sum_{i=1}^{n}\ln\left[1-(1-\theta)e^{-e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\right].$ (6.2)

The maximum likelihood estimation of the parameters are obtained by
differentiating the log-likelihood function, $\mathcal{L}$, with respect to
the parameters $\alpha,\beta$ and $\theta$ and setting the result to zero, we
have the following normal equations.

$\displaystyle\frac{\partial\mathcal{L}}{\partial\alpha}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{x_{i}^{2}}{\beta+\alpha
x_{i}^{2}}+\sum_{i=1}^{n}x_{i}-\sum_{i=1}^{n}x_{i}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}+2\sum_{i=1}^{n}\frac{(1-\theta)x_{i}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}{1-\theta-e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}}=0,$ (6.3)
$\displaystyle\frac{\partial\mathcal{L}}{\partial\beta}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{1}{\beta+\alpha
x_{i}^{2}}-\sum_{i=1}^{n}\frac{1}{x_{i}}+\sum_{i=1}^{n}\frac{1}{x_{i}}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}-2\sum_{i=1}^{n}\frac{(1-\theta)e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}{x_{i}\left[1-\theta-e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\right]}=0,$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\theta}$ $\displaystyle=$
$\displaystyle\frac{n}{\theta}+2\sum_{i=1}^{n}\frac{1}{1-\theta-e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}}=0.$ (6.5)

The MLEs can be obtained by solving the nonlinear equations previous,
(6.3)–(6.5), numerically for $\alpha,\beta$ and $\theta$.

###  6.2 Asymptotic confidence bounds

In this section, we derive the asymptotic confidence intervals when
$\alpha,\beta>0$ and $\theta>0$ as the MLEs of the unknown parameters
$\alpha,\beta>0$ and $\theta>0$ can not be obtained in closed forms, by using
variance covariance matrix $I^{-1}$ see Lawless [9], where $I^{-1}$ is the
inverse of the observed information matrix which defined as follows.

$\mathbf{I^{-1}}=\left(\begin{array}[]{ccc}-\frac{\partial^{2}\mathcal{L}}{\partial\alpha^{2}}&-\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial\beta}&-\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial\theta}\\\
-\frac{\partial^{2}\mathcal{L}}{\partial\beta\partial\alpha}&-\frac{\partial^{2}\mathcal{L}}{\partial\beta^{2}}&-\frac{\partial^{2}\mathcal{L}}{\partial\beta\partial\theta}\\\
-\frac{\partial^{2}\mathcal{L}}{\partial\theta\partial\alpha}&-\frac{\partial^{2}\mathcal{L}}{\partial\theta\partial\beta}&-\frac{\partial^{2}\mathcal{L}}{\partial\theta^{2}}\end{array}\right)^{-1}=\left(\begin{array}[]{ccc}var(\hat{\alpha})&cov(\hat{\alpha},\hat{\beta})&cov(\hat{\alpha},\hat{\theta})\\\
cov(\hat{\beta},\hat{\alpha})&var(\hat{\beta})&cov(\hat{\beta},\hat{\theta})\\\
cov(\hat{\theta},\hat{\alpha})&cov(\hat{\theta},\hat{\beta})&var(\hat{\theta})\end{array}\right).$
(6.6)

where

$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\alpha^{2}}$
$\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\frac{x_{i}^{4}}{\left(\beta+\alpha
x_{i}^{2}\right)^{2}}-\sum_{i=1}^{n}x_{i}^{2}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}+2(1-\theta)\sum_{i=1}^{n}x_{i}^{2}\mathcal{B}_{i}$
(6.7)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial\beta}$
$\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\frac{x_{i}^{2}}{\left(\beta+\alpha
x_{i}^{2}\right)^{2}}+\sum_{i=1}^{n}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}-2(1-\theta)\sum_{i=1}^{n}\mathcal{B}_{i}$ (6.8)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial\theta}$
$\displaystyle=$ $\displaystyle
2\sum_{i=1}^{n}x_{i}\mathcal{A}_{i}e^{e^{\alpha x_{i}-\frac{\beta}{x_{i}}}}$
(6.9) $\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\beta^{2}}$
$\displaystyle=$ $\displaystyle-\sum_{i=1}^{n}\frac{1}{\left(\beta+\alpha
x_{i}^{2}\right)^{2}}-\sum_{i=1}^{n}\frac{1}{x_{i}^{2}}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}+2(1-\theta)\sum_{i=1}^{n}\frac{\mathcal{B}_{i}}{x_{i}^{2}}$
(6.10)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\beta\partial\theta}$
$\displaystyle=$
$\displaystyle-2\sum_{i=1}^{n}\frac{1}{x_{i}}\mathcal{A}_{i}e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}$ (6.11)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\theta^{2}}$
$\displaystyle=$
$\displaystyle-\frac{n}{\theta^{2}}-2\sum_{i=1}^{n}\left[1-\theta-e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\right]^{-2}.$ (6.12)

where

$\mathcal{A}_{i}=e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}\left[1-\theta-e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\right]^{-2}\text{ and
}\mathcal{B}_{i}=\mathcal{A}_{i}\left[1-\theta-e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}(1-e^{\alpha x_{i}-\frac{\beta}{x_{i}}})\right].$

We can derive the $(1-\delta)100\%$ confidence intervals of the parameters
$\alpha,\beta$ and $\theta$, by using variance matrix as in the following
forms

$\hat{\alpha}\pm
Z_{\frac{\delta}{2}}\sqrt{var(\hat{\alpha})},\quad\hat{\beta}\pm
Z_{\frac{\delta}{2}}\sqrt{var(\hat{\beta})},\quad\hat{\theta}\pm
Z_{\frac{\delta}{2}}\sqrt{var(\hat{\theta})},$

where $Z_{\frac{\delta}{2}}$ is the upper $(\frac{\delta}{2})$-th percentile
of the standard normal distribution.

##  7 Application

In this section, we present the analysis of two examples for a real data sets
using the MO-FWE $(\alpha,\beta,\theta)$ model and compare it with the other
fitted models like a flexible Weibull extension (FWE), Weibull (W), linear
failure rate (LFR), exponentiated Weibull (EW), generalized linear failure
rate (GLFR), exponentiated flexible Weibull (EFW), modified Weibull (MW),
reduced additive Weibull (RAW) and Extended Weibull (EW) distributions using
Kolmogorov Smirnov (K-S) statistic, as well as Akaike Information Criterion
(AIC), [2], Akaike Information Citerion with correction (AICC), Bayesian
Information Criterion (BIC) and Hannan-Quinn information criterion (HQIC) [16]
values.

######  Example 7.1.

Consider the data have been obtained from Aarset [1], and widely reported in
many literatures. It represents the lifetimes of 50 devices, and also, possess
a bathtub-shaped failure rate property, Table 1.

Table 1: Lifetime of 50 devices, see Aarset [1].

0.1 | 0.2 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 6 |
---|---|---|---|---|---|---|---|---|---|---
7 | 11 | 12 | 18 | 18 | 18 | 18 | 18 | 21 | 32 |
36 | 40 | 45 | 46 | 47 | 50 | 55 | 60 | 63 | 63 |
67 | 67 | 67 | 67 | 72 | 75 | 79 | 82 | 82 | 83 |
84 | 84 | 84 | 85 | 85 | 85 | 85 | 85 | 86 | 86 |

Table 2 gives MLEs of parameters of the MO-FWED and K–S Statistics. The values
of the log-likelihood functions, AIC, AICC, BIC and HQIC are presented in
Table 3.

Table 2: MLEs and K–S of parameters for Aarset data [1].

Model | MLE of the parameters | K-S | P-value
---|---|---|---
Flexible Weibull |  $\hat{\alpha}$ = 0.0122, $\hat{\beta}$ = 0.7002 | 0.4386 |  4.29 $\times 10^{-9}$
Weibull |  $\hat{\alpha}$ = 0.0223, $\hat{\beta}$ = 0.949 | 0.2397 | 0.0052
Linear Failure rate |  $\hat{a}$ = 0.014, $\hat{b}$ = 2.4 $\times 10^{-4}$ | 0.1955 | 0.0370
Exponentiated Weibull |  $\hat{\alpha}$ = 0.0109, $\hat{\beta}$ = 4.69, $\hat{\gamma}$ = 0.164 | 0.1841 | 0.0590
Generalized Linear Failure rate |  $\hat{a}$ = 0.0038, $\hat{b}$ = 3.04 $\times 10^{-4}$, $\hat{c}$ = 0.533 | 0.1620 | 0.1293
Exponentiated Flexible Weibull |  $\hat{\alpha}$ = 0.0147, $\hat{\beta}$ = 0.133, $\hat{\theta}$ = 4.22 | 0.1433 | 0.2617
MO-FWE($\alpha,\beta,\theta$ ) |  $\hat{\alpha}$ = 0.017, $\hat{\beta}$ = 0.401, $\hat{\theta}$ = 9.043 | 0.1269 | 0.3756

Table 3: Log-likelihood, AIC, AICC, BIC and HQIC values of models fitted for
Aarset data [1].

Model | $\mathcal{L}$ |  -2 $\mathcal{L}$ | AIC | AICC | BIC | HQIC
---|---|---|---|---|---|---
FW($\alpha,\beta$) | -250.810 | 501.620 | 505.620 | 505.88 | 509.448 | 507.0762
W($\alpha,\beta$ ) | -241.002 | 482.004 | 486.004 | 486.26 | 489.828 | 487.4602
LFR (a, b) | -238.064 | 476.128 | 480.128 | 480.38 | 483.952 | 481.5842
EW($\alpha,\beta,\gamma$ ) | -235.926 | 471.852 | 477.852 | 478.37 | 483.588 | 480.0363
GLFR(a, b, c) | -233.145 | 466.290 | 472.290 | 472.81 | 478.026 | 474.4743
EFW($\alpha,\beta,\theta$ ) | -226.989 | 453.978 | 459.979 | 460.65 | 465.715 | 462.1623
MO-FWE($\alpha,\beta,\theta$ ) | -223.755 | 447.510 | 453.510 | 454.03 | 459.246 | 455.6943

We find that the MO-FWE distribution with three parameters provides a better
fit than the previous models flexible Weibull (FW), Weibull (W), linear
failure rate (LFR), exponentiated Weibull (EW), generalized linear failure
rate (GLFR) and exponentiated flexible Weibull (EFW). It has the largest
likelihood, and the smallest K-S, AIC, AICC, BIC and HQIC values among those
considered in this paper.

Substituting the MLE’s of the unknown parameters $\alpha,\beta$ and $\theta$
into (6.6), we get estimation of the variance covariance matrix as the
following

$I^{-1}=\left(\begin{array}[]{rrr}1.523\times 10^{-6}&-1.782\times
10^{-5}&2.177\times 10^{-3}\\\ -1.782\times 10^{-5}&0.022&-0.061\\\
2.177\times 10^{-3}&-0.061&8.458\\\ \end{array}\right)$

The approximate 95% two sided confidence intervals of the unknown parameters
$\alpha,\beta$ and $\theta$ are $\left[0.015,0.019\right]$,
$\left[0.108,0.694\right]$ and $\left[3.343,14.743\right]$, respectively.

To show that the likelihood equation have unique solution, we plot the
profiles of the log-likelihood function of $\alpha,\beta$ and $\theta$ in
Figures 7, 8.

Figure 7: The profile of the log-likelihood function of $\alpha,\beta$ for the
Aarset data.

Figure 8: The profile of the log-likelihood function of $\theta$ for the
Aarset data.

The nonparametric estimate of the survival function using the Kaplan-Meier
method and its fitted parametric estimations when the distribution is assumed
to be MO-FWE, FW, W, LFR, EW, GLFR and EFW are computed and plotted in Figure
9.

Figure 9: The Kaplan-Meier estimate of the survival function for the
Aarset(1987) data.

Figures 10 and 11 give the form of the hazard rate and cdf for the MO-FWE, FW,
W, LFR, EW, GLFR and EFW which are used to fit the Aarset(1987) data after
replacing the unknown parameters included in each distribution by their MLE.

Figure 10: The Fitted hazard rate function for the Aarset(1987) data.

Figure 11: The Fitted cumulative distribution function for the Aarset(1987)
data.

######  Example 7.2.

The data have been obtained from [13], it is for the time between failures
(thousands of hours) of secondary reactor pumps, Table 4.

Table 4: Time between failures (thousands of hours) of secondary reactor pumps
[13]

2.160 | 0.746 | 0.402 | 0.954 | 0.491 | 6.560 | 4.992 | 0.347
---|---|---|---|---|---|---|---
0.150 | 0.358 | 0.101 | 1.359 | 3.465 | 1.060 | 0.614 | 1.921
4.082 | 0.199 | 0.605 | 0.273 | 0.070 | 0.062 | 5.320 |

Table 5 gives MLEs of parameters of the MO-FWE distribution and K-S
Statistics. The values of the log-likelihood functions, AIC, AICC, BIC and
HQIC are in Table 6.

Table 5: MLEs and K–S of parameters for secondary reactor pumps.

Model | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\theta}$ | K-S
---|---|---|---|---
Flexible Weibull | 0.0207 | 2.5875 | – | 0.1342
Weibull | 0.8077 | 13.9148 | – | 0.1173
Modified Weibull | 0.1213 | 0.7924 | 0.0009 | 0.1188
Reduced Additive Weibull | 0.0070 | 1.7292 | 0.0452 | 0.1619
Extended Weibull | 0.4189 | 1.0212 | 10.2778 | 0.1057
MO-FWE | 0.2160 | 0.2350 | 1.2960 | 0.0793

Table 6: Log-likelihood, AIC, AICC, BIC and HQIC values of models fitted.

Model | $\mathcal{L}$ |  -2 $\mathcal{L}$ | AIC | AICC | BIC | HQIC
---|---|---|---|---|---|---
Flexible Weibull | -83.3424 | 166.6848 | 170.6848 | 171.2848 | 172.95579 | 171.2559
Weibull | -85.4734 | 170.9468 | 174.9468 | 175.5468 | 177.21779 | 175.5179
Modified Weibull | -85.4677 | 170.9354 | 176.9354 | 178.1986 | 180.34188 | 177.7921
Reduced Additive Weibull | -86.0728 | 172.1456 | 178.1456 | 179.4088 | 181.55208 | 179.0023
Extended Weibull | -86.6343 | 173.2686 | 179.2686 | 180.5318 | 182.67508 | 180.1253
MO-FWE | -30.2110 | 60.4220 | 66.4220 | 67.6852 | 69.8285 | 67.2787

We find that the MO-FWE distribution with the three-number of parameters
provides a better fit than the previous new modified Weibull distributions
like a flexible Weibull (FW), Weibull (W), modified Weibull (MW), reduced
additive Weibull (RAW) and extended Weibull (EW) distributions. It has the
largest likelihood, and the smallest K-S, AIC, AICC, BIC and HQIC values among
those considered in this paper.

Substituting the MLE’s of the unknown parameters $\alpha,\beta$ and $\theta$
into (6.6), we get estimation of the variance covariance matrix as the
following

$I_{0}^{-1}=\left(\begin{array}[]{rrr}1.996\times 10^{-3}&-7.744\times
10^{-4}&8.987\times 10^{-3}\\\ -7.744\times 10^{-4}&5.487\times
10^{-3}&-0.022\\\ 8.987\times 10^{-3}&-0.022&0.326\\\ \end{array}\right)$

The approximate 95% two sided confidence intervals of the unknown parameters
$\alpha,\beta$ and $\theta$ are $\left[0.128,0.304\right]$,
$\left[0.09,0.38\right]$ and $\left[0.177,2.415\right]$, respectively.

To show that the likelihood equation have unique solution, we plot the
profiles of the log-likelihood function of $\alpha,\beta$ and $\theta$ in
Figures 12 and 13.

Figure 12: The profile of the log-likelihood function of $\alpha$ and $\beta$.

Figure 13: The profile of the log-likelihood function of $\theta$.

The nonparametric estimate of the survival function using the Kaplan-Meier
method and its fitted parametric estimations when the distribution is assumed
to be MO-FWE, FW, W, MW, RAW and EW are computed and plotted in Figure 14.

Figure 14: The Kaplan-Meier estimate of the survival function for the data.

Figure 15 gives the form of the CDF for the MO-FWE, FW, W, MW, RAW and EW
which are used to fit the data after replacing the unknown parameters included
in each distribution by their MLE.

Figure 15: The Fitted cumulative distribution function for the data.

##  8 Conclusions

A new distribution MO-FWE, it’s generalized of the flexible Weibull extension
distribution based on the Marshall and Olkin’s method, has been proposed and
its properties are studied. The idea is to add parameter to a flexible Weibull
extension distribution, so that the hazard function is either increasing or
more importantly, bathtub shaped. Using Marshall and Olkin extended family by
adding a new parameter to the baseline distribution, the distribution has
flexibility to model the second peak in a distribution. We have shown that the
Marshall Olkin flexible Weibull extension distribution fits certain well-known
data sets better than existing modifications.

## References

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# Finiteness of non-constant maps over a number field

Ariyan Javanpeykar  Ariyan Javanpeykar
Institut für Mathematik
Johannes Gutenberg-Universität Mainz
Staudingerweg 9, 55099 Mainz
Germany<EMAIL_ADDRESS>
###### Abstract.

Motivated by the intermediate Lang conjectures on hyperbolicity and rational
points, we prove new finiteness results for non-constant morphisms from a
fixed variety to a fixed variety defined over a number field by applying
Faltings’s finiteness results to moduli spaces of maps.

###### Key words and phrases:

hyperbolicity, function fields, rational points, Lang-Vojta conjecture

###### 2010 Mathematics Subject Classification:

14G99 (11G35, 14G05, 32Q45)

##  1\. Introduction

We prove finiteness results motivated by Lang’s intermediate conjectures [EJR,
§2]; these conjectures generalize Lang’s conjectures [Lan86] and relate the
notion of being of general type to the behaviour of rational points. The main
idea of this note is to apply Faltings’s arithmetic finiteness results
(_formerly_ Mordell’s conjecture and the Mordell-Lang conjecture) to moduli
spaces of maps (i.e., Hom-schemes).

We work with the following conventions. If $K$ is a field, a variety over $K$
is a geometrically integral quasi-projective scheme over $K$. Moreover, a
surface over $K$ is a two-dimensional variety and a curve over $K$ is a one-
dimensional variety over $K$. If $V$ is a variety over $K$ and $L/K$ is a
field extension, we let $V_{L}$ denote
$V\times_{\operatorname{Spec}K}\operatorname{Spec}L$. Also, if
$f:V\dashrightarrow X$ is a rational map, we let $f(V)$ denote the image of
the locus of determinacy of $f$.

Recall that a variety $X$ over $\mathbb{C}$ is _Brody hyperbolic_ if every
holomorphic map $\mathbb{C}\to X^{\operatorname{an}}$ is constant, where
$X^{\operatorname{an}}$ denotes the complex-analytic space associated to $X$.
Our first result says, in particular, that a Brody hyperbolic projective
surface admits only finitely many non-constant maps defined over a fixed
number field from a given variety.

######  Theorem 1.1.

Let $X$ be a projective surface over a finitely generated field $K$ of
characteristic zero such that there is an embedding $K\to\mathbb{C}$ with
$X_{\mathbb{C}}$ Brody hyperbolic. Then, for every finitely generated
extension $L/K$ and every variety $V$ over $L$, the set of non-constant
rational maps $f:V\dashrightarrow X_{L}$ over $L$ is finite.

Note that the analogous statement over algebraically closed fields fails, so
that the assumption that $K$ is finitely generated can not be dropped in
Theorem 1.1. For example, if $C$ is a smooth projective curve of genus at
least two over $\mathbb{C}$ and $X=C\times C$, then the set of non-constant
morphisms $C\to X$ is (obviously) infinite.

Since a smooth projective variety over $\mathbb{C}$ with ample cotangent
bundle is Brody hyperbolic, we directly obtain the following finiteness
result.

######  Theorem 1.2.

Let $X$ be a smooth projective surface over a finitely generated field $K$ of
characteristic zero such that $\Omega^{1}_{X}$ is ample. Then, for every
finitely generated extension $L/K$ and every variety $V$ over $L$, the set of
non-constant rational maps $f:V\dashrightarrow X_{L}$ over $L$ is finite.

Our next result establishes a similar finiteness result for certain surfaces
of general type. The surfaces we will consider are neither Brody hyperbolic
nor have ample cotangent bundle, so that we can not appeal to the above
results. On the other hand, the surfaces we consider are ”pseudo-algebraically
hyperbolic” [Jav20, Definition 9.2], and it turns out that this suffices to
prove a similar finiteness result.

######  Theorem 1.3.

Let $X$ be a smooth projective surface of general type with
$c_{1}^{2}(T_{X})>c_{2}(T_{X})$ over a finitely generated field $K$ of
characteristic zero. Then, there is a proper closed subset $Z\subset X$ such
that, for every finitely generated extension $L/K$ and every variety $V$ over
$L$, the set of non-constant rational maps $f:V\dashrightarrow X_{L}$ over $L$
with $f(V)\not\subset Z_{L}$ is finite.

One can not take $Z$ to be the empty set in Theorem 1.3. Indeed, there exists
a smooth projective surface of general type with
$c_{1}^{2}(T_{X})>c_{2}(T_{X})$ which contains rational curves. The subset $Z$
in Theorem 1.3 will forcefully contain these rational curves. In fact, the
subset $Z$ can be taken to be the union of images of non-constant maps $E\to
X_{\overline{K}}$, where $E$ runs over all elliptic curves over
$\overline{K}$.

To prove the above three results we rely on two abstract results (Theorems 1.4
and 1.5 below). These results may be considered as the main technical results
of this note.

Before we state our results, we introduce notation for Hom-schemes and also
take the opportunity to briefly recall some fundamental properties of such
schemes. Namely, for $X$ and $Y$ projective schemes over a field $k$,
Grothendieck proved that the set-valued functor

$\mathrm{Sch}/k^{\textrm{op}}\to\mathrm{Sets}$

which associates to a $k$-scheme $T$ the set $\mathrm{Isom}_{T}(X_{T},Y_{T})$
of isomorphisms $X_{T}\to Y_{T}$ over $T$ is representable by a locally finite
type $k$-scheme $\underline{\operatorname{Hom}}_{k}(X,Y)$; Grothendieck’s
proof uses the existence of Hilbert schemes [Gro62] (see also [Nit05, Theorem
6.6] for a proof). We will refer to $\underline{\operatorname{Hom}}_{k}(X,Y)$
as the _moduli space_ (or: _Hom-scheme_) of morphisms from $X$ to $Y$; note
that the set of $k$-points $\underline{\operatorname{Hom}}_{k}(X,Y)(k)$ of
$\underline{\operatorname{Hom}}_{k}(X,Y)$ is the set of morphisms from $X$ to
$Y$ over $k$. Since the Hom-scheme $\underline{\operatorname{Hom}}_{k}(X,Y)$
is an open subscheme of the Hilbert scheme $\mathrm{Hilb}_{X\times_{k}Y/k}$ of
$X\times_{k}Y$, it is a countable disjoint union of quasi-projective schemes
over $k$ (indexed by Hilbert polynomials). We stress that
$\underline{\operatorname{Hom}}_{k}(X,Y)$ might not be quasi-projective, as it
might have infinitely many connected components (e.g.,
$\underline{\operatorname{Hom}}_{k}(\mathbb{P}^{1}_{k},\mathbb{P}^{1}_{k})$).

We define
$\underline{\operatorname{Hom}}^{c}_{k}(X,Y)\subset\underline{\operatorname{Hom}}_{k}(X,Y)$
to be the subscheme parametrizing the constant maps from $X$ to $Y$. Note that
this subscheme of $\underline{\operatorname{Hom}}_{k}(X,Y)$ is isomorphic to
$Y$. We let $\underline{\operatorname{Hom}}_{k}^{nc}(X,Y)$ be the moduli space
of non-constant morphisms from $X$ to $Y$, and note that this is the
complement of $\underline{\operatorname{Hom}}_{k}^{c}(X,Y)$. Finally, if
$Z\subset Y$ is a closed subscheme, then the natural morphism
$\underline{\operatorname{Hom}}_{k}(X,Z)\to\underline{\operatorname{Hom}}_{k}(X,Y)$
is a closed immersion. The scheme
$\underline{\operatorname{Hom}}_{k}(X,Y)\setminus\underline{\operatorname{Hom}}_{k}(X,Z)$
parametrizes morphisms $f:X\to Y$ such that $f(X)\not\subset Z$.

We can now state the first main technical result.

######  Theorem 1.4.

Let $X$ be a projective surface over an algebraically closed field $k$ of
characteristic zero and let $Z\subsetneq X$ be a proper closed subset.

Assume that, for every curve $C$ over $k$, every $c$ in $C(k)$, and every $x$
in $X(k)\setminus Z(k)$, the set of morphisms $f:C\to X$ with $f(c)=x$ is
finite.

If $C$ is a smooth projective curve over $k$ and $H$ is a positive-dimensional
irreducible (reduced) component of the scheme
$\underline{\operatorname{Hom}}^{nc}_{k}(C,X)\setminus\underline{\operatorname{Hom}}_{k}(C,Z)$,
then $H$ is birational to a smooth projective curve of genus at least two over
$k$.

Note that this result is concerned with the structure of the moduli space of
maps from a curve to a ”pseudo-geometrically hyperbolic” projective surface
[Jav20, Definition 11.2].

The proof of Theorem 1.4 relies on Kobayashi-Ochiai’s finiteness theorem for
dominant maps from a variety to a variety of general type (Theorem 2.1). We
apply it to the (a priori infinitely many) evaluation maps with domain the
moduli space of maps from a curve $C$ to $X$ to control its dimension.

Our second technical result requires a slightly stronger, but conjecturally
equivalent assumption (see (8) and (9) in [Jav20, Conjecture 12.1]), and
provides an arithmetic finiteness conclusion.

######  Theorem 1.5.

Let $X$ be a projective surface over a finitely generated field $K$ of
characteristic zero and let $Z\subsetneq X$ be a proper closed subset.

Assume that there is an embedding $K\to\mathbb{C}$ such that, for every smooth
projective curve $C$ over $\mathbb{C}$, the scheme
$\underline{\operatorname{Hom}}_{\mathbb{C}}(C,X_{\mathbb{C}})\setminus\underline{\operatorname{Hom}}_{\mathbb{C}}(C,Z_{\mathbb{C}})$
is of finite type over $\mathbb{C}$.

Then, for every finitely generated field extension $L/K$ and every variety $V$
over $L$, the set of non-constant rational maps $f:V\dashrightarrow X_{L}$
with $f(V)\not\subset Z_{L}$ is finite.

Using the terminology of [Jav20] and [EJR], Theorem 1.5 implies that, if
$X_{\mathbb{C}}$ is $1$-bounded modulo $Z_{\mathbb{C}}$ over $\mathbb{C}$,
then $X$ is $p$-Mordellic modulo $Z$ for every $p>0$.

Our proof of Theorem 1.5 relies on Faltings’s theorem (formerly Mordell’s
conjecture) and Theorem 1.4. In fact, we apply Faltings’s theorem to the
moduli space of non-constant maps from $V$ to $X$ in the case that $\dim V=1$.
We then use a cutting argument to prove the desired finiteness statement for
all varieties (by induction on the dimension of $V$). Here, it is imperative
that Theorem 1.4 holds over _all_ finitely generated fields of characteristic
zero, and does not involve any restriction to number fields.

We stress that no arithmetic methods are employed in our proofs. The main
observation is simply that using known arithmetic finiteness results (due to
Faltings) and algebro-geometric arguments, one can obtain finiteness results
on non-constant rational maps $V\dashrightarrow X$ defined over a finitely
generated field of characteristic zero.

As already explained above, all of our results (except for Theorem 1.4) rely
on Faltings’s finiteness theorems for curves (i.e., Mordell’s conjecture). By
exploiting Faltings’s finiteness theorem for closed subvarieties of abelian
varieties (i.e., the Mordell-Lang conjecture), we can also obtain new
finiteness results for ramified covers of simple abelian varieties.

More precisely, our final result is concerned with rational points on finite
surjective ramified covers $\pi:X\to A$ of an abelian variety $A$ over a
finitely generated field $K$ of characteristic zero with $X$ a normal variety
over $K$. As explained in the introduction of [CDJ+], Lang’s conjecture
implies that the $K$-rational points $X(K)$ on such a variety $X$ is not dense
in $X$. Moreover, if $A$ is assumed to be simple, then Lang’s conjecture even
predicts that $X(K)$ is finite. Neither of these conjectures are currently
known. (On the positive side, in [CDJ+] it was shown that $X$ has ”fewer”
points than the abelian variety it covers. More precisely: if $A(K)$ is dense,
then $A(K)\setminus\pi(X(K))$ is still dense.)

Our final result provides a bit of evidence for Lang’s prediction for ramified
covers of simple abelian varieties.

######  Theorem 1.6.

Let $A$ be a geometrically simple abelian variety over a finitely generated
field $K$ of characteristic zero, and let $X\to A$ be a finite surjective non-
étale morphism with $X$ a normal variety over $K$. Then, for every finite
extension $L/K$ and every variety $V$ over $L$, the set of non-constant
rational maps $f:V\dashrightarrow X_{L}$ over $L$ is finite.

Note that, in Theorem 1.6, we do not assume the variety $X$ to be two-
dimensional. Nonetheless, the arguments used to prove Theorem 1.6 are similar
to the arguments used to prove Theorems 1.4 and 1.5.

We emphasize that, in our results above, we do not prove finiteness of
”$L$-points on $X$”, but rather finiteness of ”$L$-curves on $X$” (and
$L$-surfaces, $L$-threefolds, etc.). This is, as noted above, quite natural
from the perspective of Lang’s conjectures on intermediate
(pseudo-)hyperbolicity; see [EJR, §2]. It also fits in well with predictions
made by the “Persistence Conjecture” (see Remark 2.6).

######  Acknowledgements.

We thank the referee for several useful comments.

##  2\. Proof of Theorems 1.4 and 1.5

We will use Kobayashi-Ochiai’s finiteness theorem [Kob98, Theorem 7.6.1].

######  Theorem 2.1 (Kobayashi-Ochiai).

Let $k$ be an algebraically closed field of characteristic zero. If $X$ is a
projective variety of general type over $k$ and $Y$ is a projective variety
over $k$, then the set of dominant rational maps $Y\dashrightarrow X$ is
finite.

We now prove our first technical result.

###### Proof of Theorem 1.4.

Our assumption says that $X$ is pseudo-geometrically hyperbolic [Jav20,
Definition 11.2]. In particular, since $\dim X=2$, by combining [JX,
Proposition 5.4] and [JX, Lemma 3.23], it follows that $X$ is of general type.

As in the statement of the theorem, let $H$ be a positive-dimensional
irreducible (reduced) component of the scheme
$\underline{\operatorname{Hom}}^{nc}_{k}(C,X)\setminus\underline{\operatorname{Hom}}_{k}(C,Z)$.
Then, since every connected component of
$\underline{\operatorname{Hom}}_{k}(C,X)$ is quasi-projective, the irreducible
scheme $H$ is quasi-projective over $k$.

For $c$ in $C$, let $\mathrm{ev}_{c}:H\to X$ be the evaluation-at-$c$ map. By
our assumption, this morphism has finite fibres over $X\setminus Z$. Now, let
$c$ be in $C$ be such that $\mathrm{ev}_{c}(H)\not\subset Z$. (Note that such
a point exists. Indeed, otherwise, the image of every $f$ in $H$ would be
contained in $Z$, contradicting the definition of $H$.) In particular, for
this choice of $c$, the morphism $\mathrm{ev}_{c}$ is generically finite onto
its image, so that $\dim H\leq\dim X=2$. We claim that $\dim H=1$. To prove
our claim, we argue by contradiction.

Suppose that $\dim H=2$. Then, for every $c$ in $C$, since $\mathrm{ev}_{c}$
is generically finite onto its image, the morphism $\mathrm{ev}_{c}:H\to X$ is
dominant. Thus, since $X$ is of general type, by Kobayashi-Ochiai’s finiteness
theorem (Theorem 2.1), the set $\\{\mathrm{ev}_{c}:H\to X\ |\ c\in C(k)\\}$ is
finite. This implies that every $f$ in $H$ is constant, contradicting the fact
that $H$ is contained in the moduli space of non-constant maps from $C$ to
$X$. This shows that $\dim H\leq 1$. Since $H$ is positive-dimensional by
assumption, we conclude that $H$ is an irreducible reduced (possibly singular
and non-proper) curve over $k$.

Again, let $c$ in $C(k)$ be such that $\mathrm{ev}_{c}(H)\not\subset Z$. Let
$\overline{H}$ be the smooth projective model of $H$. Then, as $\overline{H}$
is smooth and one-dimensional, we have that $\mathrm{ev}_{c}:H\to X$ induces a
morphism $\mathrm{ev}_{c}^{\prime}:\overline{H}\to X$. Now, since
$\mathrm{ev}_{c}^{\prime}:\overline{H}\to X$ does not map $\overline{H}$ into
$Z$, it follows that $\overline{H}$ has genus at least two. Indeed, every
curve of genus at most one mapping non-trivially to $X$ maps into $Z$ by our
assumption that $X$ is geometrically hyperbolic modulo $Z$. This concludes the
proof. ∎

To prove Theorem 1.5, we require several lemmas.

######  Lemma 2.2 (Pseudo-groupless).

Let $X$ be a projective variety over a field $K$ of characteristic zero, and
let $Z\subset X$ be a proper closed subset. Suppose that, for every finitely
generated field extension $L/K$ and every curve $C$ over $L$, the set of non-
constant morphisms $f:C\to X_{L}$ with $f(C)\not\subset Z_{L}$ is finite.
Then, for every abelian variety $A$ over $\overline{K}$, every non-constant
rational map $A\dashrightarrow X_{\overline{K}}$ factors over
$Z_{\overline{K}}$.

###### Proof.

We argue by contradiction. Replacing $K$ by a finite field extension if
necessary, there is an abelian variety $A$ over $K$, a dense open $U\subset A$
whose complement $A\setminus U$ is of codimension at least two in $A$, and a
non-constant morphism $U\to X$ over $K$. Let $L/K$ be a finitely generated
field extension and let $C\subset U_{L}$ be a smooth curve such that, for
every $n>0$, we have $[n]C\subset U$. (Such a field extension $L$ and curve
$C$ in $U_{L}$ exist since the codimension of $A\setminus U$ is at least two
in $A$.) Then, the morphism $f_{n}:C\to X_{L}$ given by $c\mapsto f(nc)$ is a
well-defined morphism. Note that the sequence of morphisms $f_{n}$ is an
infinite sequence of pairwise distinct non-constant morphisms from $C$ to
$X_{L}$ with $f_{n}(C)\not\subset Z_{L}$. This contradicts our assumption. ∎

######  Lemma 2.3.

Let $X$ be a projective surface over a field $K$ of characteristic zero, and
let $Z\subset X$ be a proper closed subset. Suppose that, for every finitely
generated field extension $L/K$ and every curve $C$ over $L$, the set of non-
constant morphisms $f:C\to X_{L}$ with $f(C)\not\subset Z_{L}$ is finite. Then
the following statements hold.

  1. (1)

The projective variety $X$ is of general type.

  2. (2)

If $V$ is a smooth projective variety over $K$ and $f:V\dashrightarrow X$ is a
non-constant non-dominant rational map whose image does not lie in $Z$, then
$f$ extends (uniquely) to a morphism $V\to X$.

###### Proof.

By Lemma 2.2, for every abelian variety $A$ over $\overline{K}$, every
rational map $A\dashrightarrow X_{\overline{K}}$ factors over
$Z_{\overline{K}}$. Therefore, since $X$ is two-dimensional (by assumption),
it follows that $X$ is of general type (use [JX, Corollary 3.17] and [JX,
Lemma 3.23]). This proves the first statement.

To prove the second statement, let $U\subset V$ be a dense open subset such
that $f$ is regular on $U$. Let $C:=\overline{f(U)}$ be the (Zariski-)closure
of the image of $U$. Note that $C$ is a (geometrically integral) projective
curve over $K$ (since $V\dashrightarrow X$ is non-dominant and non-constant).
Since $f$ does not factor over $Z$, we have that $C\subset X$ does not lie in
$Z$, so that the normalization of $C$ is a smooth projective curve of genus at
least one. (Here we use that every non-constant morphism
$\mathbb{P}^{1}_{\overline{K}}\to X_{\overline{K}}$ factors over
$Z_{\overline{K}}$.) Now, since $C$ does not contain any rational curves, by
[JK, §3], the rational map $V\dashrightarrow C$ extends to a morphism $V\to
C$. This concludes the proof. ∎

######  Lemma 2.4.

Let $X$ be a projective surface over a field $K$ of characteristic zero, and
let $Z\subset X$ be a proper closed subset.

Suppose that, for every finitely generated field extension $L/K$ and every
smooth projective curve $C$ over $L$, the set of non-constant morphisms
$f:C\to X_{L}$ with $f(C)\not\subset Z_{L}$ is finite.

Then, for every finitely generated field extension $M/K$ and every variety $V$
over $M$, the set of non-constant rational maps $g:V\dashrightarrow X_{M}$
with $g(V)\not\subset Z_{M}$ is finite.

###### Proof.

We may assume that $M=K$, and that $V$ is a smooth projective variety over
$K$. To prove the lemma, we argue by induction on $\dim V$.

If $\dim V=1$, let $\overline{V}$ be a smooth projective model for $V$. Then,
every rational map $V\dashrightarrow X$ extends uniquely to a morphism
$\overline{V}\to X$. Therefore, the finiteness statement holds by assumption.

Assume that $\dim V>1$. Suppose that $f_{i}:V\dashrightarrow X$ with
$i\in\mathbb{N}$ are pairwise distinct non-constant rational maps over $K$.
Since $X$ is of general type (by Lemma 2.3.(1)), we can apply the theorem of
Kobayashi-Ochiai (Theorem 2.1) to conclude that all but finitely many of the
morphisms $f_{i}$ are non-dominant. Thus, discarding finitely many of the
$f_{i}$ if necessary, each $f_{i}:V\dashrightarrow X$ is a non-constant non-
dominant rational map whose image does not lie in $Z$. Therefore, by Lemma
2.3.(2), each $f_{i}:V\dashrightarrow X$ extends to a morphism $f_{i}:V\to X$.

Let $K\subset k$ be an uncountable algebraically closed field extension. For
each $i$ and $j$, let $V^{i,j}=\\{P\in X(k)\ |\ f_{i}(P)\neq f_{j}(P)\\}$.
Also, for each $i$, let $\Delta_{i}:=f_{i}^{-1}(Z)$, and let $\Delta_{i}^{0}$
be its complement. Since $k$ is uncountable, there exists a point $P\in X(k)$
in the countable intersection of non-empty open subsets

$(\cap_{i\neq j}V^{i,j})\cap(\cap_{i}\Delta_{i}^{0}).$

Let $\mathcal{H}\subset V_{k}$ be a smooth irreducible ample divisor
containing $P$. There is a finitely generated field extension $L/K$ such that
$P$ and $\mathcal{H}\subset V_{k}$ can be defined over $L$, i.e., the point
$P$ lies in $V(L)$ and there is a smooth irreducible ample divisor $\
{H}\subset V_{L}$ containing $P$. Define $g_{i}:=f_{i}|_{H}:H\to X$ to be the
restriction of $f_{i}$ to $H$. Note that $g_{i}(P)=f_{i}(P)\neq
f_{j}(P)=g_{j}(P)$, so that the morphisms $g_{i}$ are pairwise distinct. Also,
since $P\in\Delta_{i}^{0}$, we have that $g_{i}(H)\not\subset Z_{L}$.

Note that, for each $i$, the morphism $g_{i}:H\dashrightarrow X$ is non-
constant. To prove this, suppose that $g_{i}(H)=\\{p\\}$. Choose $q\neq p$ in
$g_{i}(V)\setminus Z$ and note that the (positive-dimensional) fibre $V_{q}$
does not intersect $H$. This contradicts the ampleness of $H$. Thus, the
sequence $(g_{i}:H\to X_{L})$ is an infinite sequence of pairwise distinct
non-constant morphisms with $g_{i}(H)\not\subset Z_{L}$; this contradicts the
induction hypothesis and concludes the proof. ∎

###### Proof of Theorem 1.5.

By assumption, there is an embedding $K\to\mathbb{C}$ such that, for every
smooth projective curve $C$ over $\mathbb{C}$, the scheme
$\underline{\operatorname{Hom}}_{\mathbb{C}}(C,X_{\mathbb{C}})\setminus\underline{\operatorname{Hom}}_{\mathbb{C}}(C,Z_{\mathbb{C}})$
is of finite type over $\mathbb{C}$.

Note that $Z_{\mathbb{C}}$ contains all rational curves of $X_{\mathbb{C}}$.
Indeed, if there was a rational curve $\mathbb{P}^{1}_{\mathbb{C}}\to
X_{\mathbb{C}}$ not factoring over $Z_{\mathbb{C}}$, then the scheme
$\underline{\operatorname{Hom}}_{\mathbb{C}}(\mathbb{P}^{1}_{\mathbb{C}},X_{\mathbb{C}})\setminus\underline{\operatorname{Hom}}_{\mathbb{C}}(\mathbb{P}^{1}_{\mathbb{C}},Z_{\mathbb{C}})$
would have infinitely many connected components, contradicting the fact that
the latter scheme is of finite type. (The existence of infinitely many
components follows from the existence of self-maps of degree $>1$ on
$\mathbb{P}^{1}_{\mathbb{C}}$.) It follows from a standard specialization
argument (Lefschetz principle) that, for every algebraically closed field $k$
containing $k$, the subset $Z_{k}\subset X_{k}$ contains all rational curves
of $X_{k}$.

A standard specialization argument also shows that our assumption persists
over all algebraically closed fields of characteristic zero (Lefschetz
principle). More precisely, for every algebraically closed field $k$
containing $K$, for every smooth projective curve $C$ over $k$, the scheme
$\underline{\operatorname{Hom}}_{k}(C,Xk)\setminus\underline{\operatorname{Hom}}_{k}(C,Z_{k})$
is of finite type over $k$; we refer the reader to [BJK, §9] for details.

By Mori’s bend-and-break [Deb01, Proposition 3.1], for any algebraically
closed field $k$ containing $K$, for any smooth projective curve $C$ over $k$,
every $c$ in $C(k)$, and every $x$ in $X(k)\setminus Z(k)$, the set of pointed
morphisms $(C,c)\to(X_{k},x)$ is finite. Indeed, if this set were infinite,
then we would have an infinite sequence of pointed maps $(C,c)\to(X_{k},x)$ of
bounded degree (as
$\underline{\operatorname{Hom}}_{k}(C,X_{k})\setminus\underline{\operatorname{Hom}}_{k}(C,Z_{k})$
is of finite type over $k$), so that there is a rational curve passing through
$x$. Since $x\not\in Z_{k}$, this contradicts the fact that $Z_{k}$ contains
all rational curves of $X_{k}$.

By Theorem 1.4, for every algebraically closed field $k$ containing $K$ and
every smooth projective curve $C$ over $k$, we have that every positive-
dimensional irreducible component of

$\underline{\operatorname{Hom}}_{k}^{nc}(C,X_{k}\\})\setminus\underline{\operatorname{Hom}}_{k}(C,Z_{k})$

is birational to a smooth projective curve of genus at least two over $k$.

Let $L/K$ be a finitely generated field extension, let $k$ be an algebraically
closed field containing $L$, and let $C$ be a smooth projective curve over
$L$. To conclude the proof, by Lemma 2.4, it suffices to show that the set of
non-constant morphisms $f:C\to X_{L}$ with $f(C)\not\subset Z_{L}$ is finite.

To do so, note that, since
$\underline{\operatorname{Hom}}_{k}^{nc}(C_{k},X_{k})\setminus\underline{\operatorname{Hom}}_{k}(C_{k},Z_{k})$
is of finite type, the scheme
$\mathcal{M}:=\underline{\operatorname{Hom}}^{nc}_{L}(C,X_{L})\setminus\underline{\operatorname{Hom}}_{k}(C,Z_{L})$
is of finite type over $L$. In particular, since (the reduced closed subscheme
of) every positive-dimensional irreducible component of $\mathcal{M}_{k}$ is
birational to a smooth projective curve of genus at least two, it follows from
Faltings’s theorem [Fal84] that the set $\mathcal{M}(L)$ of non-constant
morphisms $f:C\to X_{L}$ with $f(C)\not\subset Z_{L}$ is finite, as required.
∎

The three applications stated in the introduction are now obtained by
combining Theorem 1.5 with well-known theorems of Bogomolov and Kobayashi.

###### Proof of Theorem 1.1.

Let $K\to\mathbb{C}$ be an embedding such that $X_{\mathbb{C}}$ is Brody
hyperbolic. Then, $X_{\mathbb{C}}$ is $1$-bounded over $\mathbb{C}$; see
[Kob98, Theorem 5.3.9]. Thus, the statement follows from Theorem 1.5 (with
”$Z=\emptyset$”). ∎

###### Proof of Theorem 1.2.

Let $K\to\mathbb{C}$ be an embedding. If $\Omega^{1}_{X}$ is ample, then
$\Omega^{1}_{X_{\mathbb{C}}}$ is ample. In particular, the variety
$X_{\mathbb{C}}$ is Brody hyperbolic [Kob98, Theorem 3.6.21], so that the
statement follows from Theorem 1.1. ∎

###### Proof of Theorem 1.3.

As in the statement of the theorem, let $X$ be a smooth projective surface
over $K$ with $c_{1}^{2}>c_{2}$. Let $K\to\mathbb{C}$ be an embedding. Then,
by Bogomolov’s theorem [Bog77], there is a proper closed subset $Z\subset
X_{\mathbb{C}}$ such that, for every smooth projective curve $C$ over
$\mathbb{C}$, the scheme
$\underline{\operatorname{Hom}}_{\mathbb{C}}(C,X_{\mathbb{C}})\setminus\underline{\operatorname{Hom}}_{\mathbb{C}}(C,Z)$
is of finite type. (Equivalently, the degree of a morphism $f:C\to X$ with
$f(C)\not\subset Z$ is bounded by a constant depending only on $X,Z,C$.)
Therefore, the statement follows from Theorem 1.5. ∎

######  Remark 2.5.

We can apply Theorem 1.5 to any pseudo-algebraically hyperbolic projective
surface. For example, if $X$ is a projective normal surface of general type
which has maximal Albanese dimension, then $X$ is pseudo-algebraically
hyperbolic by [Yam15] (see [JR, Theorem 3.9] for details). One can also apply
Theorem 1.5 to two-dimensional complete subvarieties of certain locally
symmetric varieties.

######  Remark 2.6 (Persistence Conjecture).

Let $X$ be a projective variety over a number field $K$ such that, for every
finite extension $L/K$, the set of $L$-rational points $X(L)$ is finite. Lang
asked [Lan86, p. 202] whether $X(M)$ is finite for every finitely generated
extension $M/K$; this question probably has a positive answer and is
formulated as the “Persistence Conjecture” in [BJK, Conjecture 1.15]. Our
finiteness results in this note are also motivated by the Persistence
Conjecture. To explain this, consider a projective surface $X$ over a number
field $K$ such that $X_{\mathbb{C}}$ is Brody hyperbolic with respect to some
embedding $K\to\mathbb{C}$. Recall that such varieties $X$ over $K$ are
conjectured to have only finitely many $L$-points for each finite extension
$L/K$. In [Jav21, Theorem 1.7] it is shown that, if $X(L)$ is finite for every
number field $L/K$, then $X(M)$ is finite for every finitely generated field
extension $M/K$ (so that Lang’s conjecture has a positive answer in this
case). Although the finiteness of $X(L)$ is currently not known for number
fields $L$, our result (Theorem 1.1) verifies that, for $M/K$ a finitely
generated extension of positive transcendence degree, the subset of ”non-
constant” elements of $X(M)$ is finite, as predicted by the Persistence
Conjecture.

##  3\. Ramified covers

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a
projective variety of general type over $k$. Lang conjectured that $X$ is
pseudo-Mordellic over $k$, i.e., there exists a proper closed subset
$\Delta\subsetneq X$ such that, for every finitely generated subfield
$K\subset k$ and every model $\mathcal{X}$ for $X$ over $K$, the set
$\mathcal{X}(K)\setminus\Delta$ is finite.

Assuming $X$ embeds into an abelian variety, this conjecture (commonly
referred to as the Mordell-Lang conjecture) was proven by Faltings [Fal91].
Faltings proved a more precise result. In fact, if $X$ is a closed subvariety
of an abelian variety $A$ over $k$, we define the special locus
$\mathrm{Sp}(X)$ of $X$ to be the union of the translates of positive-
dimensional abelian subvarieties of $A$ contained in $X$. By work of Kawamata
and Ueno [Kaw80, Uen75], the subset $\mathrm{Sp}(X)$ is a closed subset of $X$
and $X$ is of general type if and only if $\mathrm{Sp}(X)\neq X$. Faltings
proved that $X$ is Mordellic modulo $\mathrm{Sp}(X)$, i.e., for every finitely
generated field $K$ contained in $k$ and every model $\mathcal{X}$ for $X$
over $K$, the set $\mathcal{X}(K)\setminus\mathrm{Sp}(X)$ is finite.

Faltings’s more precise result implies that every proper closed subvariety of
a simple abelian variety is Mordellic (because the special locus of any such
subvariety is obviously empty). We will use this consequence of Faltings’s
work in the following form in our proof of Theorem 1.6.

######  Proposition 3.1 (Faltings).

Let $A$ be a simple abelian variety over an algebraically closed field $k$ of
characteristic zero, and let $H\to A$ be a finite morphism of schemes. If
$\dim H<\dim A$, then $H$ is Mordellic over $k$, i.e., for every finitely
generated subfield $K\subset k$ and every model $\mathcal{X}$ for $X$ over
$K$, the set $\mathcal{X}(K)\ $ is finite.

###### Proof.

Let $H^{\prime}$ be the image of $H\to A$. Note that $\dim H^{\prime}=\dim
H<\dim A$. (In particular, the morphism $H\to A$ is not surjective.) Thus, by
Faltings’s theorem, since $\dim H^{\prime}<\dim A$ and $A$ is simple, the
projective variety $H^{\prime}$ is Mordellic over $k$. Since $H$ is finite
over $H^{\prime}$ and $H^{\prime}$ is Mordellic, we conclude that $H$ itself
is Mordellic. ∎

###### Proof of Theorem 1.6.

Let $k=\overline{K}$ be an algebraic closure of $K$. Since $X_{k}$ is an
integral normal projective variety and $X_{k}\to A_{k}$ is a non-étale finite
surjective morphism, by [Yam15], the projective variety $X_{k}$ is
algebraically hyperbolic over $k$ (as defined by [Dem97], see also [JK]). Now,
let $L/K$ and $V$ be as in the statement of the theorem, and fix an embedding
$L\to k$. To prove the desired finiteness statement, we may and do assume that
$V$ is a smooth projective variety over $L$. Since $X_{k}$ has no rational
curves, every rational map $V\dashrightarrow X_{L}$ extends (uniquely) to a
morphism $V\to X_{L}$; see [JK, §3]. By [JK, Theorem 1.9], the scheme
$H:=\underline{\operatorname{Hom}}_{k}^{nc}(V_{k},X_{k})$ parametrizing non-
constant morphisms from $V_{k}$ to $X_{k}$ is a projective algebraically
hyperbolic scheme of dimension at most $\dim X-1=\dim A-1$. Also, given $v$ in
$V$, the evaluation morphism $\mathrm{ev}_{v}:H\to X$ has finite fibres [JK,
Theorem 1.8], so that (by projectivity), the evaluation morphism
$\mathrm{ev}_{v}$ is a finite morphism (see, for example, [Sta15, Tag 01WG]).
In particular, the composed morphism $H\to X\to A$ is finite, so that $H$ is a
Mordellic projective scheme over $k$ by Faltings’s theorem (Proposition 3.1).
This implies the desired finiteness statement. ∎

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