# Scale matters: The daily, weekly and monthly volatility and predictability
of Bitcoin, Gold, and the S&P 500

Nassim Dehouche

## Abstract

A reputation of high volatility accompanies the emergence of Bitcoin as a
financial asset. This paper intends to nuance this reputation and clarify our
understanding of Bitcoin’s volatility. Indeed, two distinct and non-redundant
understanding of volatility as deviation from consistency exist for a time-
series: (1) exhibiting high standard deviation, and (2) appearing highly
irregular or unpredictable. Using daily, weekly, and monthly closing prices
and log-returns data going from September 2014 to January 2021, we find that
Bitcoin is a prime example of an asset for which the two conceptions of
volatility diverge. We show that, historically, Bitcoin allies both high
volatility (high Standard Deviation) and high predictability (low Approximate
Entropy), relative to Gold and S&P 500.

Moreover, using tools from Extreme Value Theory, we analyze the convergence of
moments, and the mean excess functions of both the closing prices and the log-
returns of the three assets. We find that the closing price of Bitcoin is
consistent with a generalized Pareto distribution, when the closing prices of
the two other assets (Gold and S&P 500) present thin-tailed distributions.
However, returns for all three assets are heavy tailed and second moments
(variance, standard deviation) non-convergent. In the case of Bitcoin, lower
sampling frequencies (monthly vs weekly, weekly vs daily) drastically reduce
the Kurtosis of log-returns and increase the convergence of empirical moments
to their true value. The opposite effect is observed for Gold and S&P 500;
tails become progressively heavier and sample standard deviations higher and
less convergent, the lower the sampling frequency. Thus, the weekly log-
returns of Bitcoin present both a lower Kurtosis, Approximate Entropy,
Coefficient of Variation (Standard Deviation to Mean ratio), and faster
convergence of moments than Gold and S&P 500. These properties suggest that
Bitcoin’s volatility is essentially an intra-day and intra-week phenomenon
that is strongly attenuated on a weekly time-scale, and make it an attractive
store of value to investors and speculators, but its high standard deviation
excludes its use a currency.

##  1 Introduction and related work

Owing to its ability to act as an inflation-resistant store of value and a
decentralized medium of exchange, Bitcoin has recently reached a market
capitalization of $1 trillion [1] and cemented its emergence as a distinct
financial asset [2]. This exponential increase in value has however come with
a reputation of persistent volatility reflected in indicators based on the
standard deviation of Bitcoin’s daily price compared to traditional financial
assets (e.g. Gold, S&P 500).

For both Finance academics [13, 14] and practitioners[15], common measures of
price volatility are based on the standard deviation (e.g. in GARCH models
[7]) of the price or returns of an asset relative to its simple moving
average.

Due to the extreme in its price, Bitcoin is heterologically used as digital
asset rather than a currency. Using a GARCH model, [7] found the
cryptocurrency to possess hedging capabilities that sets the cryptocurrency in
between Gold and the US dollar. Katsiampa [9] compared different GARCH models
for the estimation of the volatility of Bitcoin’s daily price. Bariviera et
al. [8] studied the intra-day (sampling frequency: 5 hours) volatility of
Bitcoin, relative to Gold and the Euro and the volatility of Bitcoin was found
to be decreasing over time, and its long range memory is not related to market
liquidity. Moreover, this behavior across different time-scales of 5 to 12
hours was found to be essentially similar, but no longer time-scales were
considered. These properties limit its use as currency and Bitcoin is
typically regarded as a (digital) commodity. It is classified as such by the
Commodity Futures Trading Commission (CFTC) [3] and Shahzad et al. [4] find
that it shares ”safe-haven” properties with gold, and other commodities
against fluctuations in global stock market indices. The fact that the extreme
movement upward in the price of the cryptocurrency (e.g. its closing price has
increased by 14,414% between 15-09-2014 and 21-02-2021) exclude its use as
such, a currency. Our intent is to distinguish these variations from the
common parlance and finance notions of volatility.

In this paper, we argue that Bitcoin’s price volatility measured with standard
deviation-based indicators has received disproportionate attention when it is
only a partial reflection of volatility. Indeed standard deviation measures a
time series’ dispersion from its average, regardless of the predictability of
its deviations. Moreover, the variations in Bitcoin’s price are typically
studied intra-day and intra-week (Cf. the ”Price Volatility” category of
literature in the exhaustive review of Bitcoin research of Corbet et al. [5])
and its properties for these sampling frequencies are assumed to be be
inherited by weekly and monthly price variations. Using Approximate Entropy
(ApEn) [6] we propose a complementary, non-redundant perspective on Bitcoin’s
volatility, relative to two other financial assets, Gold and the S&P 500
Index. Moreover, we study both closing prices and returns for different
sampling frequencies (daily, weekly, and monthly) and find drastic changes in
the statistical properties and reduction in the volatility of Bitcoin for
weekly data, when more traditional assets show the opposite dynamic for longer
time-preferences.

The remainder of this paper is organized as follows. Section 2 discusses some
common limitations of standard deviation as measure of volatility, Section 3
describes the data and methods used in this study, and Section 4 presents
descriptive statistics for the closing price and returns of the three
considered assets. The main contributions of this paper are presented in
Section 5 and consist in an analysis of the tail properties and Approximate
Entropy of the aforementioned random variables. Finally, Section 6 concludes
this paper with a discussion of the implications and limitations of these
findings and their potential value to speculators and traders considering
Bitcoin as an investment.

##  2 Some limitations of standard deviation as a measure of volatility

###  2.1 It only measures an extent of variation, not regularity

The Cambridge definition of the word volatile is “likely to change suddenly
and unexpectedly, especially by getting worse”. According to [30], two
distinct and non-redundant forms of deviation from consistency exist for a
time-series: (1) exhibiting high standard deviation, and (2) appearing highly
irregular or unpredictable, which corresponds to the common understanding of
the term. This distinction has important consequences. Standard deviation only
measures an extent of deviation from the mean. It does not reflect the
suddenness and unexpectedness of these deviation.

Moreover, standard deviation amalgamates positive and negative deviations
when, in common parlance, volatility is understood as deviations for the
worst. Information theoretic measures, such as Approximate Entropy (ApEn) [6]
offer a way to complement these standard deviation by quantifying regularity
in a time-series. Additionally, significant increases in the Approximate
Entropy of a time-series have been found [30] to foretell major variations in
a time-series.

Let us consider the following toy example of two time-series, of mean $0$,
consisting of two different permutations of the same set of values in
$\\{-1,0,+1\\}$ :

$X=(0,+1,0,-1,0,+1,0,-1,0,+1,0,-1)$ $Y=(-1,+1,0,0,0,+1,0,-1,0,-1,1,0)$

$X$ is a perfectly predictable time-series regularly alternating from $-1$ to
$0$ to $+1$. $Y$ is a random permutation of $X$ and is, by all means, more
volatile. However, the two time series series show the same standard deviation
of $0.725$. The regularity of $X$ is, on the other hand, captured by its lower
ApEn of $-0.001$, when $Y$ presents an ApEn of $0.471$. Despite its perfect
regularity, the standard deviation of $X$ could be made arbitrarily larger
than that of $Y$ by replacing $+1$ in the time-series with any large number.
One can also very easily generate completely random $\\{-1,0,1\\}$ time-series
of an arbitrarily lower standard deviation than $X$.

###  2.2 It may be impossible to empirically estimate

For many assets, the distribution of prices and returns are known to exhibit
heavy-tailed behavior [16] that is best modeled by a generalized Pareto
distribution [17], in which empirical moments are dominated by extreme values.
This has been shown to be the case for the log-returns of the S&P 500 index
[21], Gold [22], and Bitcoin [23].

Consequently, the second moments of the corresponding random variable may not
converge and measures of volatility based on sample standard deviation may be
uninformative. Because it only quantifies the total “amount” of variation from
the average, sample standard deviation is also highly sensitive to extreme
variations. The very use of the term “standard” deviation assumes that there
is such a thing as a representative deviation, which requires these deviations
to be somehow regular. A time-series such as $(0,1,0,-1,0,1,-1,0,10000)$ would
exhibit an empirical standard deviation of $3333.33$, but this number can
decently not be considered a good estimator of the second moment of the
underlying random variable.

##  3 Data and Methods

###  3.1 Data

We use weekly and monthly Bitcoin and gold price data from 17 September 2014
(the earliest listing of Bitcoin on Yahoo Finance) to 16 January 2021 from
Yahoo Finance [10, 11, 12], which represent a total of 6942 daily, 984 weekly,
and 231 monthly observations. Log-returns are calculated by taking the natural
logarithm of the ratio of two consecutive prices.

Concerning daily price and returns variations, it should be noted that Bitcoin
trades 7 days per week, as opposed to Gold and S&P 500 only trading on
weekdays. We consider that there is no price movement during the weekend for
the two latter assets, thus somehow artificially reducing their
volatility/irregularity relative to Bitcoin, which further reinforces our
conclusions.

###  3.2 Methods

####  3.2.1 Mean Excess Functions

For a non-negative random variable $X$, with support in $D(X)$, the excess
distribution over a threshold $a\in D(X)$ is defined [18, 19, 20] as follows:

$F_{a}(x)=P(X-a\leq x|X>a),a\in D(X)$

Intuitively, its complement $1-F_{a}(x)$ measures the likelihood of $X$
exceeding $a+x$, given that $X$ has exceeded $a$. For instance, if $X$
measures a closing price, $1-F_{a}(x)$ is the likelihood of the price gaining
$x$ more units, given that it has reached $a$ monetary units so far. The Mean
Excess ($ME$) function, also known as the Mean Residual Life function, is the
expectation of this distribution for random variables of finite expectations
and is defined as

$ME(a)=E(X-a|X>a)=\frac{\int_{a}^{+\infty}(x-a)\,dF(x)}{\int_{a}^{+\infty}dF(x)},a\in
D(X)$

Thus, the empirical ME function $ME_{n}(a)$ of a sample of $n$ observations
$X_{1},X_{2},\dots,X_{n}$ can be computed by dividing the total amount of
excess over a threshold $a\in D(X)$ by the number of observations realizing
such excesses, as follow:

$ME_{n}(a)=\frac{\sum\limits_{i=1}^{n}(X_{i}-a)\cdot\mathds{1}_{(X_{i}>a)}}{\sum\limits_{i=1}^{n}\mathds{1}_{(X_{i}>a)}}$

The excess distribution and mean are the foundations for peaks over threshold
(POT) modeling [19] which fits distributions to data on excesses and has wide
applications notably in risk management, actuarial science, and project
management. Moreover, they define three classes of random variables whose mean
excess functions exhibits crucially different statistical behaviors:

  * •

A decreasing mean excess function is characteristic of thin-tailed random
variables with memory. Gaussian or Poisson random variables possess this
property.

  * •

A constant mean excess function is characteristic of memorylessness [26].
Exponential random variables and their discrete analogues, Geometric random
variables, notoriously exhibit this property.

  * •

An increasing mean excess function is characteristic of scalable heavy-tailed
random variables, among which linear ME functions characterize the Generalized
Pareto Distribution class [27], whereas convex ME functions are indicative of
log-normality [24]. Both classes of random variables can exhibit infinite or
slowly converging moments.

Chaotic perturbations are commonly observed at the extremity of plots of
empirical ME functions, as a result of finite sample bias, i.e. the fact that
points for very high order statistics in the plot are the result of very few
observations. This bias is commonly addressed by discarding points in the plot
for very high order statistics [28].

####  3.2.2 Maximum to sum ratios

For an order $p\in\\{1,2,3,4,\dots\\}$, the convergence of the ratio of the
maximum to the sum of exponent $p$ is indicative of the existence of the
moment of order $p$, and if so of the speed of convergence of the empirical
moments of order $p$ to its true value, per the Law of Large numbers.
Formally, given a sample of $n$ observations $\\{X_{1},\dots,X_{n}\\}$ of a
positive random variable $X$, let $M(n,p)=Max\\{X_{1}^{p},\dots,X_{n}^{p}\\}$
be the maximum of order $p$ and $S(n,p)=\sum\limits_{i=1}^{n}X_{i}^{p}$, the
sum of order $p$. We have the following result [27, 24, 25]:

$E(X^{p})<+\infty\Leftrightarrow\lim_{n\to+\infty}\frac{M(n,p)}{S(n,p)}=0$

Based on the previous equivalence, plots of the Maximum to Sum ratios
represent the ratio of the maximum to sum of order $p$ as a function of the
number of data points for different values of $p$ and indicate a convergence
of the moments of order $p$ to a finite value if and only if the ratio
converges to zero. The non-convergence of moments, or their very slow
convergence (which requires extremely large sample sizes), renders the
estimation of statistics such as the Mean, variance/standard deviation, or
Kurtosis (respectively, the first, second, and fourth moment) from empirical
observations uninformative.

####  3.2.3 Approximate entropy

Approximate Entropy (ApEn), introduced by Pincus [6, 29, 30], is a family of
information-theoretic statistics quantifying the ”extent of randomness” in a
continuous-state processes, given a time-series of $N$ equally-spaced in time
observations $u(1),u(2),\dots,u(N)$, and two parameters $m$, a positive
integer representing the length of successive observations to be compared, and
$r$, a positive real representing a tolerance level. Widely validated and
commonly used parameter values for the computation of $ApEn(m,r,N)$ are $m=2$
and $r=20\%$ of the Standard Deviation of the considered time-series [30].
Approximate Entropy assigns a non-negative number $ApEn(m,r,N)$ to the time-
series, with larger values corresponding to greater randomness or irregularity
and smaller values corresponding to more instances of repetitive patterns of
variation.

Formally, $ApEn(m,r,N)$ is computed using the following algorithm:

  1. 1.

Compute a sequence of real $m$-dimensional vectors $x(1),x(2),\dots
x(N-m+1)\in\mathbb{R}^{m}$ such that $x(i)=[u(i),u(i+1),\dots,u(i+m-1)]$.

  2. 2.

For each $i\in\\{1,\dots,N-m+l\\}$, compute $C_{i}^{m}(r)=$ number of
$x(j):j\in\\{1,\dots,N-m+l\\}$ such that $d(x(i),x(j))\leq\frac{r}{N-m+1}$,
where $d(x(i),x(j))$ is a distance metric between vectors $x(i)$ and $x(j)$
given by: $d(x(i),x(j))=\max\limits_{1\leq k\leq m}(|u(i+k-1)-u(j+k-1)|)$.

  3. 3.

Compute
$\Phi^{m}(r)={(N-m+1)}^{-1}\cdot{\sum\limits_{i=1}^{N-m+1}log\>C_{i}^{m}(r)}$.

  4. 4.

Compute $ApEn(m,r,N)=\Phi^{m}(r)-\Phi^{m+1}(r).$

Thus, $ApEn(m,r,N)$ measures the the logarithmic empirical likelihood that
observations that are close (within $r$) for $m$ successive observations
remain close (within the same tolerance $r$) on the next incremental
comparison [29, 31].

The presence of repetitive patterns of fluctuation in a time-series makes it
more predictable than a time series in which such patterns are absent. ApEn
reflects the likelihood that similar patterns of observations (within an
average distance of $r$) will not be followed by additional similar
observations. A low ApEn reflect a high frequency of observing previously
observed repetitive patterns and thus a more predictable process. An important
property of ApEn is that its calculation is model-independent [30]. It is able
to quantify the regularity/predictability of time-series data without making
any assumptions concerning the distribution of values and the existence of
moments. This property is particularly useful in Finance where for many assets
and market indices, the development of models that are able to produce
accurate forecasts of future returns or price movements, especially sudden
considerable variations, is typically very difficult.

Moreover, ApEn has been found to be a useful marker of system stability, with
rapid increases foreshadowing significant changes in a time-series [30].

Even if we cannot construct a relatively accurate model of the data, we can
still quantify the irregularity of data, and changes thereto,
straightforwardly. Of course, subsequent modeling remains of interest,
although the point is that this task is quite distinct from the application of
effective discriminatory tools. This perspective seems especially important
given the empirical, non-experimental nature of financial time series.

Representative Applications. Because most financial analyses and modeling
center on price increments or returns, rather than on prices, we do so below.
Given a series of prices si, we consider the prominent in evaluating “random
walk”-type hypotheses. We apply these series to a variety of assets and
indices to illustrate a breadth of this application mode. We note that in
general theoretical and empirical settings (approximate mean stationarity),
ApEn values of these series are quite similar (e.g., Fig. 1). Nonetheless,
because ApEn can discern shifts in serial characteristics, apart from the
consideration of randomness hypotheses, we expect that application of ApEn to
price series si directly will prove useful in clarifying additional changes.
Indeed, most empirical applications of ApEn have been to raw time series.

##  4 Descriptive Statistics

Tables 1 and 2 respectively present descriptive statistics For the
distribution of closing prices and log-returns. Though the closing price of
Bitcoin presents orders of magnitude higher standard deviation and coefficient
of variation than that of Gold and the S&P 500, it is significantly more
predictable and presents a lower approximate entropy. Concerning log-returns,
the three assets present a roughly similar approximate entropy.

| Daily | Weekly | Monthly
---|---|---|---
Statistic | Bitcoin | Gold | S&P 500 | Bitcoin | Gold | S&P 500 | Bitcoin | Gold | S&P 500
№ of observations | 2314 | 2314 | 2314 | 328 | 328 | 328 | 77 | 77 | 77
Mean | 5149.55 | 1346.72 | 2561.64 | 4915.74 | 1569.77 | 2563.51 | 5861.00 | 1588.53 | 2595.77
Standard deviation | 5468.10 | 218.44 | 467.27 | 4802.57 | 518.07 | 469.74 | 7111.74 | 532.30 | 494.46
coefficient of variation | 1.061 | 0.162 | 0.182 | 0.976 | 0.330 | 0.183 | 1.213 | 0.335 | 0.190
Approximate entropy | 0.095 | 0.139 | 0.137 | 0.313 | 0.514 | 0.315 | 0.370 | 0.672 | 0.417
Kurtosis | 6.961 | 0.256 | -0.656 | 1.098 | 0.366 | -0.647 | 1.671 | 3.184 | 10.234

Table 1: Descriptive statistics for closing prices

| Daily | Weekly | Monthly
---|---|---|---
Statistic | Bitcoin | Gold | S&P 500 | Bitcoin | Gold | S&P 500 | Bitcoin | Gold | S&P 500
№ of observations | 2313 | 2313 | 2313 | 327 | 327 | 327 | 76 | 76 | 76
Mean | 0.00190 | 0.00016 | 0.00027 | 0.01276 | 0.00099 | 0.00189 | 0.05771 | 0.00879 | 0.00821
Standard deviation | 0.03896 | 0.00901 | 0.00968 | 0.10386 | 0.05754 | 0.02416 | 0.20962 | 0.13171 | 0.04194
coefficient of variation | 20.498 | 53.0248 | 35.426 | 8.135 | 57.973 | 12.73147929 | 3.631 | 14.971 | 5.105
Approximate entropy | 1.616 | 1.610 | 1.484 | 1.084 | 1.178 | 1.138 | 0.474 | 0.4972 | 0.470
Kurtosis | 13.023 | 3.388 | 20.787 | 1.671 | 3.184 | 10.234 | -0.243 | 1.525 | 1.259

Table 2: Descriptive statistics for log-returns

In Figure 1 We observe that Bitcoin’s closing price presents a long right-
tail, when the closing prices of Gold and S&P 500 are thin-tailed. However,
for the three assets, the distributions of log-returns present both left and
right long tails. Bitcoin standing out with a multi-modal distribution, when
Gold and S&P 500 are both strongly unimodal with mode at zero.

Moreover, Histograms for daily log-returns in Figure 1(b), weekly log-returns
in Figure 2(b), and monthly log-returns in Figure 3(b) show an important
dynamic, for different sampling frequencies. Bitcoin’s log-returns get
progressively more thin-tailed, the longer the time-preference.

(a) Histograms of daily closing prices

(b) Histograms of daily log-returns

Figure 1:  Histograms for daily data

(a) Histograms of weekly closing prices

(b) Histograms of weekly log-returns

Figure 2:  Histograms for weekly data

(a) Histograms of monthly closing prices

(b) Histograms of monthly log-returns

Figure 3:  Histograms for monthly data

This graphical observation is confirmed by the statistics in Table 2. There is
a drastic decrease in the kurtosis of both Bitcoin’s closing price and
returns, between daily and weekly observations. For the distribution of weekly
log-returns, Bitcoin remarkably presents a significantly lower Kurtosis than
Gold and S&P 500, in addition to a lower ApEn and coefficient of variation.
Thus, over the complete time-horizon of this study, Bitcoin is less volatile
than Gold and S&P 500 for a weekly time-preference. We will confirm these
results using moving coefficients of variation and ApEn in Section 5.3, and
further study the tail properties and convergence of moments of the three
assets in Section 5.1 and Section 5.2.

##  5 Results and discussion

For different sampling frequencies (daily, weekly, and monthly), corresponding
to a progressively longer time-preference, we have studied the tail properties
of the closing prices and log-returns of Bitcoin, Gold, and the S&P 500 index,
using two widely used graphical tools from the extreme value analysis; Mean
Excess Function plots, to classify the tail behavior of distributions, and
Maximum to Sum Ratio plots to study the existence of moments. Moreover, we use
the coefficient of variation (standard deviation to mean ratio) and
approximate entropy to study the volatility and predictability of closing
prices and log-returns.

###  5.1 Tail distributions of closing prices and log-returns

Consistently with the histograms in Figure 1, for daily data, the closing
price of Bitcoin presents an increasing empirical ME function represented in
Figure , past the $\$10,000$ threshold. This indicates that, conditional of
having passed that value, the price of Bitcoin is likely to further increase.
The daily closing price of Gold and S&P 500 both exhibit decreasing ME
functions consistent with thin-tailed distributions. However, the latter
figure shows that the absolute log-returns of all three assets have increasing
ME functions consistent with heavy-tailed distributions. Interestingly, the
absolute log-returns of S&P 500 show an ME function that is even more linear
than that of Bitcoin, which is consistent with generalized Pareto behavior.

(a) Mean excess plots of daily closing prices

(b) Mean excess plots of daily absolute log-returns

Figure 4: Mean excess plots for daily data

(a) Mean excess plots of weekly closing prices

(b) Mean excess plots of weekly absolute log-returns

Figure 5: Mean excess plots for weekly data

The ME functions for weekly distributions presented in Figure 5 generally
present the same behavior as daily, with a notable exception; the ME function
of Bitcoin’s weekly absolute return becomes mainly decreasing for this
tsampling frequency, when that of S&P becomes sharply convex, indicating even
more severely heavy-tailed behavior and more disproportionate influence of
extrema on moments.

(a) Mean excess plots of monthly closing prices

(b) Mean excess plots of monthly absolute log-returns

Figure 6: Mean excess plots for monthly data

Although there only exist 76 observations of monthly closing prices and
returns since September 2014, the ME functions of monthly data represented in
Figure 6 see an accentuation of the previous observations. The distribution of
Bitcoin’s monthly absolute log-return shows thin-tailed behavior, Gold an ME
function that is consistent with an Exponential distribution and S&P 500, an
increasing ME function.

Thus, for the three sampling frequencues, Bitcoin is the only asset that shows
heavy-tailed closing prices. However, the log-returns of Bitcoin get
progressively more ”well-behaved” and thin-tailed for a longer time-
preference, when the opposite effect is observed for S&P 500’s, with returns
becoming more extremely heavy-tailed. These findings raise the question of the
finiteness of moments, particularly of second moments which are commonly used
as indicators of volatility, and of the convergence of their empirical
estimators. We address this question in the next section.

###  5.2 The convergence of empirical moments

Figure 7 presents the maximum to sum ratios of daily closing prices and
absolute log-returns. As noted in the previous section, the price of Bitcoin
shows heavy-tailed behavior and thus slower convergence of moments compared to
the other two assets. Moreover, the absolute log-returns of both Bitcoin and
S&P 500 are consistent with generalized Pareto behavior and except for the
mean, their higher moments are dominated by extrema and do not converge.
Notably, the standard deviations of the log-returns of both assets cannot be
reliably estimated from samples.

(a) Maximum to sum plots of daily closing prices

(b) Maximum to sum plots of daily absolute log-returns

Figure 7: Maximum to sum plots for daily data

Maximum to sum ratios for weekly data in Figure 8 and monthly data in Figure 9
also confirm and precise our observations based on ME functions. A lower
sampling frequency make the log-returns of Bitcoin more thin-tailed and
empirical moments convergent to their theoretical value. The variance of
absolute log-returns of Bitcoin converges even faster than Gold’s, for weekly
and monthly data, and exhibits a behavior consistent with a log-normal random
variable with finite variance for these two sampling frequencies. However, the
absolute log-returns of Gold and S&P 500 adopt a progressively more Paretian
behavior for longer time-preferences, and empirical second moments are non-
convergent for weekly and monthly data.

(a) Maximum to sum plots of weekly closing prices

(b) Maximum to sum plots of weekly absolute log-returns

Figure 8: Maximum to sum plots for weekly data

(a) Maximum to sum plots of monthly closing prices

(b) Maximum to sum plots of monthly absolute log-returns

Figure 9: Maximum to sum plots for monthly data

These results inform and nuance our analysis of volatility, in the next
section. Non-convergent second moments for the returns of Gold and S&P 500
mean that the any empirical computation of standard deviation from samples
would be dominated by maxima and should be taken as an under-estimate of the
real standard deviation of the underlying random variable.

###  5.3 Volatility

We have computed 100-day, 20-week, and 3-month rolling standard deviations of
the closing prices and log-returns for the three assets. Results are
respectively presented in Figure 10 and Figure 11. Unsurprisingly, the closing
price of Bitcoin shows the highest moving standard deviation among the three
assets in Figure 10. However, we observe a significant increase in the moving
standard deviation of Gold for lower sampling frequencies when Bitcoin’s and
S&P 500’s remain relatively stable.

(a) 100-day moving coefficient of variation of daily closing prices

(b) 20-week moving coefficient of variation of weekly closing prices

(c) 3-month moving coefficient of variation of monthly closing prices

Figure 10: Moving coefficients of variation of closing prices. For lower
sampling frequencies, sample standard deviation gets progressively higher for
Gold, when it remains relatively stable for Bitcoin and S&P 500.

The same increase is observed concerning the standard deviations of log-
returns, with S&P 500 taking the stead of Gold, while Bitcoin and Gold
maintain stable standard deviation for lower sampling frequencies. Remarkably,
the monthly log-returns of S&P 500 show an even higher rolling standard
deviation than Bitcoin’s, for several months. Furthermore, this high
volatility of monthly S&P 500 log-returns is likely under-estimate, due to the
slower convergence of this random variables empirical second moments (relative
to Bitcoin’s and Gold), discussed in Section 5.2 and summarized in Figure
9(b).

(a) 100-day moving coefficient of variation of daily log-returns

(b) 20-week moving coefficient of variation of weekly log-returns

(c) 3-month moving coefficient of variation of monthly log-returns

Figure 11: Moving coefficient of variation of log-returns. For lower sampling
frequencies, sample standard deviation gets progressively higher for S&P 500,
even exceeding Bitcoin’s for monthly log-returns, when it remains stable for
Bitcoin and Gold.

Given the slow or non-convergence of empirical second moments, as well as the
inherent limitations of standard deviation, discussed in Section 2, we have
complemented our analysis of the three assets’ volatility by computing
100-day, 20-week, and 3-month rolling Approximate Entropies of their closing
prices and log-returns. Results are respectively presented in Figure 12 and
Figure 13. Our main finding is that Bitcoin’s both the closing price shows a
significantly lower moving ApEn than Gold and S&P 500 for all sampling
frequencies. Moreover, Bitcoin’s moving ApEn is stable for different time-
preferences, when the moving ApEn of the two other assets is, once again,
sensitive to the change in the time-scale. This predictability explains the
remarkable success and accuracy of Bitcoin price prediction models based on
the Stock to Flow ratio [saifedean].

(a) 100-day moving Approximate Entropy of daily closing prices

(b) 20-week moving Approximate Entropy of weekly closing prices

(c) 3-month moving Approximate Entropy of monthly closing prices

Figure 12: Moving Approximate Entropy of closing prices

Albeit in a less marked way, the log-returns of Bitcoin also exhibit lower
ApEn than those of the two other assets. The highest discrepancy in moving
ApEn is observed for weekly log-returns. These findings, along with our study
of Bitcoin tail behavior and standard deviation, indicate that Bitcoin overall
offers a high level of predictability, along with high yield, which make it
attractive to investors and speculators, but its high standard deviation
excludes its use a currency.

(a) 100-day moving Approximate Entropy of daily log-returns

(b) 20-week moving Approximate Entropy of weekly log-returns

(c) 3-month moving Approximate Entropy of monthly log-returns

Figure 13: Moving Approximate Entropy of log-returns

##  6 Conclusion

Bitcoin still has a long way to go in order to achieve the price stability of
Gold so caution is always warranted, but based on our analyses, investors and
traders now have empirical evidence for the best time-preference to analyze a
Bitcoin investment and what to expect in terms of the behavior closing price
and returns. If the daily price volatility of Bitcoin excludes its use as a
currency, we have shown that the volatility of both its closing price and log-
returns significantly decrease for a weekly and monthly sampling frequencies,
the log-returns of Bitcoin presenting even thinner tails and comparable
coefficient of deviation to S&P 500 for weekly data. Using approximate
entropy, we have shown that, though the variations in the variation in the
price and returns of Bitcoin are wide, they exhibit a higher amount of
regularity than those of Gold and the S&P 500. Thus, in the case of Bitcoin,
there is a divergence between the Finance and dictionary definition of
volatility (“likely to change suddenly and unexpectedly, especially by getting
worse”). Bitcoin’s price and log-returns are likely to change predictably,
typically for the better. The tail behavior and unpredictability of both the
closing price and log-returns of the two traditional assets have been shown to
be sensitive to the time-scale. As a result of the disproportionate influence
of extrema in these heavy-tailed distributions, standard deviations do not
converge for the returns of all three variables and can therefore not be
accurately estimated from samples. Sample standard deviations should be taken
with caution as the Law of Large Number would require an impractically large
number of observations to apply.

## References

  * [1] Pound, J. (2021, February) Bitcoin hits $1 trillion in market value as cryptocurrency surge continues, CNBC. Retrieved on 20-02-2021 from https://www.cnbc.com/2021/02/19/bitcoin-hits-1-trillion-in-market-value-as-cryptocurrency-surge-continues.html
  * [2] Kruckeberg, S., Scholz, P. (2019) Cryptocurrencies as an Asset Class?” In Cryptofinance and Mechanisms of Exchange – The Making of Virtual Currency, edited by Stéphane Gouette, Guesmi Khaled and Samir Saadi. Springer. DOI: 10.1007/978-3-030-30738-7
  * [3] Commodity Futures Trading Commission (2021) Bitcoin Retrieved on 20-02-2021 from https://www.cftc.gov/Bitcoin/index.htm
  * [4] Shahzad, S., Hussain, S. J., Bouri, E. et al. (2019) Is Bitcoin a better safe-haven investment than gold and commodities?, International Review of Financial Analysis, 63(C): 322-330.
  * [5] Corbet, S., Lucey, B., Urquhart, A., Yarovaya, L. (2018). Cryptocurrencies as a Financial Asset: A systematic analysis. International Review of Financial Analysis. doi:10.1016/j.irfa.2018.09.003
  * [6] Pincus, S. M. (1991) Approximate entropy as a measure of system complexity. Proceedings of the National Academy of Science USA, 188: 2297–2301.
  * [7] Dyhrberg, A. H. (2016). Bitcoin, gold and the dollar - A GARCH volatility analysis. Financial Research Letters, 16:85-92.
  * [8] Bariviera, A. F., Basgall, M. J., Hasperué, W., Naiouf, M. (2017) Some stylized facts of the Bitcoin market. Physica A. 484:82-90.
  * [9] Katsiampa, P. (2017) Volatility estimation for Bitcoin: A comparison of GARCH models. Economics Letters (2017), 158:3-6. http://dx.doi.org/10.1016/j.econlet.2017.06.023
  * [10] Yahoo Finance, Bitcoin-US Dollar quotes. Retrieved on 20-01-2021 from https://finance.yahoo.com/quote/BTC-USD/history?p=BTC-USD
  * [11] Yahoo Finance, COMEX - COMEX Delayed Price. Retrieved on 20-01-2021 from https://finance.yahoo.com/quote/GC%3DF/history?p=GC%3DF
  * [12] Yahoo Finance, SNP - SNP Real Time Price. Retrieved on 20-01-2021 from https://finance.yahoo.com/quote/%5EGSPC/history?p=%5EGSPC
  * [13] Glantz, M., Kissell, R. (2014) Price Volatility in Multi-Asset Risk Modeling, Pages 119-154, Academic Press, ISBN 9780124016903, https://doi.org/10.1016/B978-0-12-401690-3.00004-4.
  * [14] Kissell, R. (2014) Price Volatility in The Science of Algorithmic Trading and Portfolio Management, Academic Press, https://doi.org/10.1016/B978-0-12-401689-7.00006-4.
  * [15] Fidelity Learning Center (2020) Standard Deviation, Technical Indicator Guide. Retrieved from https://www.fidelity.com/learning-center/trading-investing/technical-analysis/technical-indicator-guide/standard-deviation on 15/01/2021.
  * [16] Malevergne, Y. Pisarenko, V., Sornette, D. (2005). Empirical distributions of stock returns: between the stretched exponential and the power law?, Quantitative Finance, 5(4):379-401, DOI: 10.1080/14697680500151343
  * [17] Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer.
  * [18] Queensley, C., Chukwudum, P. M., Mung’atu, J. K. Optimal threshold determination based on the mean excess plot, Communications in Statistics - Theory and Methods, In Press, 2019. https://doi.org/10.1080/03610926.2019.1624772
  * [19] Ghosh, S., Resnick, S. A discussion on mean excess plots. Stochastic Processes and their Applications, 120:1492–1517, 2010.
  * [20] Ghosh, S., Resnick, S. When Does the Mean Excess Plot Look Linear? Stochastic Models, 27(4), 2011.
  * [21] Hajizadeh, E., Seifi, A., Fazel Zarandi, A. H., Turksen, A. B. (2012) A hybrid modeling approach for forecasting the volatility of S&P 500 index return. Expert Systems with Applications, 39(1): 431-436.
  * [22] Aggarwal, R., Soenen, L. (1988) The Nature And Efficency Of The Gold Market. Journal of Portfolio Management, 14(3): 18-21.
  * [23] Lahmiri, S., Bekiros, S. (2018) Chaos, randomness and multi-fractality in Bitcoin market. Chaos, Solitons & Fractals, 106: 28-34.
  * [24] Cirillo, P., Taleb, N. N. Tail risk of contagious diseases. Nature Physics, 16:606–613, 2020.
  * [25] Bonetti, M., Cirillo, P., Musile Tanzi, P., Trinchero, E. (2016) An Analysis of the Number of Medical Malpractice Claims and Their Amounts. PLoS ONE 11(4): e0153362. doi:10.1371/journal.pone.0153362
  * [26] Feller, W. (1971) Introduction to Probability Theory and Its Applications, Vol II (2nd edition), Wiley. Section I.3 ISBN 0-471-25709-5.
  * [27] Embrechts, P., Mikosch, T., Kluppelberg, C. Modelling Extremal Events: for Insurance and Finance. Springer-Verlag, London, UK, 1997.
  * [28] Ghosh, S., Resnick, S. A discussion on mean excess plots. Stochastic Processes and their Applications, 120(8):1492-1517, 2010.
  * [29] Pincus, S. M., Huang, W.-M. (1992) Approximate entropy: Statistical properties and applications. Communications in Statistics - Theory and Methods, 21(11): 3061-3077.
  * [30] Pincus, S. M., Kalman, E. (2004) Irregularity, volatility, risk, and financial market time series. Proceedings of the National Academy of Science USA, 101(38): 13709-13714.
  * [31] Delgado-Bonal, A., Marshak, A. (2019) Approximate Entropy and Sample Entropy: A Comprehensive Tutorial. Entropy. 21(6): 541.
  * [32] Ammous, S. (2018) The Bitcoin Standard: The Decentralized Alternative to Central Banking. Wiley. ISBN: 978-1-119-47386-2.


# On pluricanonical systems of algebraic varieties of general type

Meng Chen  Institute of Mathematics, Fudan University, Shanghai, 200433, China
<EMAIL_ADDRESS>
###### Abstract.

We extend Kollár’s technique to look for an explicit function $h(n)$ with
$\varphi_{m}$ birational onto its image for all integers $m\geq h(n)$ and for
all $n$-dimensional nonsingular projective varieties of general type.

Supported by National Outstanding Young Scientist Foundation (#10625103) and
NNSFC Key project (#10731030)

##  1\. Introduction

One of the fundamental problems in birational geometry is to find a constant
$r_{n}>0$ such that the $r_{n}$-canonical map is an Iitaka fibration for any
$n$-dimensional projective variety with positive Kodaira dimension. It is
well-known that one may take $r_{2}=5$ (see Bombieri [2]) for surfaces of
general type. The existence of $r_{n}$ for varieties of general type was
proved by Hacon-M${}^{\text{c}}$kernan [10], Takayama [19] and Tsuji [20] and,
recently, $r_{3}\leq 73$ was proved by Chen-Chen [6, Theorem 1.1]. Some other
relevant results with regard to the existence of $r_{n}$ have been already
proved by Chen-Hacon [3], Pacienza [15] and Viehweg-Zhang [23]. The following
problem, however, is still open:

######  Problem 1.1.

To find an explicit constant $\mu_{n}$ ($n\geq 4$) such that the m-canonical
map $\varphi_{m}$ is birational onto its image for all $m\geq\mu_{n}$ and for
all $n$-dimensional projective varieties of general type.

Let $V$ be a $n$-dimensional nonsingular projective variety of general type.
Denote by $K_{V}$ a canonical divisor on $V$. A reasonable strategy for
studying Problem 1.1 is composed of two steps:

  * [a

] To find a positive integer $m_{0}$ such that $h^{0}(V,m_{0}K_{V})\geq 2$;

  * [b

] To find an explicit function $g(m_{0},n)$ such that the $m$-canonical map
$\varphi_{m}$ is birational for all $m\geq g(m_{0},n)$.

This strategy works well in dimension 3 (see, for instance, [5, 6]). In fact,
since Reid [17] has found the Riemann-Roch formula for minimal 3-folds, we [5,
Theorem 1.1] managed to prove $h^{0}(V,m_{0}K_{V})\geq 2$ by utilizing that
formula. On the other hand, Kollár [14] and myself [8] have an effective
formula in dimension 3 for Step (b). Generally, for Step [a], since the
classification to 4-dimensional terminal singularities is still incomplete,
there is no known Riemann-Roch formula for $\chi(mK)$ on minimal varieties.
Thus to compute $P_{m}:=h^{0}(V,m_{0}K_{V})$ is still expected. Though Kollár
[14] has essentially solved Step [b], what we are more concerned here is the
stable birationality, to be worked out by an improved and generalized
technique, of linear systems $|mK+\lceil{Q}\rceil|$ where $Q$ is any nef
${\mathbb{Q}}$-divisor. Hence this paper can be regarded as a remark or
complementary to Kollár’s method. We will build up some new results about
induced fibrations from pluricanonical systems.

Assume $P_{m_{0}}(V):=h^{0}(V,{\mathcal{O}}_{V}(m_{0}K_{V}))\geq 2$ for some
positive integer $m_{0}$. Set $\varphi_{m_{0}}:=\Phi_{|m_{0}K_{V}|}$, the
$m_{0}$-canonical map of $V$. We define the $m_{0}$-canonical dimension
$\iota:=\dim\overline{\varphi_{m_{0}}(V)}$. Clearly $1\leq\iota\leq n$. In
order to formulate our statements, we introduce the following:

######  Definition 1.2.

Define $\lambda(V)$ to be the smallest positive integer such that
$P_{\lambda(V)}(V)\geq 2$ for a given $n$-dimensional projective variety $V$
of general type. Define $\lambda_{n}:=\text{sup}\\{\lambda(V)|\dim(V)=n\\}$.

######  Remark 1.3.

According to Hacon-M${}^{\text{c}}$kernan [10], Takayama [19] and Tsuji [20],
one knows $\lambda_{n}<+\infty$. Therefore an assumption like $P_{m_{0}}\geq
2$ is reasonable and natural.

The main result of this paper is the following which, at least, induces new
results for the case $n=4$:

######  Theorem 1.4.

Let $V$ be a $n$-dimensional ($n\geq 3$) nonsingular projective variety of
general type. Let $Q$ be a nef ${\mathbb{Q}}$-divisor on $V$. Assume
$P_{m_{0}}\geq 2$ for some positive integer $m_{0}$. Then the linear system
$|mK_{V}+\lceil{Q}\rceil|$ defines a birational map onto its image for all
integers $m\geq\varepsilon(\iota)$ where $\varepsilon(\iota)$ is a function as
follows:

  * (1)

when $\iota\geq n-2$,
$\varepsilon(\iota)=\text{min}\\{4m_{0}+4,57\\}\cdot(2m_{0}+1)^{n-3}+m_{0}(n-2)+2$;

  * (2)

when $\iota=n-3$, $\varepsilon(\iota)=75(2m_{0}+1)^{n-3}+m_{0}(n-3)+2$;

  * (3)

when $\iota\leq n-4$,
$\varepsilon(\iota)=(2m_{0}+1)^{\iota-1}w_{n-\iota+1}+m_{0}(\iota-1)+2$ where
$w_{n-\iota+1}$ can be obtained by the number sequence
$\\{w_{t}\\}_{t=4}^{n-\iota+1}$ with
$w_{i}=\widetilde{\lambda}_{i}+w_{i-1}(2\widetilde{\lambda}_{i}+1)$,
$w_{4}=151\widetilde{\lambda}_{4}+75$, $\widetilde{\lambda}_{n-\iota+1}=m_{0}$
and, for all other $i$, $\widetilde{\lambda}_{i}=\lambda_{i}$.

By taking $Q=0$, $\varphi_{m}$ is birational for all integers
$m\geq\varepsilon(\iota)$.

In particular we have the following:

######  Corollary 1.5.

Let $V$ be a nonsingular projective n-dimensional ($n\geq 4$) variety of
general type. Then $\varphi_{m}$ is birational for all integers $m\geq
w_{n}+2$ where $w_{n}$ is obtained by the number sequence
$\\{w_{t}\\}_{t=4}^{n}$ with $w_{i}=\lambda_{i}+w_{i-1}(2\lambda_{i}+1)$ and
$w_{4}=151\lambda_{4}+75$.

######  Corollary 1.6.

(= Corollary 4.3) Let $V$ be a nonsingular projective 4-fold of general type
with $P_{m_{0}}\geq 2$ for some integer $m_{0}>0$. Let $Q$ be any nef
${\mathbb{Q}}$-divisor on $V$. Then $\Phi_{|mK_{V}+\lceil{Q}\rceil|}$ (in
particular, $\varphi_{m}$) is birational for all $m\geq 151m_{0}+77.$

Theorem 1.4 also implies the following:

######  Corollary 1.7.

An explicit constant $\mu_{n}$ mentioned in Problem 1.1 can be found by means
of Theorem 1.4 if and only if explicit constants $\rho_{k}$ for all $k\leq n$
can be found such that the pluri-genus $P_{\rho_{k}}\geq 2$ for all
$k$-dimensional projective varieties of general type.

From this point of view, a Riemann-Roch formula for $\chi({\mathcal{O}}(mK))$
is of key importance just like what Reid has done in [17, last section] for
threefolds.

This paper is organized as follows. First we fix the notation for the map
$\varphi_{m_{0}}$. Then we systematically study the property, of the induced
fibration $f\colon X^{\prime}\longrightarrow B$, which generalizes known
inequalities in 3-dimensional case. In Section 3, we will improve known
results on surfaces and 3-folds. Theorems in Section 4 are original. We will
prove the main theorem by induction in the last section.

We always use the symbol $\equiv$ to denote numerical equivalence while $\sim$
means linear equivalence.

##  2\. Properties of canonically induced fibrations

In this paper, $V$ is always a $n$-dimensional nonsingular projective variety
of general type. Let $m_{0}$ be a positive integer. Assume that
$\Lambda\subset|m_{0}K_{V}|$ is a sub-linear system such that
$\Phi_{\Lambda}(V)=1$ where $\Phi_{\Lambda}$ is the rational map defined by
$\Lambda$. We call $\Lambda$ a pencil contained in $|m_{0}K_{V}|$. Note that
such a pencil always exists. For instance, a 2-dimensional sub-space of
$H^{0}(V,m_{0}K_{V})$ corresponds to a pencil. We will study properties of the
rational map $\Phi_{\Lambda}$ in this section.

######  2.1.

Existence of minimal models. By recent works of Birkar-Cascini-
Hacon-M${}^{\text{c}}$kernan [1], Hacon-M${}^{\text{c}}$kernan[12] and Siu
[18], $V$ has a minimal model. Again by [12], any fibration $f\colon
Y\longrightarrow B$, from a nonsingular variety $Y$ of general type to a
smooth curve $B$, has a relative minimal model. According to the established
Minimal Model Program (MMP), one may always assume that a minimal model has at
worst ${\mathbb{Q}}$-factorial terminal singularities.

From this point of view, it suffices to study $\varphi_{m}$ on minimal models.

(${\ddagger}$) Throughout $X$ always denotes a minimal model of $V$.

Because $\Phi_{\Lambda}$ can be defined on a Zariski open subset of $X$, we
may also regard $\Lambda$ as a pencil on the minimal model $X$.

######  2.2.

Set up for $\Phi_{\Lambda}$. Denote by $\mu\colon V\dashrightarrow X$ the
birational contraction map. Because $P_{m_{0}}(X)=P_{m_{0}}(V)\geq 2$, we may
fix an effective Weil divisor $K_{m_{0}}\sim m_{0}K_{X}$ on $X$ and a divisor
$\widetilde{K}_{m_{0}}\sim m_{0}K_{V}$ on $V$. Take successive blow-ups
$\pi\colon X^{\prime}\rightarrow X$ along nonsingular centers, such that the
following conditions are satisfied:

  * (i)

$X^{\prime}$ is smooth;

  * (ii)

there is a birational morphism $\pi_{V}\colon X^{\prime}\rightarrow V$ such
that $\mu\circ\pi_{V}=\pi$;

  * (iii)

the movable part $M_{0}$ of $\pi^{*}_{V}(\Lambda)$ is base point free and so
that $g:=\Phi_{\Lambda}\circ\pi_{V}$ is a non-constant morphism;

  * (iv)

$\pi^{*}(K_{m_{0}})\cup\pi^{*}_{V}(\widetilde{K}_{m_{0}})$ has simple normal
crossing supports;

  * (v)

for certain purpose $\pi$ even satisfies a couple of extra conditions by
further modifying $X^{\prime}$. (This condition will be specified in explicit
whenever we need it.)

We have a morphism $g\colon X^{\prime}\longrightarrow
W^{\prime}\subseteq{\mathbb{P}}^{N}$. Let
$X^{\prime}\overset{f}{\longrightarrow}B\overset{s}{\longrightarrow}W^{\prime}$
be the Stein factorization of $g$. We have the following commutative diagram:

$V$$X^{\prime}$$W^{\prime}$$B$-----------$f$$s$$\pi_{V}$$\varphi_{\Lambda}$$g$

Denote by $r(X)$ the Cartier index of $X$. We can write
$r(X)K_{X^{\prime}}=\pi^{*}(r(X)K_{X})+E_{\pi}$ where $E_{\pi}$ is a sum of
exceptional divisors. Recall that

$\pi^{*}(K_{X}):=K_{X^{\prime}}-\frac{1}{r(X)}E_{\pi}.$

Clearly, whenever we take the round-up of $m\pi^{*}(K_{X})$ for $m>0$, we
always have $\lceil{m\pi^{*}(K_{X})}\rceil\leq mK_{X^{\prime}}.$

Denote by $M_{k,X^{\prime}}$ the movable part of $|kK_{X^{\prime}}|$ for any
positive integer $k>0$. We may write
$m_{0}K_{X^{\prime}}=_{\mathbb{Q}}\pi^{*}(m_{0}K_{X})+E_{\pi,m_{0}}=M_{m_{0},X^{\prime}}+Z_{m_{0}},$
where $Z_{m_{0}}$ is the fixed part of $|m_{0}K_{X^{\prime}}|$ and
$E_{\pi,m_{0}}$ an effective ${\mathbb{Q}}$-divisor which is a
${\mathbb{Q}}$-sum of distinct exceptional divisors with regard to $\pi$.

Since $M_{0}\leq M_{m_{0},X^{\prime}}\leq\pi^{*}(m_{0}K_{X})$, we can write
$\pi^{*}(m_{0}K_{X})=M_{0}+E_{\Lambda}^{\prime}$ where $E_{\Lambda}^{\prime}$
is an effective $\mathbb{Q}$-divisor. By our assumption ($\Lambda$ is a
pencil), $B$ is a nonsingular complete curve. By Bertini’s theorem, a general
fiber $F$ of the fibration $f\colon X^{\prime}\longrightarrow B$ is a
$(n-1)$-dimensional nonsingular projective variety of general type.

Once a fibration $f\colon X^{\prime}\longrightarrow B$ is obtained, we may
take the relative minimal model $f_{0}\colon X_{0}\longrightarrow B$ of $f$.
Then we can remodify $\pi$ and $\pi_{V}$ again such that a new birational
model $X^{\prime\prime}$ dominates $X^{\prime}$ and $X_{0}$. Namely assume
$\pi^{\prime}\colon X^{\prime\prime}\longrightarrow X^{\prime}$ and
$\pi^{\prime\prime}\colon X^{\prime\prime}\longrightarrow X_{0}$ are the
birational morphisms, set $f^{\prime\prime}:=f\circ\pi^{\prime}$, then
$f^{\prime\prime}=f_{0}\circ\pi^{\prime\prime}$.

Therefore we can make the following:

######  2.3.

Assumption on $X^{\prime}$. To avoid too complicated notations, we may assume
from the beginning that $X^{\prime}=X^{\prime\prime}$ by further birational
modifications, i.e. there is a contraction morphism
$\theta:X^{\prime}\longrightarrow X_{0}$ such that $f=f_{0}\circ\theta$ and
that $f_{0}$ is a relative minimal model of $f$.

With this assumption, we pick up a general fiber $F_{0}$ of $f_{0}$ and set
$\sigma:=\theta|_{F}$, then $\sigma\colon F\longrightarrow F_{0}$ is a
birational morphism onto the minimal model.

Set $b:=g(B)$. We will study the geometry of $f$ according to the value of
$b$. In fact there are two cases:

  * (i)

$b>0$, $M_{0}\sim\underset{i=1}{\overset{p}{\sum}}F_{i}\equiv pF$ where the
$F_{i}$’s are different smooth fibers of $f$ for all $i$ and $p\geq 2$;

  * (ii)

$b=0$, $M_{0}\sim pF\leq m_{0}\pi^{*}(K_{X})$ with $p\geq 1$.

######  2.4.

Reduction to problems on $X^{\prime}$. As we have seen, there is a birational
morphism $\pi_{V}\colon X^{\prime}\longrightarrow V$. Let $m$ be a positive
integer and $Q$ a nef ${\mathbb{Q}}$-divisor on $V$. Since

${\pi_{V}}_{*}{\mathcal{O}}_{X^{\prime}}(mK_{X^{\prime}}+\pi_{V}^{*}(\lceil{Q}\rceil))\cong{\mathcal{O}}_{V}(mK_{V}+\lceil{Q}\rceil)$

and $mK_{X^{\prime}}+\pi_{V}^{*}(\lceil{Q}\rceil)\geq
mK_{X^{\prime}}+\lceil{Q^{\prime}}\rceil$ where $Q^{\prime}:=\pi_{V}^{*}(Q)$
is nef on $X^{\prime}$, the birationality of
$\Phi_{|mK_{X^{\prime}}+\lceil{Q^{\prime}}\rceil|}$ implies that of
$\Phi_{|mK_{V}+\lceil{Q}\rceil|}$. Furthermore the fact $mK_{X^{\prime}}\geq
m\pi^{*}(K_{X})$ allows us to consider a smaller linear system on $X^{\prime}$
like:

$|K_{X^{\prime}}+\lceil{(m-1)\pi^{*}(K_{X})+Q^{\prime}}\rceil|.$

The 3-dimensional version of the next lemma has appeared as [9, Lemma 3.4].

######  Lemma 2.5.

Let $f_{Y}\colon Y\dashrightarrow B_{0}$ be a rational map onto a smooth curve
$B_{0}$ where $Y$ is a normal projective minimal variety (i.e. $K_{Y}$ nef)
with at worst terminal singularities. Let $\pi_{Y}\colon Y^{\prime}\rightarrow
Y$ be any birational modification from a nonsingular projective model
$Y^{\prime}$ such that $g_{Y}:=f_{Y}\circ\pi_{Y}\colon
Y^{\prime}\longrightarrow B_{0}$ is a proper morphism. Denote by $F_{b}$ any
irreducible component in a general fiber of $g_{Y}$. Assume $g(B_{0})>0$. Then

${\mathcal{O}}_{F_{b}}(\pi_{Y}^{*}(K_{Y})|_{F_{b}})\cong{\mathcal{O}}_{F_{b}}({\sigma^{\prime}}^{*}(K_{F_{b,0}}))$

where $F_{b,0}$ is a minimal model of $F_{b}$ and there is a contradiction
morphism $\sigma^{\prime}\colon F_{b}\longrightarrow F_{b,0}$.

###### Proof.

One has a morphism $g_{Y}\colon Y^{\prime}\longrightarrow B_{0}$. By theorems
of Shokurov [16] and Hacon-M${}^{\text{c}}$kernan [11], each fiber of
$\pi_{Y}\colon Y^{\prime}\longrightarrow Y$ is rationally chain connected.
Therefore, $g_{Y}(\pi_{Y}^{-1}(y))$ is a point for all $y\in Y$. Considering
the image $G\subset(Y\times B_{0})$ of $Y^{\prime}$ via the morphism
$(\pi_{Y}\times g_{y})\circ\triangle_{Y^{\prime}}$ where
$\triangle_{Y^{\prime}}$ is the diagonal map $Y^{\prime}\longrightarrow
Y^{\prime}\times Y^{\prime}$, one knows that $G$ is a projective variety. Let
$g_{1}\colon G\longrightarrow Y$ and $g_{2}\colon G\longrightarrow B_{0}$ be
two projection maps. Since $g_{1}$ is a projective morphism and even a
bijective map, $g_{1}$ must be both a finite morphism of degree 1 and a
birational morphism. Since $Y$ is normal, $g_{1}$ must be an isomorphism. So
$g_{Y}$ factors as $f_{1}\circ\pi_{Y}$ where $f_{1}:=g_{2}\circ
g_{1}^{-1}\colon Y\rightarrow B_{0}$ is a well-defined morphism. Let
$Y\overset{f_{0}}{\longrightarrow}B^{\prime}\overset{s^{\prime}}{\longrightarrow}B_{0}$
be the Stein factorization of $f_{1}$. Set $f^{\prime}:=f_{0}\circ\pi_{Y}$.
Then $F$ is a general fiber of $f^{\prime}$. Denote by $F_{b,0}$ a general
fiber of $f_{0}$. Clearly $K_{F_{b,0}}\sim{K_{Y}}|_{F_{b,0}}$ is nef and so
$F_{b,0}$ is minimal. So it is clear that
$\pi_{Y}^{*}(K_{Y})|_{F_{b}}\sim\sigma^{\prime*}(K_{F_{b,0}})$ where we set
$\sigma^{\prime}:={\pi_{Y}}|_{F_{b}}\colon F_{b}\longrightarrow F_{b,0}$. ∎

The above lemma clearly applies to our situation with $b>0$.

Now we begin to study the case $b=0$. According to our definition, $M_{m,F}$
denotes the movable part of $|mK_{F}|$ for any $m>0$.

Since $B\cong{\mathbb{P}}^{1}$, one has ${\mathcal{O}}_{B}(1)\hookrightarrow
f_{*}\omega_{X^{\prime}}^{\otimes m_{0}}$. Then there is the inclusion:

$f_{*}\omega_{X^{\prime}/B}^{\otimes m}\hookrightarrow
f_{*}\omega_{X^{\prime}}^{\otimes m(2m_{0}+1)}$

for all integers $m>0$. Because $f_{*}\omega_{X^{\prime}/B}^{\otimes m}$ is
semi-positive (see Viehweg [22]) and is thus a direct sum of line bundles of
non-negative degree, so it is generated by global sections. Therefore any
local section of $f_{*}\omega_{X^{\prime}/B}^{\otimes m}$ can be extended to a
global one of $f_{*}\omega_{X^{\prime}}^{\otimes m(2m_{0}+1)}$. This already
means

$m(2m_{0}+1)\pi^{*}(K_{X})|_{F}\geq M_{m(2m_{0}+1),X^{\prime}}|_{F}\geq
M_{m,F}.$

Whenever $m$ is divisible by the Cartier index of $F_{0}$, the Base Point Free
Theorem says that $M_{m,F}\sim m\sigma^{*}(K_{F_{0}})$. Thus we get

$None$
$\pi^{*}(K_{X})|_{F}\geq\frac{1}{m(2m_{0}+1)}M_{m(2m_{0}+1),X^{\prime}}|_{F}\geq\frac{1}{2m_{0}+1}\sigma^{*}(K_{F_{0}}).$

######  Lemma 2.6.

Keep the same notation as in 2.2. Assume $b=0$ and that a general fiber of $f$
has a Gorenstein minimal model. Then there exists a sequence of positive
rational numbers $\\{\beta_{t}\\}$, with $\beta_{t}<\frac{p}{m_{0}+p}$ and
$\beta_{t}\underset{t\mapsto+\infty}{\mapsto}\frac{p}{m_{0}+p}$, such that

$\pi^{*}(K_{X})|_{F}-\beta_{t}\sigma^{*}(K_{F_{0}})$

is ${\mathbb{Q}}$-linearly equivalent to an effective ${\mathbb{Q}}$-divisor
for all integers $t>0$.

###### Proof.

When $n=\dim(V)=3$, this is nothing but Theorem [9, Lemma 3.3]. Here we
generalize it for any $n$. One has
$\mathcal{O}_{B}(p)\hookrightarrow{f}_{*}\omega_{X^{\prime}}^{m_{0}}$ and
therefore
${f}_{*}\omega_{X^{\prime}/B}^{t_{0}p}\hookrightarrow{f}_{*}\omega_{X^{\prime}}^{t_{0}p+2t_{0}m_{0}}$
for any positive integer $t_{0}$.

Note that ${f}_{*}\omega_{X^{\prime}/B}^{t_{0}p}$ is generated by global
sections since it is semi-positive according to Viehweg [22]. So any local
section can be extended to a global one. On the other hand, whenever $t_{0}$
is bigger, $|t_{0}p\sigma^{*}(K_{F_{0}})|$ is free by Base Point Free Theorem
and is exactly the movable part of $|t_{0}pK_{F}|$ by the ordinary projection
formula. Clearly one has the following relation:

$a_{0}\pi^{*}(K_{X})|_{F}\geq M_{t_{0}p+2t_{0}m_{0},X^{\prime}}|_{F}\geq
b_{0}\sigma^{*}(K_{F_{0}})$

where $a_{0}:=t_{0}p+2t_{0}m_{0}$ and $b_{0}:=t_{0}p$. This means that there
is an effective $\mathbb{Q}$-divisor $E_{F}^{\prime}$ on $F$ such that

$a_{0}\pi^{*}(K_{X})|_{F}=_{{\mathbb{Q}}}b_{0}\sigma^{*}(K_{F_{0}})+E_{F}^{\prime}.$

Thus
$\pi^{*}(K_{X})|_{F}=_{{\mathbb{Q}}}\frac{p}{p+2m_{0}}\sigma^{*}(K_{F_{0}})+E_{F}$
with $E_{F}=\frac{1}{a_{0}}E_{F}^{\prime}$.

Let us consider the case $p\geq 2$ first.

Assume we have defined $a_{l}$ and $b_{l}$ such that the following is
satisfied with $l=t:$

$a_{l}\pi^{*}(K_{X})|_{F}\geq b_{l}\sigma^{*}(K_{F_{0}}).$

We will define $a_{t+1}$ and $b_{t+1}$ inductively such that the above
inequality is satisfied with $l=t+1$. One may assume from the beginning
(modulo necessary blow-ups) that the fractional part of the support of
$a_{t}\pi^{*}(K_{X})$ is of simple normal crossing. Then the Kawamata-Viehweg
vanishing theorem gives the surjective map

$H^{0}(K_{X^{\prime}}+\lceil{a_{t}\pi^{*}(K_{X})}\rceil+F)\longrightarrow
H^{0}(F,K_{F}+\lceil{a_{t}\pi^{*}(K_{X})}\rceil|_{F}).$

One has the relation

$\displaystyle|K_{X^{\prime}}+\lceil{a_{t}\pi^{*}(K_{X})}\rceil+F||_{F}$
$\displaystyle=$ $\displaystyle|K_{F}+\lceil{a_{t}\pi^{*}(K_{X})}\rceil|_{F}|$
$\displaystyle\supset$ $\displaystyle|K_{F}+b_{t}\sigma^{*}(K_{F_{0}})|$
$\displaystyle\supset$ $\displaystyle|(b_{t}+1)\sigma^{*}(K_{F_{0}})|.$

Denote by $M_{a_{t}+1,X^{\prime}}^{\prime}$ the movable part of
$|(a_{t}+1)K_{X^{\prime}}+F|$. Applying [7, Lemma 2.7], one has
$M_{a_{t}+1,X^{\prime}}^{\prime}|_{F}\geq(b_{t}+1)\sigma^{*}(K_{F_{0}}).$ Re-
modifying our original $\pi$ such that $|M_{a_{t}+1,X^{\prime}}^{\prime}|$ is
base point free. In particular, $M_{a_{t}+1,X^{\prime}}^{\prime}$ is nef.
Since $X$ is of general type $|mK_{X}|$ gives a birational map whenever $m$ is
big enough. Thus we see that $M_{a_{t}+1,X^{\prime}}^{\prime}$ is big if we
fix a very big $t_{0}$ in advance.

Now the Kawamata-Viehweg vanishing theorem again gives

$\displaystyle|K_{X^{\prime}}+M_{a_{t}+1,X^{\prime}}^{\prime}+F||_{F}$
$\displaystyle=$ $\displaystyle|K_{F}+M_{a_{t}+1,X^{\prime}}^{\prime}|_{F}|$
$\displaystyle\supset$ $\displaystyle|K_{F}+(b_{t}+1)\sigma^{*}(K_{F_{0}})|$
$\displaystyle\supset$ $\displaystyle|(b_{t}+2)\sigma^{*}(K_{F_{0}})|.$

Repeat the above procedure and denote by $M_{a_{t}+u,X^{\prime}}^{\prime}$ the
movable part of $|K_{X^{\prime}}+M_{a_{t}+u-1,X^{\prime}}^{\prime}+F|$ for
integers $u\geq 2$. For the same reason, we may assume
$|M_{a_{t}+u,X^{\prime}}^{\prime}|$ is base point free and is thus nef and
big. Inductively, for any $u>0$, one has:

$M_{a_{t}+u,X^{\prime}}^{\prime}|_{F}\geq(b_{t}+u)\sigma^{*}(K_{F_{0}}).$

Applying the vanishing theorem once more, one has

$\displaystyle|K_{X^{\prime}}+M_{a_{t}+u,X^{\prime}}^{\prime}+F||_{F}$
$\displaystyle=$ $\displaystyle|K_{F}+M_{a_{t}+u,X^{\prime}}^{\prime}|_{F}|$
$\displaystyle\supset$ $\displaystyle|K_{F}+(b_{t}+u)\sigma^{*}(K_{F_{0}})|$
$\displaystyle\supset$ $\displaystyle|(b_{t}+u+1)\sigma^{*}(K_{F_{0}})|.$

Take $u=p-1$. Noting that

$|K_{X^{\prime}}+M_{a_{t}+p-1,X^{\prime}}^{\prime}+F|\subset|(a_{t}+p+m_{0})K_{X^{\prime}}|$

and applying [7, Lemma 2.7] again, one has

$a_{t+1}\pi^{*}(K_{X})|_{F}\geq M_{a_{t}+p+m_{0},X^{\prime}}|_{F}\geq
M^{\prime}_{a_{t}+p,X^{\prime}}|_{F}\geq b_{t+1}\sigma^{*}(K_{F_{0}})$

where $a_{t+1}:=a_{t}+p+m_{0}$ and $b_{t+1}=b_{t}+p$. Set
$\beta_{t}=\frac{b_{t}}{a_{t}}.$ Clearly
$\lim_{t\mapsto+\infty}\beta_{t}=\frac{p}{m_{0}+p}$.

The case $p=1$ can be considered similarly with a simpler induction. We leave
it as an exercise. ∎

######  Lemma 2.7.

Let $\widetilde{\Lambda}\subset|L|$ be a pencil on $V$ where $L$ is a divisor
on $V$. Let $R$ be a nef and big ${\mathbb{Q}}$-divisor on $V$. Assume there
is a birational modification
$\pi_{\widetilde{\Lambda}}:V^{\prime}\longrightarrow V$ such that:

  * (1)

the fractional part of $\pi_{\widetilde{\Lambda}}^{*}(R)$ has simple normal
crossing supports and the movable part of
$\pi_{\widetilde{\Lambda}}^{*}(\widetilde{\Lambda})$ is base point free;

  * (2)

$|K_{V^{\prime}}+\lceil{\pi_{\widetilde{\Lambda}}^{*}(R)}\rceil|\neq\emptyset.$

Then $|K_{V}+\lceil{R}\rceil+L|$ distinguishes different irreducible elements
in the movable part of $\widetilde{\Lambda}$.

###### Proof.

Noting that

${\pi_{\widetilde{\Lambda}}}_{*}{\mathcal{O}}_{V^{\prime}}(K_{V^{\prime}}+\pi_{\widetilde{\Lambda}}^{*}(\lceil{R}\rceil)+\pi_{\widetilde{\Lambda}}^{*}(L)|)\cong{\mathcal{O}}_{V}(K_{V}+\lceil{R}\rceil+L)$

and that
$\pi_{\widetilde{\Lambda}}^{*}(\lceil{R}\rceil)\geq\lceil{\pi_{\widetilde{\Lambda}}^{*}(R)}\rceil$,
we only need to study the smaller linear system
$|K_{V^{\prime}}+\lceil{\pi_{\widetilde{\Lambda}}^{*}(R)}\rceil)+M_{\widetilde{\Lambda}}|$
where $|M_{\widetilde{\Lambda}}|$ is the movable part of
$\pi_{\widetilde{\Lambda}}^{*}(\widetilde{\Lambda})$. By our assumption,
$|M_{\widetilde{\Lambda}}|$ is composed with a pencil and
$f_{\widetilde{\Lambda}}:V^{\prime}\longrightarrow B_{\widetilde{\Lambda}}$ is
an induced fibration of $\Phi_{|M_{\widetilde{\Lambda}}|}$, where
$B_{\widetilde{\Lambda}}$ is a smooth curve.

If $g(B_{\widetilde{\Lambda}})=0$, then
$|K_{V^{\prime}}+\lceil{\pi_{\widetilde{\Lambda}}^{*}(R)}\rceil+M_{\widetilde{\Lambda}}|$
distinguishes different general fibers of $f_{\widetilde{\Lambda}}$ because
$|K_{V^{\prime}}+\lceil{\pi_{\widetilde{\Lambda}}^{*}(R)}\rceil|\neq\emptyset.$

If $g(B_{\widetilde{\Lambda}})>0$, we pick up two general fibers
$F_{\widetilde{\Lambda}}^{\prime}$ and
$F_{\widetilde{\Lambda}}^{\prime\prime}$. Then
$M_{\widetilde{\Lambda}}-F_{\widetilde{\Lambda}}^{\prime}-F_{\widetilde{\Lambda}}^{\prime\prime}$
is nef and the Kawamata-Viehweg vanishing theorem [13, 21] gives

$H^{1}(V^{\prime},K_{V^{\prime}}+\lceil{\pi_{\widetilde{\Lambda}}^{*}(R)}\rceil+M_{\widetilde{\Lambda}}-F_{\widetilde{\Lambda}}^{\prime}-F_{\widetilde{\Lambda}}^{\prime\prime})=0.$

Thus follows the following surjective map:

$\displaystyle
H^{0}(V^{\prime},K_{V^{\prime}}+\lceil{\pi_{\widetilde{\Lambda}}^{*}(R)}\rceil+M_{\widetilde{\Lambda}})$
$\displaystyle\longrightarrow$ $\displaystyle
H^{0}(F_{\widetilde{\Lambda}}^{\prime},(K_{V^{\prime}}+\lceil{\pi_{\widetilde{\Lambda}}^{*}(R)}\rceil)|_{F_{\widetilde{\Lambda}}^{\prime}})\oplus
H^{0}(F_{{\widetilde{\Lambda}}^{\prime\prime}},(K_{V^{\prime}}+\lceil{\pi_{\widetilde{\Lambda}}^{*}(R)}\rceil)|_{F_{\widetilde{\Lambda}}^{\prime\prime}}).$

Again the assumption
$|K_{V^{\prime}}+\lceil{\pi_{\widetilde{\Lambda}}^{*}(R)}\rceil|\neq\emptyset$
implies that
$|K_{V^{\prime}}+\lceil{\pi_{\widetilde{\Lambda}}^{*}(R)}\rceil+M_{\widetilde{\Lambda}}|$
(and thus $|K_{V}+\lceil{R}\rceil+L|$) distinguishes different irreducible
elements in the movable part of $\widetilde{\Lambda}$. We are done. ∎

The following lemma is tacitly used in our context.

######  Lemma 2.8.

Let $\bar{Q}$ be any ${\mathbb{Q}}$-divisor on a nonsingular projective
variety $Z$. Let $\widetilde{\pi}:\widetilde{Z}\longrightarrow Z$ be any
birational modification. Assume that
$|K_{\widetilde{Z}}+\lceil{\widetilde{\pi}^{*}(\bar{Q})}\rceil|$ gives a
birational map. Then $|K_{Z}+\lceil{\bar{Q}}\rceil|$ gives a birational map.

###### Proof.

This is clear due to the fact:

$\widetilde{\pi}_{*}{\mathcal{O}}_{\widetilde{Z}}(K_{\widetilde{Z}}+\widetilde{\pi}^{*}(\lceil{\bar{Q}}\rceil))\cong{\mathcal{O}}_{Z}(K_{Z}+\lceil{\bar{Q}}\rceil)$

and
$\widetilde{\pi}^{*}(\lceil{\bar{Q}}\rceil)\geq\lceil{\widetilde{\pi}^{*}(\bar{Q})}\rceil$.
∎

##  3\. ${\mathbb{Q}}$-divisors on surfaces and threefolds

We leave the proof for the next two results on surfaces as an exercise which
is really a standard ${\mathbb{Q}}$-divisor argument.

######  Lemma 3.1.

Let $S$ be a nonsingular projective surface of general type. Denote by
$\sigma:S\longrightarrow S_{0}$ the birational contraction onto the minimal
model $S_{0}$. For any nef and big ${\mathbb{Q}}$-divisor $Q_{2}$ on $S$, one
has

$h^{0}(S,K_{S}+m\sigma^{*}(K_{S_{0}})+\lceil{Q_{2}}\rceil)>1$

under one of the following situations:

(1) $m\geq 2$;

(2) $m=1$ and $p_{g}(S)>0$.

######  Theorem 3.2.

Keep the same notation as in Lemma 3.1. Then the rational map
$\Phi_{|K_{S}+m\sigma^{*}(K_{S_{0}})+\lceil{Q_{2}}\rceil|}$ is birational in
either of the following cases:

(1) $m\geq 4$;

(2) $m\geq 3$ and $p_{g}(S)>0$.

######  Proposition 3.3.

Assume $\dim(V)=3$ and $P_{m_{0}}(V)\geq 2$ for some positive integer $m_{0}$.
Keep the same notation as in 2.2. Let $Q_{3}$ be a nef ${\mathbb{Q}}$-divisor
on $V$ and $Q^{\prime}_{3}$ a nef ${\mathbb{Q}}$-divisor on $X^{\prime}$. Then
$K_{X^{\prime}}+\lceil{q_{3}\pi^{*}(K_{X})+Q^{\prime}_{3}}\rceil+F$ is
effective for all rational numbers $q_{3}>2m_{0}+2$. Consequently
$mK_{V}+\lceil{Q_{3}}\rceil$ is effective for all integers $m\geq 3m_{0}+4$.

###### Proof.

The last statement is a direct application of the first one due to 2.4. We
prove the first statement.

> ($\pounds$) Take further necessary modifications to $X^{\prime}$ such that
> the supports of the fractional parts of $Q_{3}^{\prime}$ and
> $\pi_{V}^{*}(Q_{3})$ are of simple normal crossing. For simplicity we still
> use $X^{\prime}$ to denote the final birational model dominating $V$.

We have a fibration $f:X^{\prime}\longrightarrow B$ induced from $|m_{0}K|$ as
in 2.2. We consider the linear system

$|K_{X^{\prime}}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil+F|\subset|(\lceil{q_{3}}\rceil+m_{0}+1)K_{X^{\prime}}+\lceil{Q_{3}^{\prime}}\rceil|.$

Because $q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}$ is nef and big and the fractional
part of $Q_{3}^{\prime}$ is of simple normal crossing by our assumption, the
vanishing theorem says

$H^{1}(X,K_{X^{\prime}}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil)=0.$

Thus one has the surjective map:

$H^{0}(X,K_{X^{\prime}}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil+F)\longrightarrow
H^{0}(F,K_{F}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil|_{F}).$

Note that in our case a general fiber of $f$ is a surface of general type
which has a Gorenstein minimal model. Thus the conditions in both Lemma 2.5
and Lemma 2.6 are satisfied. If $b>0$, Lemma 2.5 says that
$\pi^{*}(K_{X})|_{F}\sim\sigma^{*}(K_{F_{0}})$ where $\sigma:F\longrightarrow
F_{0}$ is the contraction map. If $g(B)=0$, Lemma 2.6 says that one can find a
very big number $s$ such that

$\pi^{*}(K_{X})|_{F}\geq\beta_{s}\sigma^{*}(K_{F_{0}})$

and $\beta_{s}$ is sufficiently near $\frac{p}{m_{0}+p}\geq\frac{1}{m_{0}+1}$
and $\beta_{s}<\frac{p}{m_{0}+p}$.

Let us put $\alpha_{s}:=\frac{p}{m_{0}+p}-\beta_{s}$. Then $\alpha_{s}\mapsto
0$ whenever $s\mapsto+\infty$.

Whenever $q_{3}>2m_{0}+2$, one has

$\displaystyle q_{3}\pi^{*}(K_{X})|_{F}$ $\displaystyle\geq$ $\displaystyle
q_{3}\beta_{s}\sigma^{*}(K_{F_{0}})=q_{3}(\frac{p}{m_{0}+p}-\alpha_{s})\sigma^{*}(K_{F_{0}})$
$\displaystyle=$
$\displaystyle(2+(\frac{q_{3}p-2m_{0}-2p}{m_{0}+p}-q_{3}\alpha_{s}))\sigma^{*}(K_{F_{0}}).$

When $s$ is big enough, one sees
$\frac{q_{3}p-2m_{0}-2p}{m_{0}+p}-q_{3}\alpha_{s}>0$. We may assume that
$q_{3}\pi^{*}(K_{X})|_{F}-q_{3}\beta_{s}\sigma^{*}(K_{F_{0}})$ is
${\mathbb{Q}}$-linearly equivalent to an effective divisor $R_{q_{3},s}$ on
$F$. Then

$\displaystyle\widetilde{R_{q_{3},s}}$ $\displaystyle:=$ $\displaystyle
q_{3}\pi^{*}(K_{X})|_{F}-R_{q_{3},s}-2\sigma^{*}(K_{F,0})$
$\displaystyle\equiv$
$\displaystyle(\frac{q_{3}p-2m_{0}-2p}{m_{0}+p}-q_{3}\alpha_{s})\sigma^{*}(K_{F_{0}})$

is nef and big since $\frac{q_{3}p-2m_{0}-2p}{m_{0}+p}>0$. Therefore

$H^{0}(K_{F}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil|_{F})\supset
H^{0}(K_{F}+2\sigma^{*}(K_{F_{0}})+\lceil{\widetilde{R}_{q_{3},n}+Q_{3}^{\prime}}\rceil)\neq
0$

by Lemma 3.1. And in fact
$h^{0}(K_{F}+2\sigma^{*}(K_{F_{0}})+\lceil{\widetilde{R}_{q_{3},n}+Q_{3}^{\prime}}\rceil)>1$.
∎

######  Remark 3.4.

In the proof of Proposition 3.3, if the general fiber of $f$ is a surface with
$p_{g}>0$, then, according to Lemma 3.1,
$K_{X^{\prime}}+\lceil{q_{3}\pi^{*}(K_{X})+Q^{\prime}_{3})}\rceil+F$ is
effective for all rational numbers $q_{3}>m_{0}+1$. And accordingly
$mK_{V}+\lceil{Q_{3}}\rceil$ is effective for all integers $m\geq 2m_{0}+3$.

######  Theorem 3.5.

Assume $\dim(V)=3$ and $P_{m_{0}}(V)\geq 2$ for some positive integer $m_{0}$.
Keep the same notation as in 2.2. Let $Q_{3}$ be a nef ${\mathbb{Q}}$-divisor
on $V$ and $Q_{3}^{\prime}$ a nef ${\mathbb{Q}}$-divisor on $X^{\prime}$. Then

  * (1)

$\Phi_{|K_{X^{\prime}}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil+M_{0}|}$
is birational for all rational numbers $q_{3}>4m_{0}+4$. In particular
$\Phi_{|mK_{V}+\lceil{Q_{3}}\rceil|}$ is birational for all integers $m\geq
5m_{0}+6$;

  * (2)

if the general fiber $F$ of $f\colon X^{\prime}\longrightarrow B$ has positive
geometric genus,
$\Phi_{|K_{X^{\prime}}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil+M_{0}|}$
is birational for all rational numbers $q_{3}>3m_{0}+3$. In particular
$\Phi_{|mK_{V}+\lceil{Q_{3}}\rceil|}$ is birational for all integers $m\geq
4m_{0}+5$.

###### Proof.

According to Proposition 3.3 and Remark 3.4,

$K_{X^{\prime}}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil$

is always effective under each situation since $m_{0}\pi^{*}(K_{X})\geq
M_{0}$. Therefore Lemma 2.7 says that
$|K_{X^{\prime}}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil+M_{0}|$ can
distinguish different generic irreducible elements of $|M_{0}|$. Thus it
suffices to prove the birationality of
$\Phi_{|K_{X^{\prime}}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil+M_{0}|}|_{F}$
for a general fiber $F$ of $f$. The proofs for statements (1) and (2) are
similar. We only consider (1) while omitting the proof for (2).

Of course, the first step in utilizing the vanishing theorem is to make the
support of the fractional part of $\\{Q_{3}^{\prime}\\}$ to be simple normal
crossing. This can be done by re-modifying $X^{\prime}$. For simplicity we may
assume, from now on, that our $X^{\prime}$ has the property stated in
$(\pounds)$ (see the proof of Proposition 3.3).

The Kawatama-Viehweg vanishing theorem implies, noticing $F|_{F}\sim 0$, that

$|K_{X^{\prime}}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil+M_{0}||_{F}=|K_{F}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil|_{F}|.$

We study a smaller system
$|K_{F}+\lceil{q_{3}\pi^{*}(K_{X})|_{F}+{Q_{3}^{\prime}}|_{F}}\rceil$. We have
already $\pi^{*}(K_{X})|_{F}\geq\beta_{s}\sigma^{*}(K_{F_{0}})$ and
$0<\alpha_{s}:=\frac{p}{m_{0}+p}-\beta_{s}$, $\alpha_{s}\mapsto 0$ whenever
$s\mapsto+\infty$. When $q_{3}>4m_{0}+4$,

$\displaystyle q_{3}\pi^{*}(K_{X})|_{F}$ $\displaystyle\geq$ $\displaystyle
q_{3}\beta_{s}\sigma^{*}(K_{F})$ $\displaystyle=$ $\displaystyle
4\sigma^{*}(K_{F_{0}})+t_{s}\sigma^{*}(K_{F_{0}})$

where $t_{s}:=\frac{q_{3}p-4m_{0}-4p}{m_{0}+p}-q_{3}\alpha_{s}>0$ whenever $s$
is big enough. Therefore, by Theorem 3.2 and Lemma 2.8,
$|K_{S}+\lceil{4\sigma^{*}(K_{F_{0}})+t_{s}\sigma^{*}(K_{F_{0}})+{Q_{3}^{\prime}}}\rceil|_{F}|$
gives a birational map. Being a bigger linear system,

$|K_{F}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil|_{F}|$

also gives a birational map. We are done. ∎

######  3.6.

Threefolds $V$ with $\chi({\mathcal{O}}_{V})>1$. Keep the same notation as in
2.2. As seen in [6, Lemma 2.32], if $\chi({\mathcal{O}}_{V})>1$ and $q(V)=0$,
then a general fiber $F$ of $f:X^{\prime}\longrightarrow B$ has the geometric
genus $p_{g}(F)>0$.

######  Lemma 3.7.

Assume $\dim(V)=3$. Then

$\lambda(V)\leq\begin{cases}18&\text{if}\ \chi({\mathcal{O}}_{V})>1\
\text{and}\ q(V)=0;\\\ 10&otherwise.\end{cases}$

###### Proof.

The first statement $\lambda(V)\leq 18$ is due to [6, Theorem 4.8].

When $q(V)>0$, $\lambda(V)\leq 3$ by Chen-Hacon [4].

When $\chi({\mathcal{O}}_{V})=1$, $\lambda(V)\leq 10$ by [6, Corollary 3.13].

Finally when $\chi({\mathcal{O}}_{V})<0$, $\lambda(V)\leq 3$ is a direct
consequence of Reid’s plurigenus formula by Reid [17] and by Chen-Zuo [9,
Lemma 4.1]. ∎

From now on, we classify a 3-fold $V$ into two types:

  * (I)

$\chi({\mathcal{O}}_{V})>1$ and $q(V)=0$;

  * (II)

either $\chi({\mathcal{O}}_{V})\leq 1$ or $q(V)>0$.

##  4\. Point separation on 4-folds

######  Proposition 4.1.

Assume $n=\dim(V)=4$ and $P_{m_{0}}(V)\geq 2$ for some positive integer
$m_{0}$. Keep the same notation as in 2.2. Let $Q_{4}$ be a nef
${\mathbb{Q}}$-divisor on $V$ and $Q_{4}^{\prime}$ a nef
${\mathbb{Q}}$-divisor on $X^{\prime}$. Then

  * (1)

$K_{X^{\prime}}+\lceil{q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime}}\rceil+F$ is an
effective divisor for all rational numbers $q_{4}>74m_{0}+37$;

  * (2)

$mK_{V}+\lceil{Q}\rceil$ is effective for all integers $m\geq 75m_{0}+39.$

###### Proof.

(2) is a direct result from (1) according to 2.4. We only prove (1). Similar
to assumption ($\pounds$) in the proof of Proposition 3.3, we may assume that
$X^{\prime}$ is good enough.

Because $q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime}$ is nef and big, and its
fractional part has normal crossing supports by assumption, the Kawamata-
Viehweg vanishing theorem gives the surjective map:

$H^{0}(X^{\prime},K_{X^{\prime}}+\lceil{q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime}}\rceil+F)\longrightarrow
H^{0}(F,K_{F}+\lceil{q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime}}\rceil|_{F}).$

By Lemma 2.5 and inequality (1), one has

$\displaystyle\lceil{q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime}}\rceil|_{F}$
$\displaystyle\geq$
$\displaystyle\lceil{(q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime})|_{F}}\rceil$
$\displaystyle\geq$
$\displaystyle\lceil{\frac{q_{4}}{2m_{0}+1}\sigma^{*}(K_{F_{0}})+{Q_{4}^{\prime}}|_{F}}\rceil.$

If $F$ is of type (I), then, by Remark 3.4, 3.6 and Lemma 3.7, we need to set
$\frac{q_{4}}{2m_{0}+1}>37\geq(\lambda(F)+1)+\lambda(F)$, i.e.
$q_{4}>74m_{0}+37$, so that
$K_{F}+\lceil{\frac{q_{4}}{2m_{0}+1}\sigma^{*}(K_{F_{0}})+{Q_{4}^{\prime}}|_{F}}\rceil$
is an effective divisor on $F$.

If $F$ is of type (II), then by Proposition 3.3 and Lemma 3.7 we need
$\frac{q_{4}}{2m_{0}+1}>32\geq(2\lambda(F)+2)+\lambda(F)$ so that
$K_{F}+\lceil{\frac{q_{4}}{2m_{0}+1}\sigma^{*}(K_{F_{0}})+{Q_{4}^{\prime}}|_{F}}\rceil$
is effective.

In a word, when $q_{4}>74m_{0}+37$,
$K_{X^{\prime}}+\lceil{q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime}}\rceil+F$ is
effective. ∎

######  Theorem 4.2.

Assume $n=\dim(V)=4$ and $P_{m_{0}}\geq 2$ for some positive integer $m_{0}$.
Keep the same notation as in 2.2. Let $Q_{4}$ be a nef ${\mathbb{Q}}$-divisor
on $V$ and $Q_{4}^{\prime}$ a nef ${\mathbb{Q}}$-divisor on $X^{\prime}$. Then

  * (1)

$|K_{X^{\prime}}+\lceil{q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime}}\rceil+M_{0}|$
gives a birational map for all rational numbers $q_{4}>150m_{0}+75$;

  * (2)

$\Phi_{|mK_{V}+\lceil{Q}\rceil|}$ is birational for all integers $m\geq
151m_{0}+77.$

###### Proof.

Similar to assumption $(\pounds)$, we may assume that $X^{\prime}$ is good
enough (after a necessary modification). Also (2) is a direct result of (1).
We only prove (1).

By Proposition 4.1, we see that
$K_{X^{\prime}}+\lceil{q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime}}\rceil\geq 0$. Lemma
2.7 tells us that we only need to verify the birationality of

$\Phi_{|K_{X^{\prime}}+\lceil{q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime}}\rceil+M_{0}|}|_{F}$

for a general fiber $F$ of $f$. The vanishing theorem gives

$|K_{X^{\prime}}+\lceil{q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime}}\rceil+M_{0}||_{F}=|K_{F}+\lceil{q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime}}\rceil|_{F}|$

noticing ${M_{0}}|_{F}\sim 0$. Lemma 2.5 and inequality (1) imply

$\lceil{q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime}}\rceil|_{F}\geq\lceil{\frac{q_{4}}{2m_{0}+1}\sigma^{*}(K_{F_{0}})+{Q_{4}^{\prime}}|_{F}}\rceil.$

Noting that $F$ is a threefold of general type, we still use a similar
argument to that in the proof of Proposition 4.1.

If $F$ is of type (I), then, by Theorem 3.5(2), 3.6, Lemma 3.7 and Lemma 2.8,
$|K_{F}+\lceil{\frac{q_{4}}{2m_{0}+1}\sigma^{*}(K_{F_{0}})+{Q_{4}^{\prime}}|_{F}}\rceil|$
gives a birational map when

$\frac{q_{4}}{2m_{0}+1}>75\geq(3\lambda(F)+3)+\lambda(F),$

i.e. $q_{4}>150m_{0}+75$.

If $F$ is of type (II), by Theorem 3.5(1) and Lemma 3.7,

$\Phi_{|K_{F}+\lceil{\frac{q_{4}}{2m_{0}+1}\sigma^{*}(K_{F_{0}})+{Q_{4}^{\prime}}|_{F}}\rceil|}$

is birational when $\frac{q_{4}}{2m_{0}+1}>54\geq(4\lambda(F)+4)+\lambda(F)$,
i.e. $q_{4}>108m_{0}+54$.

To make a conclusion,
$|K_{X^{\prime}}+\lceil{q_{4}\pi^{*}(K_{X})+Q_{4}^{\prime}}\rceil+M_{0}|$
gives a birational map for all rational numbers $q_{4}>150m_{0}+75$. ∎

A direct result of Theorem 4.2 is the following:

######  Corollary 4.3.

Assume $\dim(V)=4$ and $P_{m_{0}}\geq 2$ for some positive integer $m_{0}$.
Then $\varphi_{m}$ is birational onto its image for all integers $m\geq
151m_{0}+77$.

##  5\. Proof of the main theorem

We organize the proof according to the value of $\iota$.

First we consider the case $\iota\geq n-2$.

######  Proposition 5.1.

Assume $n=\dim(V)\geq 3$, $P_{m_{0}}(V)\geq 2$ for some positive integer
$m_{0}$ and $\iota\geq n-2$. Keep the same notation as in 2.2. Let $Q_{n}$ be
any nef ${\mathbb{Q}}$-divisor on $V$ and $Q_{n}^{\prime}$ any nef
${\mathbb{Q}}$-divisor on $X^{\prime}$. Then

  * (1)

$K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(n-3)M_{0}+F$
is effective for all rational numbers
$q_{n}>\text{min}\\{2m_{0}+2,22\\}\cdot(2m_{0}+1)^{n-3}$;

  * (2)

$mK_{V}+\lceil{Q_{n}}\rceil$ is effective for all integers
$m\geq\text{min}\\{2m_{0}+2,22\\}\cdot(2m_{0}+1)^{n-3}+m_{0}(n-2)+2$.

###### Proof.

Noting that (2) is a direct application of (1), we only prove (1). We are
going to do an induction on $n$.

When $n=3$, Proposition 3.3 says that the statement is true when
$q_{3}>2m_{0}+2=(2m_{0}+2)\cdot(2m_{0}+1)^{n-3}+m_{0}(n-3)$. On the other
hand, we may replace $m_{0}$ with $\lambda(V)$. In fact, Lemma 3.7 gives
$\lambda(V)\leq 18$ for type (I) and $\lambda(V)\leq 10$ for type (II). Thus,
by Proposition 3.3 and Remark 3.4,
$K_{X^{\prime}}+\lceil{q_{3}\pi^{*}(K_{X})+Q_{3}^{\prime}}\rceil+F$ is
effective whenever $q_{3}>22$. Therefore statement (1) is true for
$q_{3}>\text{min}\\{2m_{0}+2,22\\}\cdot(2m_{0}+1)^{n-3}+m_{0}(n-3)$.

Assume that statement (1) is correct for varieties of dimension $\leq n-1$.
Starting with a good model $X^{\prime}$ satisfying 2.3, we can do the
induction. Noticing that $F$ is of dimension $n-1$, we hope to reduce the
problem onto $F$. Since $\iota\geq n-2$, we have
$\dim\varphi_{m_{0}}(F)\geq\dim\Phi_{|M_{0}|}(F)\geq n-3=\dim(F)-2$ by the
simple additivity property. Because ${M_{0}}|_{F}\leq
m_{0}{K_{X^{\prime}}}|_{F}\sim m_{0}K_{F}$, we know
$\Phi_{|m_{0}K_{F}|}(F)\geq\Phi_{|{M_{0}}|_{F}|}(F)\geq\dim(F)-2$. Because
$q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}$ is nef and big and has simple normal
crossing fractional parts, the vanishing theorem gives the surjective map:

$\displaystyle
H^{0}(X^{\prime},K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(n-3)M_{0}+F)$
$\displaystyle\rightarrow$ $\displaystyle
H^{0}(F,K_{F}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil|_{F}+(n-3){M_{0}}|_{F})$
$\displaystyle\supset$ $\displaystyle
H^{0}(F,K_{F}+\lceil{q_{n}\pi^{*}(K_{X})|_{F}+{Q_{n}^{\prime}}|_{F}}\rceil+(n-3){M_{0}}|_{F})$
$\displaystyle\supset$ $\displaystyle
H^{0}(F,K_{F}+\lceil{q_{n}^{\prime}\sigma^{*}(K_{F_{0}})+{Q_{n}^{\prime}}|_{F}}\rceil+(n-4){M_{0}}|_{F}+{M_{0}}|_{F})\
\ \ \ (4)$

where
$q_{n}^{\prime}\geq\frac{q_{n}}{2m_{0}+1}>\text{min}\\{2m_{0}+2,22\\}\cdot(2m_{0}+1)^{(n-1)-3}$
by Lemma 2.5 and inequality (1). Because ${M_{0}}|_{F}\leq m_{0}K_{F}$ and by
taking those pencil $\Lambda_{F}\subset|M_{0}||_{F}$, the induction and Lemma
2.8 tell us that
$K_{F}+\lceil{q_{n}^{\prime}\sigma^{*}(K_{F_{0}})+{Q_{n}^{\prime}}|_{F}}\rceil+((n-1)-3){M_{0}}|_{F}+{M_{0}}|_{F}$
is effective whenever
$q_{n}^{\prime}>\text{min}\\{2m_{0}+2,22\\}\cdot(2m_{0}+1)^{(n-1)-3}$. We are
done. ∎

######  Theorem 5.2.

Assume $n=\dim(V)\geq 3$, $P_{m_{0}}(V)\geq 2$ for some positive integer
$m_{0}$ and $\iota\geq n-2$. Keep the same notation as in 2.2. Let $Q_{n}$ be
any nef ${\mathbb{Q}}$-divisor on $V$ and $Q_{n}^{\prime}$ any nef
${\mathbb{Q}}$-divisor on $X^{\prime}$. Then

  * (1)

$|K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(n-2)M_{0}|$
gives a birational map for all rational numbers

$q_{n}>\text{min}\\{4m_{0}+4,57\\}\cdot(2m_{0}+1)^{n-3};$

  * (2)

$|mK_{V}+\lceil{Q_{n}}\rceil|$ gives a birational map for all integers

$m\geq\text{min}\\{4m_{0}+4,57\\}\cdot(2m_{0}+1)^{n-3}+m_{0}(n-2)+2.$

###### Proof.

Again since (2) is a direct application of (1), we only prove (1). We are
going to do an induction on $n$.

Under the assumption $q_{n}>\text{min}\\{4m_{0}+4,57\\}\cdot(2m_{0}+1)^{n-3}$,
since $m_{0}\pi^{*}(K_{X})\geq M_{0}$, we see

$\displaystyle
K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(n-3)M_{0}$
$\displaystyle\geq$ $\displaystyle
K_{X^{\prime}}+\lceil{\widetilde{q}_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(n-3)M_{0}+F\geq
0$

by Proposition 5.1 since
$\widetilde{q}_{n}:=q_{n}-m_{0}>\text{min}\\{2m_{0}+2,22\\}\cdot(2m_{0}+1)^{n-3}$.
According to Lemma 2.7,
$|K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(n-2)M_{0}|$
can distinguish different fibers of $f$. We are left to show the birationality
of the rational map given by
$|K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(n-2)M_{0}||_{F}$
for a general fiber $F$ of $f$.

When $n=3$ and $q_{3}>4m_{0}+4=(4m_{0}+4)\cdot(2m_{0}+1)^{n-3}$, the statement
is nothing but Theorem 3.5(1). On the other hand, we may replace $m_{0}$ with
$\lambda(V)$. In fact, Lemma 3.7 gives $\lambda(V)\leq 18$ for type (I) and
$\lambda(V)\leq 10$ for type (II). Then, by Theorem 3.5(1) and (2),
$|K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(n-2)M_{0}|$
gives a birational map for all rational numbers
$q_{3}>\text{max}\\{57,44\\}=57$. Therefore statement (1) is true whenever
$q_{3}>\text{min}\\{4m_{0}+4,57\\}$.

Assume that statement (1) is correct for varieties of dimension $\leq n-1$.
Starting with a good model $X^{\prime}$ satisfying 2.3, we can do the
induction again. Still, we see
$\Phi_{|m_{0}K_{F}|}(F)\geq\Phi_{|{M_{0}}|_{F}|}(F)\geq\dim(F)-2$. We hope to
reduce to the problem on $F$. According to the relation (4), we only need to
study

$|J_{n-1}|:=|K_{F}+\lceil{q_{n}^{\prime}\sigma^{*}(K_{F_{0}})+{Q_{n}^{\prime}}|_{F}}\rceil+(n-4){M_{0}}|_{F}+{M_{0}}|_{F}|$

where
$q_{n}^{\prime}\geq\frac{q_{n}}{2m_{0}+1}>\text{min}\\{4m_{0}+4,57\\}\cdot(2m_{0}+1)^{(n-1)-3}$
by Lemma 2.5 and inequality (1). Because ${M_{0}}|_{F}\leq m_{0}K_{F}$ and by
taking those pencil $\Lambda_{F}\subset|M_{0}||_{F}$, the induction and Lemma
2.8 tell us that
$|K_{F}+\lceil{q_{n}^{\prime}\sigma^{*}(K_{F_{0}})+{Q_{n}^{\prime}}|_{F}}\rceil+((n-1)-2){M_{0}}|_{F}|$
gives a birational map. Thus
$|K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(n-2)M_{0}||_{F}$
gives a birational map. We are done. ∎

Next we discuss the case $\iota=n-3$.

######  Proposition 5.3.

Assume $n=\dim(V)\geq 4$, $P_{m_{0}}(V)\geq 2$ for some positive integer
$m_{0}$ and $\iota\geq n-3$. Keep the same notation as in 2.2. Let $Q_{n}$ be
any nef ${\mathbb{Q}}$-divisor on $V$ and $Q_{n}^{\prime}$ any nef
${\mathbb{Q}}$-divisor on $X^{\prime}$. Then

  * (1)

$K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(n-4)M_{0}+F$
is effective for all rational numbers $q_{n}>37(2m_{0}+1)^{n-3}$;

  * (2)

$mK_{V}+\lceil{Q_{n}}\rceil$ is effective for all integers $m\geq
37(2m_{0}+1)^{n-3}+m_{0}(n-3)+2$.

###### Proof.

Statement (1) implies (2). So we only prove (1). Similar to the assumption
$(\pounds)$, we may assume that $X^{\prime}$ is good enough (modulo blow-ups)
for our purpose. We prove by an induction on $n$.

When $n=4$, (1) is exactly Proposition 4.1(1).

Assume that (1) is correct for all varieties of dimension $n-1$. Pick a
general fiber $F$ of $f$. Because ${M_{0}}|_{F}\leq
m_{0}{K_{X^{\prime}}}|_{F}\sim m_{0}K_{F}$, we know
$\Phi_{|m_{0}K_{F}|}(F)\geq\Phi_{|{M_{0}}|_{F}|}(F)\geq n-4=\dim(F)-3$. As
long as we take those pencils $\Lambda_{F}\subset|M_{0}||_{F}$ on $F$, the
induction works on $F$. Thus we restrict everything onto $F$. By the vanishing
theorem, we may get the similar relation to (4):

$\displaystyle|K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(n-4)M_{0}+F||_{F}$
$\displaystyle\supset$
$\displaystyle|K_{F}+\lceil{q_{n}^{\prime}\sigma^{*}(K_{F_{0}})+{Q_{n}^{\prime}}|_{F}}\rceil+((n-1)-4){M_{0}}|_{F}+{M_{0}}|_{F})|$

where $q_{n}^{\prime}\geq\frac{q_{n}}{2m_{0}+1}>37(2m_{0}+1)^{(n-1)-3}$ by
Lemma 2.5 and inequality (1). The later linear system is non-empty by
induction. Therefore
$K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(n-4)M_{0}+F$
is effective for all rational numbers $q_{n}>37(2m_{0}+1)^{n-3}$. ∎

######  Theorem 5.4.

Assume $n=\dim(V)\geq 4$, $P_{m_{0}}(V)\geq 2$ for some positive integer
$m_{0}$ and $\iota\geq n-3$. Keep the same notation as in 2.2. Let $Q_{n}$ be
any nef ${\mathbb{Q}}$-divisor on $V$ and $Q_{n}^{\prime}$ any nef
${\mathbb{Q}}$-divisor on $X^{\prime}$. Then

  * (1)

$K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(n-4)M_{0}+F$
is effective for all rational numbers $q_{n}>75(2m_{0}+1)^{n-3}$;

  * (2)

$mK_{V}+\lceil{Q_{n}}\rceil$ is effective for all integers $m\geq
75(2m_{0}+1)^{n-3}+m_{0}(n-3)+2$.

###### Proof.

The proof is parallel to that of Proposition 5.3. To avoid unnecessary
redundancy, we omit the details. ∎

######  Definition 5.5.

The sequences $\\{u_{t}\\}_{t=4}^{n}$ and $\\{w_{t}\\}_{t=4}^{n}$ are defined
by the following rules:

  * •

$\widetilde{\lambda}_{n}=m_{0}$ and, for all $i<n$,
$\widetilde{\lambda}_{i}=\lambda_{i}$;

  * •

$u_{4}=75\widetilde{\lambda}_{4}+37$ and
$w_{4}=151\widetilde{\lambda}_{4}+75$;

  * •

for all $i$,
$u_{i}=\widetilde{\lambda}_{i}+u_{i-1}(2\widetilde{\lambda}_{i}+1)$ and
$w_{i}=\widetilde{\lambda}_{i}+w_{i-1}(2\widetilde{\lambda}_{i}+1)$.

Finally we study the case $\iota\leq n-4$. We begin with the case $\iota=1$.

######  Theorem 5.6.

Assume $n=\dim(V)\geq 4$ and $P_{m_{0}}\geq 2$ for some positive integer
$m_{0}$ and $\iota\geq 1$. Keep the same notation as in 2.2. Let $Q_{n}$ be
any nef ${\mathbb{Q}}$-divisor on $V$ and $Q_{n}^{\prime}$ any nef
${\mathbb{Q}}$-divisor on $X^{\prime}$. Then

  * (1)

$K_{X^{\prime}}+\lceil{\widetilde{q}_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil$
is effective for all rational numbers $\widetilde{q}_{n}>u_{n}$.

  * (2)

$|K_{X^{\prime}}+\lceil{\widetilde{q}_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil|$
gives a birational map for all rational numbers $\widetilde{q}_{n}>w_{n}$.

  * (3)

$|mK_{V}+\lceil{Q_{n}}\rceil|$ gives a birational map for all integers $m\geq
w_{n}+2.$

###### Proof.

Statement (3) is a direct result of (2). So we have to prove (1) and (2).

When $n=4$, the conditions in (1) and (2) read
$\widetilde{q}_{4}>u_{4}:=75m_{0}+37$ and
$\widetilde{q}_{4}>w_{4}:=151m_{0}+75$. Both the statements are nothing but
Proposition 4.1(1) and Theorem 4.2(1), noting that $m_{0}\pi^{*}(K_{X})\geq
M_{0}\geq F$. Besides if we take $m_{0}=\lambda(V)$, the statements are true
for $\widetilde{q}_{4}>u_{4}:=75\lambda(V)+37$ and
$\widetilde{q}_{4}>w_{4}:=151\lambda(V)+75$.

Assume that the statements are correct for $n-1$ dimensional varieties. By
definition, $m_{0}\geq\lambda(V)$ and $\lambda_{n}\geq\lambda(V)$. Because
$\widetilde{q}_{n}\pi^{*}(K_{X})\geq(\widetilde{q}_{n}-m_{0})\pi^{*}(K_{X})+M_{0}$,
we will study
$|K_{X^{\prime}}+\lceil{(\widetilde{q}_{n}-m_{0})\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+M_{0}|$.
Now the vanishing theorem gives:

$\displaystyle|K_{X^{\prime}}+\lceil{(\widetilde{q}_{n}-m_{0})\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+M_{0}||_{F}$
$\displaystyle=$
$\displaystyle|K_{F}+\lceil{(\widetilde{q}_{n}-m_{0})\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil|_{F}|$
$\displaystyle\supset$
$\displaystyle|K_{F}+\lceil{(\widetilde{q}_{n}-m_{0})\pi^{*}(K_{X})|_{F}+{Q_{n}^{\prime}}|_{F}}\rceil$
$\displaystyle\supset$
$\displaystyle|K_{F}+\lceil{\frac{\widetilde{q}_{n}-m_{0}}{2m_{0}+1}\sigma^{*}(K_{F_{0}})+{Q_{n}^{\prime}}|_{F}}\rceil|.$

Clearly $\dim(F)=n-1$, the induction hypothesis says

$K_{F}+\lceil{\frac{\widetilde{q}_{n}-m_{0}}{2m_{0}+1}\sigma^{*}(K_{F_{0}})+{Q_{n}^{\prime}}|_{F}}\rceil$

is effective when $\frac{\widetilde{q}_{n}-m_{0}}{2m_{0}+1}>u_{n-1}$ and
$|K_{F}+\lceil{\frac{\widetilde{q}_{n}-m_{0}}{2m_{0}+1}\sigma^{*}(K_{F_{0}})+{Q_{n}^{\prime}}|_{F}}\rceil|$
gives a birational map when
$\frac{\widetilde{q}_{n}-m_{0}}{2m_{0}+1}>w_{n-1}$. Both conditions can be
replaced by
$\widetilde{q}_{n}>u_{n}=\widetilde{\lambda}_{n}+u_{n-1}(2\widetilde{\lambda}_{n}+1)$
and
$\widetilde{q}_{n}>w_{n}=\widetilde{\lambda}_{n}+w_{n-1}(2\widetilde{\lambda}_{n}+1)$,
where $\widetilde{\lambda}_{n}=m_{0}$. Note however it is enough to take
$\widetilde{\lambda}_{i}=\lambda_{i}$ for all $i<n$. We are done. ∎

######  Theorem 5.7.

Assume $n=\dim(V)\geq 5$ and $P_{m_{0}}\geq 2$ for some positive integer
$m_{0}$ and $\iota\leq n-4$. Keep the same notation as in 2.2. Let $Q_{n}$ be
any nef ${\mathbb{Q}}$-divisor on $V$ and $Q_{n}^{\prime}$ any nef
${\mathbb{Q}}$-divisor on $X^{\prime}$. Then

  * (1)

$K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(\iota-1)M_{0}$
is effective for all rational numbers
$q_{n}>(2m_{0}+1)^{\iota-1}u_{n-\iota+1}$.

  * (2)

$|K_{X^{\prime}}+\lceil{q_{n}\pi^{*}(K_{X})+Q_{n}^{\prime}}\rceil+(\iota-1)M_{0}|$
gives a birational map for all rational numbers
$q_{n}>(2m_{0}+1)^{\iota-1}w_{n-\iota+1}$.

  * (3)

$|mK_{V}+\lceil{Q_{n}}\rceil|$ gives a birational map for all integers
$m\geq(2m_{0}+1)^{\iota-1}w_{n-\iota+1}+m_{0}(\iota-1)+2.$

###### Proof.

Statement (3) is direct from (2). So we only need to prove (1) and (2).

When $n=5$, one necessarily has $\iota=1$ and the statements are nothing but
those in Theorem 5.6.

First We consider statement (1). We may restrict the problem to $F$ by the
vanishing theorem. Then, since $M_{0}|_{F}\leq m_{0}\sigma^{*}(K_{F_{0}})\leq
m_{0}K_{F}$, we may study the linear system

$None$
$|K_{F}+\lceil{\frac{q_{n}}{2m_{0}+1}\sigma^{*}(K_{F_{0}})+{Q_{n}^{\prime}}|_{F}}\rceil+(\iota-2){M_{0}}|_{F}|.$

Note that $\dim\Phi_{|m_{0}K_{F}|}(F)\geq\Phi_{|M_{0}|}(F)\geq\iota-1$. Then
we can do an induction and repeat this program for finite times. Finally we
are reduced to study the non-emptyness of the linear system on $W$ of
dimension $n-\iota+1$:

$None$
$|K_{W}+\lceil{\frac{q_{n}}{(2m_{0}+1)^{\iota-1}}\tau^{*}(K_{W_{0}})+{Q_{n}^{\prime}}|_{W}}\rceil|$

where $\tau\colon W\rightarrow W_{0}$ is a contraction morphism to the minimal
model. Furthermore $\dim\Phi_{|m_{0}K_{W}|}(W)\geq\dim\Phi_{|M_{0}|}(W)\geq
1$. Now Theorem 5.6(1) says that
$\frac{q_{n}}{(2m_{0}+1)^{\iota-1}}>u_{n-\iota+1}$ is enough to secure the
non-emptyness of the linear system (6), where $u_{n-\iota+1}$ is obtained by
the sequence $\\{u_{t}\\}_{t=4}^{n-\iota+1}$ with
$u_{i}=\widetilde{\lambda}_{i}+u_{i-1}(2\widetilde{\lambda}_{i}+1)$,
$u_{4}=75\widetilde{\lambda}_{4}+37$, $\widetilde{\lambda}_{n-\iota+1}=m_{0}$
and, for all other $i$, $\widetilde{\lambda}_{i}=\lambda_{i}$. Therefore
statement (1) is correct.

Statement (1) and Lemma 2.7 allow us to reduce the problem onto lower
dimensional varieties. Thus what we are left to do is similar to that for
statement (1). So after one step restriction, we get the linear system (5) on
$F$. After successive restrictions and inductions, we may obtain the linear
system (6) on $W$ of dimension $n-\iota+1$. Now we may apply Theorem 5.6(2) to
get the condition $\frac{q_{n}}{(2m_{0}+1)^{\iota-1}}>w_{n-\iota+1}$ where
$w_{n-\iota+1}$ is obtained by the sequence $\\{w_{t}\\}_{t=4}^{n-\iota+1}$
with $w_{i}=\widetilde{\lambda}_{i}+w_{i-1}(2\widetilde{\lambda}_{i}+1)$,
$w_{4}=151\widetilde{\lambda}_{4}+75$, $\widetilde{\lambda}_{n-\iota+1}=m_{0}$
and, for all other $i$, $\widetilde{\lambda}_{i}=\lambda_{i}$. We are done. ∎

Finally we propose the following:

######  Problem 5.8.

As we have seen, inequality (1) in Section 2 is the key step to get optimal
birationality. Can one find a better constant $\gamma>\frac{1}{2m_{0}+1}$ such
that $\pi^{*}(K_{X})|_{F}\geq\gamma\sigma^{*}(K_{F_{0}})$? When $\dim(V)=3$,
$\gamma=\frac{p}{m_{0}+p}$ is nearly optimal by virtue of our previous work.

######  5.9.

Acknowledgment. This note grew out of discussions with Jun Li to whom I feel
considerably indebted. I would like to thank both Jun Li and the Mathematics
Research Center of Stanford University for the support of my visit in the
Spring of 2007\. Thanks are also due to Jungkai A. Chen, Christopher D. Hacon,
Yujiro Kawamata, Eckart Viehweg, De-Qi Zhang and Kang Zuo for their generous
helps and stimulating discussions. Finally I am grateful to the referee for
several technical suggestions.

## References

  * [1] C. Birkar, P. Cascini, C. D. Hacon, J. McKernan, Existence of minimal models for varieties of log general type, arXiv: math/0610203
  * [2] E. Bombieri, Canonical models of surfaces of general type. Inst. Hautes Etudes Sci. Publ. Math. 42 (1973), 171-219.
  * [3] J. A. Chen, C. D. Hacon, Pluricanonical maps of varieties of maximal Albanese dimension. Math. Ann. 320 (2001), 367-380
  * [4] J. A. Chen, C. D. Hacon, Pluricanonical systems on irregular 3-folds of general type. Math. Z. 255(2007), no. 2, 343-355
  * [5] J. A. Chen, M. Chen, Explicit birational geometry of 3-folds of general type, I, Ann. Sci. École Norm. Sup. (to appear). arXiv: 0810.5041
  * [6] J. A. Chen, M. Chen, Explicit birational geometry of 3-folds of general type, II, arXiv: 0810.5044
  * [7] M. Chen, Canonical stability in terms of singularity index for algebraic threefolds, Math. Proc. Camb. Phil. Soc. 131 (2001), 241-264.
  * [8] M. Chen, On the ${\bf Q}$-divisor method and its application. J. Pure Appl. Algebra 191 (2004), 143–156.
  * [9] M. Chen, K. Zuo, Complex projective 3-folds with non-negative canonical Enler-Poincare characteristic, Comm. Anal. Geom. 16 (2008), 159-182.
  * [10] C. D. Hacon, J. M${}^{\text{\rm c}}$Kernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math. 166 (2006), no. 1, 1-25
  * [11] C. D. Hacon, J. McKernan, Shokurov’s rational connectedness conjecture, Duke Math. J. 138 (2007), no. 1, 119-136
  * [12] C. D. Hacon, J. M${}^{\text{c}}$kernan, Existence of minimal models for varieties of log general type, II, preprint.
  * [13] Y. Kawamata, A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann. 261 (1982), 43-46.
  * [14] J. Kollár, Higher direct images of dualizing sheaves I, Ann. Math. 123 (1986), 11-42.
  * [15] G. Pacienza, On the uniformity of the Iitaka fibration. preprint. arXiv: 0709.0310.
  * [16] V. V. Shokurov, On rational connectedness, Math. Notes 68 (2000), 652-660.
  * [17] M. Reid, Young person’s guide to canonical singularities, Proc. Symposia in pure Math. 46(1987), 345-414.
  * [18] Y. T. Siu, A general non-vanishing theorem and an analytic proof of the finite generation of the canonical ring, arXiv:math/0610740
  * [19] S. Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math. 165 (2006), no. 3, 551-587.
  * [20] H. Tsuji, Pluricanonical systems of projective varieties of general type. I. Osaka J. Math. 43 (2006), no. 4, 967–995
  * [21] E. Viehweg, Vanishing theorems, J. reine angew. Math. 335 (1982), 1-8.
  * [22] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces. Proc. Algebraic Varieties and Analytic Varieties, Tokyo 1981. Adv. Studies in Math. 1, Kinokunya-North-Holland Publ. 1983, 329-353
  * [23] E. Viehweg, D.-Q. Zhang, Effective Iitaka fibrations, J. Algebraic Geom. 18 (2009), 711-730. arXiv:0707.4287