---
abstract: 'We prove an analogue of Fujino and Mori’s “bounding the denominators” [@fm Theorem 3.1] in the log canonical bundle formula (see also [@shok Theorem 8.1]) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a klt pair $(X,\Delta)$ of Kodaira codimension one and dimension at most three such that the coefficients of $\Delta$ are in a DCC set $\mathcal{A}$, there is a natural number $N$ that depends only on $\mathcal{A}$ for which ${\lfloor N(K_X+\Delta) \rfloor}$ induces the Iitaka fibration. We also prove a birational boundedness result for klt surfaces of general type.'
author:
- 'Gueorgui Tomov Todorov[^1]'
title: 'Effective log Iitaka fibrations for surfaces and threefolds.'
---

Introduction.
=============

Let us start by recalling Kodaira’s canonical bundle formula for a minimal elliptic surface $f\colon S\to C$ defined over the complex number field: $$K_S=f^*(K_C+B_C+M_C).$$ The [*moduli part*]{} $M_C$ is a ${\mathbb{Q}}$-divisor such that $12M_C$ is integral and ${{\mathcal O}}_C(12M_C)\simeq J^*{{\mathcal O}}_{{\mathbb P}^1}(1)$, where $J\colon C\to {\mathbb P}^1$ is the $J$-invariant function. The [*discriminant*]{} $B_C=\sum_P b_PP$, supported by the singular locus of $f$, is computed in terms of the local monodromies around the singular fibers $S_P$. Kawamata [@kaw97; @kaw98] proposed an equivalent definition, which does not require classification of the fibers: $1-b_P$ is the [*log canonical threshold*]{} of the log pair $(S,S_P)$ in a neighborhood of the fiber $S_P$.

A higher dimensional analogue consists of a log klt pair $(X,\Delta)$ and surjective morphism such that the Kodaira dimension of $K_X+\Delta$ restricted to the general fibre is zero. For now let us assume that $K_X+\Delta=f^*D$ for some ${\mathbb{Q}}$-divisor $D$ on $Y$. Then we can define the *discriminant* or divisorial part on $Y$ for $K_X+\Delta$ to be the ${\mathbb{Q}}$-Weil divisor $B_Y:=\sum_P b_P P$, where $1-b_P$ is the maximal real number $t$ such that the log pair $(X,\Delta+tf^*(P))$ has log canonical singularities over the generic point of $P$. The sum runs over all codimension one points of $Y$, but it has finite support. The *moduli part* or *J-part* is the unique ${\mathbb{Q}}$-Weil divisor $M_Y$ on $Y$ satisfying $$K_X+\Delta= f^*(K_Y+B_Y+M_Y).$$

According to Kawamata [@kaw Theorem 2](see also Ambro [@ambroSBP Theorem 0.2 (ii)] and Fujino [@fujino]) we know that on some birational model $\mu:Y'{\longrightarrow}Y$ the moduli divisor $M_{Y'}$ is nef.

Some of the main questions concerning the moduli part are the following.

\[conj\] ([@shok Conjecture 7.12]) Let $(X,\Delta)$ and $f:X{\longrightarrow}Y$ be as above and let us write as before $$K_X+\Delta=f^*(K_Y+B_Y+M_Y).$$

Then we have the following

(1)

:   (Log Canonical Adjunction) There exists a birational contraction $\mu:Y'{\longrightarrow}Y$ such that after base change the induced moduli divisor $M_{Y'}$ on $Y'$ is semiample.

(2)

:   (Particular Case of Effective Log Abundance Conjecture). Let $X_\eta$ be the generic fibre of $f$. Then $I(K_{X_\eta}+\Delta_{X_\eta})\sim 0$, where $I$ depends only on $\dim X_\eta$ and the horizontal multiplicities of $\Delta$.

(3)

:   (Effective Adjunction) There exist a positive integer depending only on the dimension of $X$ and the horizontal multiplicities of $\Delta$ such that $IM_{Y'}$ is base point free on some model $Y'/Y$.

There is a proof of the above conjecture by Shokurov and Prokhorov in the case in which the relative dimension of $f$ is one (Theorem 8.1 of [@shok]). For results towards [**(1)**]{} see Ambro [@ambro3]. Here we prove that there exist a positive integer $I$ depending only on the dimension of $X$ and the horizontal multiplicities of $\Delta$ such that $IM$ is integral when the relative dimension is one using ideas of Mori and Fujino [@fm] (see also [@ko]). The main advantage of our proof is that the number $I$ that we produce is explicitly computable.

Our main interest in Conjecture \[conj\] is because of its applications towards boundedness results for Iitaka fibrations. When $X$ is of general type the existence of a natural number $N$ such that $|NK_X|$ induces the Iitaka fibration is know by results of C. Hacon and J. M$^\textrm{c}$Kernan(cf. [@chris1]) and Takayama(cf. [@tak]) following ideas by Tsuji. Similar results in low dimension when $X$ is not of general type appear in the recent preprints [@vie; @ringler; @pacienza]. Here we address the boundedness of Iitaka fibrations in the log case.

\[appl\] Let $(X,\Delta)$ be a klt log pair of Kodaira codimension one and dimension at most three. Then there is a natural number $ N$ depending only on the coefficients of $\Delta$ such that $|{\lfloor N(K_X+\Delta) \rfloor}|$ induces the Iitaka fibration.

The proof of the above Theorem in dimension two relies on the existence of $I$ as in the Conjecture and follows the strategy in Section 6 of [@fm]. For the proof of the Theorem in dimension three we need to bound the smallest positive number $N$ such that $|N(K_X+\Delta)|$ induces a birational map for any log surface of general type with the coefficients of $\Delta$ in a DCC set $\mathcal{A}$ as a function of the DCC set only (i.e. $N=N(\mathcal{A})$). This is an interesting question in its own right (cf. [@vie]) and we address it in the last section. We can show that:

Let $(X,\Delta)$ be a klt surface and assume that the coefficients of $\Delta$ are in a DCC set $\mathcal{A}$ . Then there is a number $N$ depending only on $\mathcal{A}$ such that ${\lceil m(K_X+\Delta) \rceil}$ (and ${\lfloor m(K_X+\Delta) \rfloor}$) defines a birational map for $m\ge N$.

The above two Theorems complete the boundedness of Iitaka fibrations of klt pairs of dimension two ( for the case of Kodaira dimension zero see [@alex]). The proof is based on the fact that by a result of [@alex] (see also [@almo]) for these surfaces we have a lower bound of the volume which allows us to produce centres of log canonical singularities of a controlled multiple of $K_X+\Delta$. Using standard techniques we reduce to the case where the centres are isolated points. In order to achieve this, using ideas of M$^\textrm{c}$Kernan [@mac] and Tsuji, we produce a morphism to a curve and we use this morphism to produce the required sections (cf. [@tod]).

Preliminaries.
==============

Notations and Conventions.
--------------------------

We will work over the field of complex numbers $\mathbb{C}$. A ${\mathbb{Q}}$-Cartier divisor $D$ is nef if $D\cdot C \ge0$ for any curve $C$ on $X$. We call two ${\mathbb{Q}}$-divisors $D_1, D_2$ ${\mathbb{Q}}$-linearly equivalent $D_1\sim_{\mathbb{Q}}D_2$ if there exists an integer $m>0$ such that $mD_i$ are integral and linearly equivalent. We call two ${\mathbb{Q}}$-Cartier divisors $D_1, D_2$ numerically equivalent $D_1\equiv D_2$ if $(D_1-D_2)\cdot C=0$ for any curve $C$ on $X$. A log pair $(X,\Delta)$ is a normal variety $X$ and an effective ${\mathbb{Q}}$-Weil divisor $\Delta$ such that ${K_X}+\Delta$ is ${\mathbb{Q}}$-Cartier. A projective morphism $\mu:Y {\longrightarrow}X$ is a log resolution of the pair $(X,\Delta)$ if $Y$ is smooth and $\mu^{-1}(\Delta)\cup\{\textrm{exceptional set of } \mu\}$ is a divisor with simple normal crossing support. For such $\mu$ we write $\mu^*({K_X}+\Delta)  =K_Y+\Gamma$, and $\Gamma=\Sigma a_i\Gamma_i$ where $\Gamma_i$ are distinct integral divisors. A pair is called klt (resp. lc) if there is a log resolution $\mu:Y {\longrightarrow}X$ such that in the above notation we have $a_i <1$ (resp. $a_i\le 1$). The number $1-a_i$ is called log discrepancy of $\Gamma_i$ with respect to the pair $(X,\Delta)$. We say that a subvariety $V \subset X$ is a log canonical centre if it is the image of a divisor of log discrepancy at most zero. A log canonical place is a valuation corresponding to a divisor of log discrepancy at most zero. A log canonical centre is pure if ${K_X}+\Delta$ is log canonical at the generic point of $V$. If moreover there is a unique log canonical place lying over the generic point of V, then we say that $V$ is exceptional. LCS$(X,\Delta,x)$ is the union of all log canonical centres of $(X,\Delta)$ through the point $x$. We will denote by LLC$(X,\Delta,x)$ the set of all log canonical centres containing a point $x \in X$.

Generalities on cyclic covers.
------------------------------

\[cyc\]

Let $X$ be a smooth variety and $L$ a line bundle on $X$ and $D$ an integral divisor. Assume that $L^m\sim {\mathcal{O}_X}(D)$. Let $s$ be any rational section and $1_D$ the constant section of ${\mathcal{O}_X}(D)$. Then $1_D/s^m$ is a rational function which gives a well defined element of the quotient group $k(X)^*/(k(X)^*)^m$, thus a well defined degree $m$ field extension $k(X)(^m\sqrt{1_D/s^m})$. Let $\pi:X'{\longrightarrow}X$ denote the normalization of $X$ in the field $k(X)(^m\sqrt{1_D/s^m})$. Then

(1)

:   $\pi_*{\mathcal{O}}_{X'}=\sum_{i=0}^{m-1}L^{-i}({\lfloor iD/m \rfloor})$, and

(2)

:   $\pi_*\omega_{X'}=\sum_{i=0}^{m-1}\omega_X\otimes L^i(-{\lfloor iD/m \rfloor})$.

In particular, if $E$ is any integral divisor then the normalized cyclic cover obtained from $L^m\sim {\mathcal{O}_X}(D)$ is the same as the normalized cyclic cover obtained from $(L(E))^m\sim {\mathcal{O}_X}(D+mE)$. If $D$ has simple normal crossing support then $X'$ has only rational singularities.

DCC sets
--------

A subset $\mathcal{A}$ of $\mathbb{R}$ is said to satisfy the descending chain condition if any strictly decreasing subsequence of elements of $\mathcal{A}$ is finite. In this case we also say that $\mathcal{A}$ is a DCC set.

For the general properties of DCC sets we refer to Section 2 of [@almo].

A sum of $n$ sets $\mathcal{A}_1,\mathcal{A}_2,...,\mathcal{A}_n$ is defined as $$\sum_{i=1}^n\mathcal{A}_i=\{a_1+a_2+...+a_n|a_i\in\mathcal{A}_i\}.$$ Define also $$\mathcal{A}_\infty=\{0\}\cup\bigcup_{n=1}^{\infty}\sum_{i=1}^n\mathcal{A}.$$

If $\mathcal{A}$ is a DCC set and it contains only non-negative numbers then it is easy to see that $\mathcal{A}_\infty$ is also a DCC set.

For $\mathcal{A}\subset [0,1]$ we define the derivative set $$\mathcal{A}'=\{\frac{n-1+a_\infty}{n}|n\in\mathbb{N},a_\infty\in\mathcal{A}_\infty\cap[0,1]\}\cup\{1\}.$$

It is easy to verify that if $\mathcal{A}$ is a DCC set then so is $\mathcal{A}'$.

Bounding the moduli part.
=========================

We start by describing the moduli part as it appears in [@ko]. Let $f:(X,R){\longrightarrow}Y$ be a proper morphism of normal varieties with generic fibre $F$ and $R$ a ${\mathbb{Q}}$-divisor such that $K_X+R$ is ${\mathbb{Q}}$-Cartier and assume that $(F,R_{|F})$ is lc and that $K_F+R_{|F}\sim_{{\mathbb{Q}}}0$. Let $Y^0\subset Y$ and $X^0=f^{-1}(Y^0)$ be open subsets such that $K_{X^0}+R^0\sim_{{\mathbb{Q}}} 0$ where $R^0:=R_{|X^0}$ (cf. [@ko Lemma 8.3.4]). Write $R^0=D^0+\Delta^0$ with $D$ integral and ${\lfloor \Delta \rfloor}=0$.

Assume that $X^0, Y^0$ are smooth and $R^0$ is relative simple normal crossing over $Y^0$.

Define $V={\mathcal{O}}_{X^0}(-K_{X^0}-D^0)$. Let $m$ be (the smallest) positive integer such that $m\Delta^0$ is an integral divisor. Then we have an isomorphism $$V^{\otimes m}\cong {\mathcal{O}}_{X^0}(m\Delta^0),$$ which defines a local system $\mathbb{V}$ on $X^0\setminus R^0$ (cf. [@ko Definition 8.4.6]).

Assume also that $Y$ is smooth, $Y\setminus Y^0$ is a simple normal crossing divisor and that $R^{\dim F}f_{*}\mathbb{V}$ has only unipotent monodromies. Then the bottom piece of the Hodge filtration of $R^{\dim F}f_{*}\mathbb{V}$ has a natural extension to a line bundle $J$. Set $J(X/Y,R)$ to be the divisor class corresponding to $J$.

If the smoothness, normal crossing, and unipotency assumptions above are not satisfied, take a generically finite morphism $\pi:Y'{\longrightarrow}Y$ and a resolution of the main component $f':X'{\longrightarrow}X\times_{Y} Y'{\longrightarrow}Y'$ for which the assumptions hold and $R'$ the corresponding divisor. Then define $$J(X/Y,R)=\frac{1}{\deg \pi}\pi_*J(X'/Y',R').$$

We need the following definition.

([@ko Definition 8.4.2]) Assume that $(X,R)$ is lc and $K_X+R\sim_{\mathbb{Q}}0$ and write $R=R_{\ge 0}-R_{\le 0}$ as the difference of its positive and negative parts. Define $$p_g^+(X,R):=h^0(X,{\mathcal{O}}_X(\lceil R_{\le 0} \rceil)).$$

\[kodform\]([@ko Theorem 8.5.1]) Let $X, Y$ be normal projective varieties and let $f:X{\longrightarrow}Y$ a dominant morphism with generic fibre $F$. Let $R$ be a ${\mathbb{Q}}$-divisor on $X$ such that $K_X+R$ is ${\mathbb{Q}}$-Cartier and $B$ a reduced divisor on $Y$. Assume that

(1)

:   $K_X+R\sim f^*($some ${\mathbb{Q}}$-Cartier divisor on $Y),$

(2)

:   $p_g^+(F,R_{|F})=1$, and

(3)

:   $f$ has slc fibres in codimension 1 over $Y\setminus B$ (cf. [@ko]).

Then one can write $$K_X+R\sim_{\mathbb{Q}}f^*(K_Y+J(X/Y,R)+B_R),\textrm{   where}$$

(i)

:   $J(X/Y,R)$ is the moduli part defined above,

(ii)

:   $B_R$ is the unique ${\mathbb{Q}}$-divisor supported on $B$ for which there is a codimension $\ge 2$ closed subset $Z \subset Y$ such that $(X\setminus f^{-1}(Z),R+f^*(B-B_R))$ is lc and every irreducible component of $B$ is dominated by a log canonical centre of $(X,R+f^*(B-B_R))$.

Let $(X_1,R_1)$, $f_1:X_1{\longrightarrow}Y_1$ and $B_1$ be a pair satisfying the assumptions of Theorem \[kodform\] and $R_1$ effective on the general fibre. Assume furthermore that the relative dimension of $f_1$ is one and that $(X_1,R_1)$ is klt. Then the following holds.

\[main\] There exist an integer $N$ depending only on the horizontal multiplicities of $R_1$ such that the divisor $NJ(X_1/Y_1,R_1)$ is integral.

We are going to prove the theorem in the case when the restriction of $\Delta$ to the general fibre of $f$ is non-trivial. When the restriction is trivial the Theorem follows as in [@fm Theorem 3.1].

[**Step I.**]{} We start with some harmless reductions. Cutting by hyperplanes we can reduce to the case when $Y_1$ is a curve. From Step 1 of the proof of [@ko Theorem 8.5.1], we can reduce to the case with normal crossing assumptions, that is we can assume that $X_1, Y_1$ are smooth, $R_1+f_1^*B_1$ and $B_1$ are snc divisors, $f_1$ is smooth over $Y_1\setminus  B_1$ and $R_1$ is relative snc divisor over $Y_1\setminus  B_1$.

By Step 2 of the same proof we can assume also that $B_1=B_{R_1}$.

[**Step II - Galois cover of $Y_1$.**]{} By [@mo (4.6) and (4.7)] there is a finite Galois cover $\pi:Y{\longrightarrow}Y_1$ with Galois group $G$, such that for the induced morphism $f:X {\longrightarrow}Y$ every possible local system $R^{\dim F}f_{*}\mathbb{V}_j$ (the difference being given by a choice of isomorphism between two line bundles, compare [@ko Remark 8.4.7]) has unipotent monodromies around every irreducible component of $B$. Note that we can also arrange that $G$ acts on $X$.

Thus by [@ko Theorem 8.5.1] we have that the moduli part $L:=J(X/Y,R)$ is integral. Here $\pi_{X}^*(K_{X_1}+R_1)=K_X+R$ and $R=(\pi_{X})_*R_1$.

[**Step III - Constructing the right cyclic cover.**]{} There is a unique way to write $$R-f^*B=\Delta-G+E,$$ where $\Delta$ is effective, ${\lfloor \Delta \rfloor}=0$, the divisors $E$ and $G$ are integral and vertical.

Let $M=f^*(K_Y+L+B)-(K_X+E-G+f^*B)$. Notice that $M$ is integral and that $\Delta\sim_{\mathbb{Q}}M$. Pick $m>0$ such that $m\Delta$ is integral on the general fibre $F$. Such $m$ depends only on the horizontal multiplicities of $\Delta$. Since $m\Delta_{|F}\sim m M_{|F}$ there is an integral divisor $D$ on $X$ such that $D\sim mM$ and $D_{|F}=m\Delta_{|F}$.

Construct the cyclic covering $h:Z{\longrightarrow}X$ corresponding to $D \sim mM$ and let $X^0$ be as in Definition \[cyc\]. After possibly changing the birational model of $X$ we can assume that $D$ is simple normal crossing and $Z$ has rational singularities. We have the following diagram. $$\xymatrix{
  &               &Z\ar[dl]_h \ar@/^/[ddl]^{h'}\\
X_1\ar[d]^{f_1}&X\ar[l]_{\pi_X} \ar[d]^f&\\
Y_1           &\ar[l]_{\pi} Y&
}$$ The restriction of $\pi_X$ to $X^0$ gives one of the cyclic covers used in the construction of the local systems $R^{\dim F}f_*\mathbb{V}_j$.

We have that $$h_*\omega_{Z}=\sum_{i=0}^{m-1}{\mathcal{O}}_X(K_X+iM-{\lfloor iD/m \rfloor}).$$

[**Step IV - The G-action on $h'_*(\omega_{Z/Y})$.**]{}

We now proceed as [@fm 3.8]. By the pull-back property [@ko Proposition 8.4.9 (3)] we have that $L=\pi^*J(X_1/Y_1,R_1)$.

Let $P\in Y$ and localize everything in a neighborhood of $P_1=\pi(P)$ and $P$, and let $e$ be the ramification index at $P$. Let $z_1$ be a local coordinate for the germ $(Y_1,P_1)$ and $z=(z_1)^{1/e}$ for $(Y,P)$. Since the divisor $D$ is $\mu_e$-equivariant over an open set $Y_0\subset Y$ there is a group $G_0$ acting on $Z_{|Y_0}$ which fits in the sequence $0\rightarrow\mu_m\rightarrow G_0\rightarrow\mu_e\rightarrow 0$. In fact if locally $X$ is Spec$A$ then $Z$ is Spec$A[\phi^{1/m}]$ where $\phi$ is a local equation of $D$. Since locally $D$ is $\mu_e$-equivariant $\mu_e$ acts on $\phi$ by multiplication by $e$-th root of unity $\epsilon$ and $\mu_m$ acts on $\phi^{1/m}$ by a multiplication by an $m$-th root of unity $\varepsilon$ there is $\mu_m\rtimes\mu_e$ action on $Z$. Thus we can define a $\mu_{er}$-action on the local systems $R^1h'_*\mathbb{V}_j$ where $r=m/(m,e)$ and hence on the canonical extension $h'_*\omega_{Z/Y}\otimes\mathbb{C}(P)$. The action on the summand $
L\otimes\mathbb{C}(P)\subset h'_*\omega_{Z/Y}\otimes\mathbb{C}(P)$ is by a character $\chi_{P}$.By Lemma \[trivial\], $NJ(X_1/Y_1,R_1)$ is a divisor if and only if the character $\chi_P^N$ is trivial for every $P\in Y$.

Let $E$ be the general fibre of $h'$. Then by \[cyc\] we have that $$h^0(E,\omega_E)=h^0(F,\sum_{i=0}^{m-1}\omega_F^{1-i}(-{\lfloor i\Delta_{|F} \rfloor}))\le (m-1)^2.$$

Reasoning as in [@fm 3.8] if $l$ is the order of $\chi_P$, then $\varphi(l)\le (m-1)^2$, where $\varphi(l)$ is the Euler function. Set $N(x)=\textrm{lcm}\{l|\varphi(l)\le x\}$. Then for $N_1=N((m-1)^2)$, the divisor $N_1J(X_1/Y_1,R_1)$ is integral.

[*Remark.*]{} Note that the number above is easy to compute explicitly. This is then main advantage of our approach.

Auxiliary Lemma.
----------------

Let $Y$ be a smooth curve and let $h:Y'{\longrightarrow}Y$ be a finite Galois cover with group $G$. Let $D$ be a ${\mathbb{Q}}$-divisor on $Y$ such that $D'=h^*D$ is Cartier. For $p'\in Y'$ let $G_{p'}$ be the stabilizer. We have that $G_{p'}$ acts on ${\mathcal{O}}_{Y'}(D')\otimes {\mathcal{O}}_{P'}$ via a character $\chi_{p'}:G_{p'}{\longrightarrow}\mathbb{C}$. In this setting we have the following lemma due to Fujino and Mori [@fm].

\[trivial\](cf. [@fm]) For an integer $N$ the divisor $ND$ is integral if and only if for each $p'\in Y'$ the character $\chi_{p'}^{N}$ is trivial.

Iitaka fibrations for surfaces of log Kodaira dimension one.
============================================================

In this section we prove Theorem \[appl\] in dimension two. We start with the following lemma.

\[simDCC\] Let $(X,\Delta)$ be a klt pair of dimension $n$ where the coefficients of $\Delta$ are in a DCC set $\mathcal{A}\subset [0,1]$. Let $f:X{\longrightarrow}Y$ be a surjective projective morphism such that for the general fibre $F\cong{\mathbb{P}^1}$ we have that $(K_X+\Delta)_{|F}\sim_{\mathbb{Q}}0$. Then the set $\mathcal{B}$ of coefficients of the horizontal components of $\Delta$ is finite. In particular there is an integer $m=m(\mathcal{B})$ that clears all the denominators of the horizontal components.

We can describe $\mathcal{B}$ as the set $\{b\in\mathcal{A}|b+a=2,\textrm{for some }a\in\mathcal{A}_\infty\}$. $\mathcal{B}$ is a subset of a bounded DCC set, so it is itself a bounded DCC set. If $\mathcal{B}$ is infinite, then there is an increasing infinite sequence. But this would give a decreasing infinite sequence in $\mathcal{A}_\infty$, which is impossible since $\mathcal{A}_\infty$ is a DCC set.

\[surf\] Let $(X,\Delta)$ be a klt pair of dimension two and assume that the coefficients of $\Delta$ are in a DCC set of rational numbers $\mathcal{A}\subset [0,1]$. Assume that $\kappa(K_X+\Delta)=1$. Then there is an explicitly computable constant $N$ depending only on the set $\mathcal{A}$ such that ${\lfloor  N(K_X+\Delta)  \rfloor}$ induces the Iitaka fibration.

To prove the theorem we are free to change the birational model of $(X,\Delta)$ (without changing the coefficients of $\Delta$). So after running the Log Minimal Model Program we can assume that $K_X+\Delta$ is nef. Log abundance for surfaces implies that $K_X+\Delta$ is semiample. Therefore there exists a positive integer $k$ such that $|{\lfloor k(K_X+\Delta) \rfloor}|
$ defines the Iitaka fibration $f:X{\longrightarrow}Y$. The morphism $f:X{\longrightarrow}Y$ for $K_X+\Delta$ satisfies the hypothesis of Theorem \[kodform\], and hence we can write $$K_X+\Delta\sim_{\mathbb{Q}}f^*(K_Y+B+J).$$ By replacing the morphism $f:X {\longrightarrow}Y$ by an appropriate model we can assume that we have an isomorphism $$H^0(X,{\lfloor n(K_X+\Delta) \rfloor})\cong H^0(Y,{\lfloor n(K_Y+B+J) \rfloor})$$ for every natural number $n$ divisible by $m$ as of Lemma \[simDCC\] and $\Delta$ is simple normal crossing over the generic point of $Y$ ( cf. [@fm Theorem 4.5]). Here $Y$ is a smooth curve. The coefficients of $B$ are in a DCC set depending only on $\mathcal{A}$ (cf. [@ambroAC Remark 3.1.4]). We follow the argument in Section 6 of [@fm] to compute an integer $N$ depending only on $\mathcal{A}$ for which ${\lfloor N(K_Y+B+J) \rfloor}$ is an ample divisor. By Theorem \[main\] there is an integer $m$, depending only on the DCC set $\mathcal{A}$ by Lemma \[simDCC\], for which $mJ$ is integral. Also note that ${\lfloor B \rfloor}\ge 0$.

We treat three cases.

Case 1

:   ($g\ge2$). For $N=3m$ we obtain that deg${\lfloor N(K_Y+B+J) \rfloor}\ge 2g+1$ and so the divisor in question is ample.

Case 2

:   ($g=1$). We have that $\deg(J+B)>0$ and the coefficients of $m(J+B)$ are of the form integer plus an element in a fixed DCC set. Hence there is a positive constant $c=c(\mathcal{A})$ such that the multiplicity at of $m(J+B)$ at some point is greater than $c$. Then for $N>\frac{3}{c}$ we have that $\deg{\lfloor N(J+B) \rfloor}\ge3$.

Case 3

:   ($g=0$). In this case we have to find an integer $N$ such that deg${\lfloor N(J+B) \rfloor}-2N>0$. This follows immediately form Lemma \[DCCr\].

\[DCCr\] For any set of elements $a_i$ in a DCC set $\mathcal{A}\subset (0,1)$ such that $-2+\sum_{i=1}^na_i>0$ there is an integer $N=N(\mathcal{A})$ such that $-2N+\sum_{i=1}^n{\lfloor Na_i \rfloor}>0$.

We proceed by induction on $n$. Let $c$ be any number $0<c<\min\mathcal{A}$ and $k$ such that $0<k<\min\{\mathcal{A}_\infty\cap(2,\infty)\}-2$. The base case is $n=3$ and then it is enough to take $N>\frac{4}{k}$. In fact $${\lfloor Na_1 \rfloor}+{\lfloor Na_2 \rfloor}+{\lfloor Na_3 \rfloor}\ge
{\lfloor Na_1 \rfloor}+{\lfloor Na_2 \rfloor}+{\lfloor 2N \rfloor}-{\lfloor N(2-a_3) \rfloor}-1.$$ But $N(a_1+a_2+a_3-2)>4$ hence ${\lfloor Na_1 \rfloor}+{\lfloor Na_2 \rfloor}-{\lfloor N(2-a_3) \rfloor}>2$ and so the desired inequality follows.

For the inductive step suppose that $\sum_{i=1}^na_i\ge 3$ and order the $a_i$ so that $a_i\le a_{i+1}$. Then $\sum_{i=1}^{n-1}a_i>2$ and the assertion follows by induction. If not we have that $\sum_{i=1}^na_i < 3$ and hence $n<\frac{3}{c}$. It suffices to take $N>\frac{3+c}{ck}>\frac{n+1}{k}$ since then $$\sum_{i=1}^n{\lfloor Na_i \rfloor}-2N\ge {\lfloor \sum_{i=1}^nNa_i-2N \rfloor}-n+1\ge 2.$$

Iitaka fibration for threefolds of log Kodaira dimension two.
=============================================================

In this section we complete the proof of Theorem \[appl\] by proving it in dimension three.

\[3folds\] Let $(X,\Delta)$ be a klt pair of dimension three and assume that the coefficients of $\Delta$ are in a DCC set of rational number $\mathcal{A}\subset [0,1]$. Assume that $\kappa (K_X+\Delta)=2$. Then there is a constant $N$ depending only on the set $\mathcal{A}$ such that ${\lfloor  N(K_X+\Delta)  \rfloor}$ induces the Iitaka fibration.

Performing the same type of reductions in the proof of Theorem \[surf\] we assume that we are in the case when we have a morphism $f:X{\longrightarrow}Y$ where $Y$ is a surface, $\Delta_{|F}$ is non-trivial and we have an isomorphism $
H^0(X,{\lfloor n(K_X+\Delta) \rfloor})\cong H^0(Y,{\lfloor n(K_Y+B+M) \rfloor})
$ for every $n$ sufficiently divisible. Here the divisor $K_Y+B+M$ is big, the coefficients of $B$ are in a DCC set depending only on $\mathcal{A}$ (cf. [@ambroAC Remark 3.1.4]), $M$ is nef. Now take $n$ also divisible by by $l$ where $l$ is an integer such that $lM$ is integral and $|lM|$ is base point free. The integer $l$ depends only on the DCC set $\mathcal{A}$. Such $l$ exists by the case of Conjecture \[conj\] that is proven in [@shok Theorem 8.1].

Notice that $(Y,B)$ is klt by [@ambroSBP Theorem 3.1] and also by [@shok Corollary 7.17]. The divisor $lM$ is base point free so we can replace it with a linearly equivalent divisor in $M'$, such that the the pair $(Y,B+\frac{1}{l}M')$ is klt and $H^0(Y,{\lfloor n (K_Y+B+\frac{1}{l}M' \rfloor}) = H^0(Y,{\lfloor n(K_Y+B+M) \rfloor})$ for every natural number $n$ divisible by $l$.

Now define the DCC set $\mathcal{B}=\mathcal{A'}\cup\{\frac{1}{l}\}$. Observe that $\mathcal{B}$ depends only on $\mathcal{A}$. Define $B_1=B+\frac{1}{l}M'=\sum_i b_iB_i$ where $B_i$ are distinct irreducible divisors. By [@almo Theorem 4.6] there is a computable constant $\beta$ that depends only on $\mathcal{B}$ such that $K_Y+(1-\beta)B_1$ is a big divisor. Let $b$ be the minimum of the set $\mathcal{B}$ and let $k={\lceil \frac{1}{b\beta} \rceil}$. Then define $B'=\sum_ib_i'B_i$ where $b_i'=\frac{{\lfloor kb_i \rfloor}}{k}$. We have that the divisor $K_Y+B'$ is big with coefficients in the DCC set $\mathcal{C}=\{\frac{i}{k}|i=1,\ldots, k-1\}$. Also we have the inclusion $H^0(Y,{\lfloor m(K_Y+B') \rfloor}) \subset H^0(Y, {\lfloor m(K_Y+B+\frac{1}{l}M') \rfloor})$ for every $m$.

Now Theorem \[birsurf\] implies that there is a number $N'$ depending only on $\mathcal{A}$ such that ${\lceil m(K_Y+B') \rceil}$ defines a birational map for $m\ge N'$. Define $N=kN'$. Then we have that $H^0(Y,{\lceil N(K_Y+B') \rceil})=H^0(Y,{\lfloor N(K_Y+B') \rfloor})\subset H^0(Y,{\lfloor N(K_Y+B+\frac{1}{l}M') \rfloor})$ and hence the theorem follows.

Birational boundedness for log surfaces of general type.
========================================================

In this section we prove that for a surface pair $(X,\Delta)$ of log general type with the coefficients of $\Delta$ in a DCC set $\mathcal{A}$ there is a number $N$ depending only on $\mathcal{A}$ such that the linear system $|{\lceil N(K_X+\Delta) \rceil}|$ gives a birational map. Again by [@alex] or [@almo Theorem 4.8] we have that vol$(K_X+\Delta)>\alpha^2$ for some $\alpha$ depending only on the DCC set $\mathcal{A}$. We are going to use this lower bound of the volume to create a log canonical centre. The good case is when the volume of the restriction of $K_X+\Delta$ to the log canonical centre is large. Then we can proceed by cutting down the log canonical centre to a point and we generate a section of an appropriate multiple of $K_X+\Delta$. If the volume of the restriction is smaller then we are going to proceed as in [@tod].

\[birsurf\] Let $(X,\Delta)$ be a klt surface of log general type and assume that the coefficients of $\Delta$ are in a DCC set $\mathcal{A}\subset{\mathbb{Q}}$. Then there is a number $N$ depending only on $\mathcal{A}$ such that ${\lceil m(K_X+\Delta) \rceil}$ defines a birational map for $m\ge N$.

Consider a log resolution $f:X'{\longrightarrow}X$ of $(X,\Delta)$ and write $f^*(K_X+\Delta)=K_{X'}+(f^{-1})_*\Delta+\sum_ie_iE_i$ with $E_i$ exceptional. There is a natural number $n$ such that $e_i<1-\frac{1}{n}$ for every $i$. Define $\Delta'=(f^{-1})_*\Delta+\sum(1-\frac{1}{n})E_i$. Since we have the inclusion $H^0(X',{\lceil m(K_{X'}+\Delta') \rceil})\subset H^0(X,{\lceil m(K_X+\Delta) \rceil})$ by replacing the $\mathcal{A}$ with the DCC set $\mathcal{A}\cup\{1-\frac{1}{n}|n\in\mathbb{N}\}$ we can assume that $X$ is smooth.

By [@alex] or [@almo Theorem 4.8] we have that vol$(K_X+\Delta)>\alpha^2$ for some $\alpha$ depending only on the DCC set $\mathcal{A}$. Take a Zariski decomposition $K_X+\Delta \sim_{\mathbb{Q}}A +E $ with $A$ nef and $E$ effective and $A$ orthogonal to each component of $E$. We have that vol$(K_X+D)=$vol$(A)>\alpha^2$

Choose two general points $x_1,x_2 \in X$. Arguing as in [@tak Lemma 5.4 and Lemma 5.5] we can produce a divisor $D_1 \sim a_1 A$, with $a_1 < \frac {\sqrt{2}}{\alpha}$ such that there is a non-empty subset $I_1$ of $\{1,2\}$ with the following property:

(\*)

:   $(X,D_1)$ is lc but not klt at $x_i$ for $i \in I_1$ and not lc at $x_i$ for $i \notin I_1$.

With this choice of $a_1$ we can furthermore assume that either codim Nklt$(X,D_1)=2$ at $x_i$ for $i\in I_1$ or Nklt$(X,D_1)=Z\cup Z_+$ such that $Z$ is irreducible curve and $x_i$ is in $Z$ but not in $Z_+$ for $i \in I_1$.

Assuming that $Z\cdot A > c$ for some constant $c$ and still following [@tak Lemma 5.8] we can produce a divisor $D_2\sim a_2 A$ with $a_2 <c+\epsilon+a_1$ such that there is a subset $I_2$ of $\{1,2\}$ with the property that $(X,D_2)$ is lc but not klt at $x_i\in I_2$ and not lc at $x_i$ for $i \notin I_2$ and codim Nklt$(X,D_2)=2$ at $x_i$ for $i \in I_2$.

Now if we set $G=D_2+(m-1-a_2-\epsilon)A+(m-1)E+F$ where $0<\epsilon\ll 1$ and $F={\lceil (m-1)K_X+m\Delta \rceil}-(m-1)K_X-(m-1)\Delta$ we observe that ${\lceil (m-1)K_X+m\Delta \rceil}-G\sim_{\mathbb{Q}}\epsilon A$. Since $A$ is an ample divisor Kawamata-Viehweg vanishing implies that $H^1(X,{\lceil m(K_X+\Delta) \rceil}\otimes{\mathcal{J}}(G))=0$ for $m>a_2+1$ and hence the linear system $|{\lceil m(K_X+\Delta) \rceil}|$ gives a birational map onto its image (cf. [@raz Chapter 9]).

Thus we can now assume that for every general point $x\in X$ we have a pair $(D_x,V_x)$, such that $D_x \sim a_1 A$, $V_x$ is a pure log canonical centre of $D_x$, and dim $V_x = 1$. By [@mac Lemma 3.2] we have a diagram $$\xymatrix{
X'\ar[d]^f \ar[r]^\pi & X \\
B&
}$$ where $\pi$ is dominant and generically finite morphism of normal projective varieties, and the image of the general fibre of $f$ is $V_x$ for some $x\in X$.

Arguing as in section 3 of [@tod] we can assume that the map $\pi$ is birational. In fact if $\pi$ is not birational we have at least two centres of log canonical singularities through a general point. Replacing each such pair of centres with a *minimal* centre we may assume that the dimension of the centres is zero and this way ${\lceil m(K_X+\Delta) \rceil}$ gives a birational map onto its image for $m>3a_1+1$ (compare [@tod page 11]).

Thus we consider the case when $\pi$ is birational. We replace $X$ with a model on which $K_X+\Delta$ is nef and big. To complete the proof we will show that the degree of the restriction of $K_X+\Delta$ to a log canonical centre through a general point on an appropriate model is bounded from below by a constant that depends only on the DCC set $\mathcal{A}$. This is enough since we can apply Kawamata-Viehweg vanishing as before before to produce sections with the desired properties and hence a birational map.

If $X {\longrightarrow}B$ is not a morphism (over a general point $b \in B$ then there is a point $x\in X$ such that we have at least two pairs $(D_1,V_1)$ and $(D_2,V_2)$, such that $D_i \sim a_1 (K_X+\Delta)$, $x\in V_i$ a pure log canonical centre of $K_X+\Delta+D_i$ of dimension 1 and $V_1\neq V_2$ corresponding to two general fibres of $f$. If $x$ is a smooth point, we have that $$(K_X+\Delta)\cdot V_1=\frac{1}{a_1}D_2\cdot V_1 \ge \frac{1}{a_1} V_2\cdot V_1 \ge\frac{1}{a_1}$$ since $V_1^2\ge 0$. If $x$ is not smooth then $(X,\Delta)$ is not terminal at $x$ and so there is a projective birational morphism $\pi:X'{\longrightarrow}X$ extracting a divisor of discrepancy less than or equal to zero. Therefore $\pi^*(K_X+\Delta)=K_{X'}+\Delta'$ where $\Delta'\ge 0$ and $K_{X'}+\Delta'$ is still nef and big. Since there are only finitely many divisors of non-positive discrepancy after finitely many extractions as above we may assume that there is a morphism $f:X'{\longrightarrow}B$. Thus we may write $\pi^*(K_X+\Delta)=K_{X'}+\Delta'$. Here $\Delta'$ is effective and $K_{X'}+\Delta'$ is nef and big. Now let $\beta=\beta(\mathcal{A})$ be as defined in 3.5 of [@almo]. We can assume that every $\pi$-exceptional divisor that dominates $B$ appears with a coefficient grater than $1-\beta$ in $K_{X'}+\Delta'$. In fact suppose that this is not the case for an exceptional divisor $E$. Away from the intersection of $E$ with the other components of $\Delta'$ the divisor $E$ intersects two general fibres $F_1$ and $F_2$ corresponding to two log canonical centres as before. With this choice there are no other log canonical places of $K_X+\Delta+D_1+D_2$ lying over $\pi(E)$ connecting the intersection of $E$ with $F_1$ and $F_2$. Then by the Connectedness Principle $E$ is a log canonical place for $K_X+\Delta+D_1+D_2$. In particular mult$_E\pi^*D_i\ge \frac{\beta}{2}$ for at least one of the $D_i$, say $D_1$, and hence $(K_X+\Delta)\cdot V_1>\frac{\beta}{2a_1}$.

Now take a log resolution $g:X''{\longrightarrow}X'$ of $(X',\Delta')$ and let $f'=f\circ g$ and write $K_{X''}+\Delta''+\sum e_iE_i+N_1=g^*(K_{X'}+\Delta')+N_2$ where $\Delta'''+\sum e_iE_i+N_1$ and $N_2$ are effective with no common components, $\Delta''$ is the strict transform of $\Delta$ and the $E_i$ are the strict transforms of the $\pi$-exceptional divisors that dominate $B$ with $g_*(\Delta''+\sum e_iE_i+N_1)=\Delta'$ ($N_1$ and $N_2$ do not intersect the general fibre of $f'$). The divisor $K_{X''}+\Delta''+\sum E_i+{\lceil N_1 \rceil}$ is big and $(X'', \Delta''+\sum E_i+{\lceil N_1 \rceil})$ is lc with the coefficients in a DCC set that depends only on $\mathcal{A}$ and so by [@almo Theorem 4.6] the divisor $K_{X''}+\Delta''+(1-\beta)\sum E_i+{\lceil N_1 \rceil}$ is still big. Hence for the general fibre $F'$ of $f'$ we have that deg$(K_{X''}+\Delta''+(1-\beta)\sum E_i+{\lceil N_1 \rceil})_{|F'}=\deg (K_{X''}+\Delta''+(1-\beta)\sum E_i)_{|F'}\ge c>0$ where $c$ depends only on $\mathcal{A}$. Now $g_*(\Delta''+(1-\beta)\sum E_i)\le \Delta'$ and so $(K_{X'}+\Delta')_{|F}\ge c$. Since $K_{X'}+\Delta'=\pi_*(K_X+\Delta)$ it follows that $(K_X+\Delta)\cdot V_1\ge c$.

Let $(X,\Delta)$ be a surface klt pair of log general type and assume that the coefficients of $\Delta$ are in a DCC set $\mathcal{A}$. Then there is a number $N$ depending only on $\mathcal{A}$ such that ${\lfloor N(K_X+\Delta) \rfloor}$ defines a birational map.

Change the coefficients as in the last part of the proof of Theorem \[3folds\] and reduce to the case in which all the denominators of the coefficients of $\Delta$ are the same. Then by taking an appropriate multiple proceed with integral divisors only.

[\[G-K-MP98\]]{}

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[^1]: The author would like to thank Professor Christopher Hacon for suggesting the problem and many useful conversations and suggestions.
