On accumulation points of volumes of log surfaces
Valery Alexeev Wenfei Liu



Valery Alexeev
Department of Mathematics
 University of Georgia
 Athens
GA 30605
 USA
valery@math.uga.edu

Wenfei Liu 
School of Mathematical 
Sciences
 Xiamen University
Siming South Road 422
Xiamen
 Fujian 361005
 P. R. China
wliu@xmu.edu.cn
[2010]Primary 14J29; Secondary 14J26, 14R05




  Let ⊂(0,1] be a set satisfying the descending chain
  condition. We show that any accumulation point of volumes of log
  canonical surfaces (X, B) with coefficients in  can be
  realized as the volume of a log canonical surface with big and nef
  K_X+B and coefficients in ∪{1}, with at least
  one coefficient in ()∪{1}.  As a corollary, if
  ⊂ then all accumulation points of volumes
  are rational numbers, solving a conjecture of Blache.  For the
  set of standard coefficients
  _2={1-1/n| n∈}∪{1} we prove
  that the minimal accumulation point is between
  1/7^2· 42^2 and 1/42^2.

On accumulation points of volumes of log surfaces
    Wenfei Liu
    March 25, 2018
=================================================






§ INTRODUCTION

Let (X, B) be a projective log surface with log canonical 
singularities. The volume (K_X+B) measures the
asymptotic growth of the space of global sections of pluri-log-canonical
divisors:

    h^0(X, ⌊ m(K_X+B)⌋) = (K_X+B)/2 m^2 + o(m^2)

One says
that K_X+B is big if (K_X+B)>0.  If this is the case, let
π (X, B)→ (X_, B_) be the contraction to
the log canonical model. Then π^*(K_X_+B_) is the
positive part of K_X+B in the Zariski decomposition and
 
    (K_X+ B)=(K_X_+B_)=(K_X_+B_)^2.
 The log canonical divisor K_X_+B_ is ample, but not Cartier in general. Hence (K_X+B), being positive, is not necessarily an integer. 
 
 Assume that ⊂ (0,1] and let () be the set of log
 canonical projective surfaces (X, B) such that the coefficients of
 B belong to . Using the minimal model program for log
 surfaces, one has the following equalities of sets of volumes

    () :   ={((K_X+B) | (X, B) ∈(), K_X+B  big}
       ={(K_X+B)^2 | (X, B) ∈(), K_X+B   big and nef}
        = {(K_X+B)^2 | (X, B) ∈(), K_X+B  ample}


The three most commonly used sets of coefficients are
_0=∅, _1 = {1}, and
_2={1-1/n| n∈}∪{1}. The latter set is
called the set of standard coefficients. It naturally appears when one
studies groups of automorphisms of smooth varieties and in the
adjunction formula. An easy observation is that these three sets
satisfy the descending chain condition (DCC).
 
It is a fundamental result of the first author that if  is a DCC
set then the set () is also DCC
(<cit.>).  The DCC is an
assertion about the accumulation points of ().  Blache
<cit.> and Kollár
<cit.> have constructed several
examples of accumulation points of volumes of log surfaces without
boundary, and recently Urzúa and Yáñez
<cit.> exploited Kollár's construction
to show that all natural numbers are actually accumulation points of
(_0). Yet the overall picture of the accumulation points is
unknown even for _0.

We recall the definition of an nklt center of (X,B) in
Section <ref>. We call an nklt center
accessible
if it is not a simple elliptic singularity P∈ X or
a smooth curve B_0⊂ B disjoint from (X) and the rest of
B.  The first main result of this paper is the following.


  Suppose that ⊂(0,1] satisfies the descending chain
  condition.  Then v_∞∈_>0 is an accumulation point of
  () if and only if there exists a log canonical surface 
  (X, B)∈(∪{1}) such that
 

  * K_X+B is ample, and v_∞=(K_X+B)^2.

  * One of the following conditions is satisfied:


  * The set of codiscrepancies of divisors over X with
  respect to (X, B) contains an accumulation point of .

  * (X,B) has an accessible nklt center. 


  * If 1 is not in the closure  of   then each irreducible component of ⌊ B⌋ has geometric
  genus at most 1. 




  An equivalent condition for (ii.a) is that there exists a log
  surface (X,B)∈(∪ 1) with big and nef K_X+B and a
  coefficient of B is in (). 


In Section <ref> we show the "if part" of
Theorem <ref> by constructing a sequence of log canonical
surfaces with volumes converging to v_∞. The harder part of the
theorem is the "only if part". The proof is carried out in
Section <ref>. 
There are several interesting consequences of Theorem <ref>.

Let ⊂(0,1] be a DCC set. Then (())⊂(∪{1}), where (·) denotes the set of
accumulation points. In particular, if ⊂ then
(())⊂.



  Theorem <ref> gives an easy way to find many accumulation
  points of (). E.g., it easily follows that
  _>0⊂((_0)), cf. Example <ref>.

  Corollary <ref> and Example <ref> prove one half of a
  conjecture of Blache <cit.>.  On the other hand, the second
  half <cit.> saying
  that min((_0))= 1 is false. For example, the surface
  in <cit.> of volume 1/462
  has a non-empty accessible nklt locus as in Theorem <ref>(ii.b), and one infers that 1/462 is an accumulation
  point of (_0).

  The fact that _>0⊂((_0)) and that there are
  many accumulation points smaller than one was also established in
  the recent work of Urzúa and Yáñez
  <cit.>.





  Let ⊂(0,1] be a DCC set such that () = {1},
  e.g. =_2.  Then (())=_(),
  where
  
    _() = {(K_X+B)^2 | (X, B)
      ∈(),  K_X+B  ample, (X, B)≠∅}.

  If  is finite, e.g. =_0 or _1,
  then (()) ⊂_().


Our second main result is an explicit lower bound for
((_2)) for the standard coefficient set
_2 = {1-1/n,  n∈}∪{1}. In Section <ref> we prove:

  One has 
  min(_2) =
  min_(_2)≥1/86436 = 1/7^2·42^2. 


On the other hand, 1/42^2 is an accumulation point of
(_2) by applying Theorem <ref>(ii.b) to
<cit.>. Overall, our bounds for
the minimal accumulation points of the three most commonly used sets
are as follows:


  The following is true:
  

    
  * For =_0, _1 and _2, ones has
    min(())= min_().

  
  * One has
    1/86436 = 1/7^2·42^2≤min_(_2) ≤1/1764 = 1/42^2.
    
  
  * For =_0 or _1, one has
    1/86436 = 1/7^2·42^2≤min_() ≤1/462 = 1/11· 42.    
  




  <cit.> gives an effective lower
  bound for () for any DCC set , which is however too
  small to be realistic.  For the sets _0,_1,_2 it works
  out to about 10^-3· 10^10,
  cf. <cit.>.  Our bounds for
  ((_2)) are considerably "larger".


In Section <ref> we discuss the iterated accumulation
points of ^2(), i.e. accumulation points of accumulation
points etc. In particular, we establish that its “accumulation
complexity” is unbounded already for the simplest set
_0=∅.


  The work of the first author was partially supported by NSF under
  DMS-1603604. He would like to thank Christopher Hacon and Chenyang
  Xu for useful discussions.
The second author was partially supported by the NSFC
  (No. 11501012, No. 11771294) and by the Recruitment Program for
  Young Professionals. He benefited from discussions with Professor
  Meng Chen during a visit to Fudan University in September
  2017. Thanks go to Giancarlo Urzúa for sending us the manuscript
  <cit.>.




§ PRELIMINARIES

 

Throughout the paper, we work over an algebraically closed field of
arbitrary characteristic.
We recall the standard definitions of the MMP, see
e.g. <cit.>. 
We will only need them for surfaces.
Given an -divisor B=∑_j b_j B_j on a normal surface X and  a real number a we write

  B^= a = ∑_b_j= a b_jB_j,   B^>a = ∑_b_j>a b_jB_j,  B^<a = ∑_b_j<a b_jB_j.   


A log surface consists of a normal projective surface X and
effective -divisor B
such that K_X+B is -Cartier.  Let
f Y→ X be a morphism from a normal surface Y.  Then we
frequently denote by B_Y the -divisor on Y defined by the
identity K_Y+B_Y = f^*(K_X+B).

A log resolution is a resolution of singularities
f Y→ X such that (f)∪ f^-1_*B has a
simple normal crossing support, where (f) is the exceptional
locus of f and f^-1_*B is the strict transform of B on Y. We
can write

    K_Y = f^*(K_X+B) +∑_i a_iE_i.

One usually writes a_i=a(E_i, X,B) and calls it the discrepancy
of E_i with respect to (X,B). One says that (X, B) is log
  canonical (resp. Kawamata log terminal) if a_i≥ -1
(resp. a_i>-1) for every i. The numbers
b(E_i, X, B):=-a(E_i, X, B) are called codiscrepancies. We
use abbreviations lc and klt for log canonical and
Kawamata log terminal respectively.
Thus, (X,B) is lc, resp. klt, if the
coefficients in B_Y are ≤1, resp. are <1.

A nonklt, or nklt center of a log surface (X,B) is an image
of a divisor E on Y whose coefficient in B_Y is ≥1,
i.e. =1 in the log canonical case. In other words, E is either a
component of B=∑ b_jB_j with b_j≥ 1 or an image of an
exceptional divisor with  a(E, X,B)≤ -1. This definition does not
depend on a log resolution.
The nklt locus of (X,B), denoted by (X, B),
is the union of all nklt centers.

A log surface (X, B) is dlt if it is log canonical, and there exists
a finite collection of points S⊂ X such that
(X∖ S,  B∖ S) is a smooth normal crossing
pair, and there are no nklt centers contained in S.


  Let (X, B = ∑ b_jB_j) be a log surface, b_i≥0. We do not
  assume that it is 
  lc and we allow some coefficients to be
  b_i>1, but K_X+B is -Cartier.  A dlt blowup of
  (X,B)
  is a partial resolution of singularities f Y→ X such that
  
  
  * Y is -factorial.
  
  * The pair 
    (Y, ∑_j min(1,b_j)f_* B_j + ∑_i E_i) is dlt. Here,
    E_i are the exceptional divisors of f.
  
  * One has B_Y≥ f_* B + ∑ E_i, i.e. all discrepancies
    are a(E_i,X,B)≤ -1.
  
  If (X,B) is lc then it follows that   B_Y = f_* B+∑ E_i.


It is a result of Hacon that dlt blowups exist in any dimension, see
e.g. <cit.>.  For
surfaces, this is an elementary fact: a dlt blowup is obtained by
taking a log resolution followed by contracting back the exceptional
curves with discrepancy a(E,X,B)>-1. The configuration of such
curves is log terminal, hence rational, and the contraction exists by
<cit.>.


  We will say that f Y→ X is an effective resolution
  of a log surface (X,B) if Y is smooth and B_Y≥0, i.e. the
  discrepancies are a_i≤0. For example, the minimal resolution of
 surface singularities is effective.



  Let (X,B) be a log canonical surface, and let Z be an nklt
  center (so a point or a curve). Let f Y→ X be a log
  resolution, and E an irreducible component of B_Y^=1 on Y such that Z=f(E).  We say
  that Z is an inaccessible nklt center if
  E is disjoint from the rest of B_Y^>0. We call Z a
  accessible nklt center otherwise.  It is easy to see that
  this definition does not depend on the choices of the log resolution f and of the component E.



One sees easily that a log canonical surface (X, B) has nonempty accessible nklt locus if and only if B_Y^>0 is singular at some point of B_Y^=1.


From the classification of log canonical singularities (see
e.g. <cit.>) it follows that an nklt center
is inaccessible iff it is a point p∈ X which is a simple
elliptic singularity with p∉ B, or if it is a smooth curve B_j
which appears in B with coefficient b_j=1, lies in the smooth part of
X, and is disjoint from the rest of B.


  Let Z be an accessible nklt center of a log canonical surface
  (X,B). Then there exists an effective resolution of singularities
  f Y→ X such that Z=f(B_1), B_Y=∑_i=1^n b_iB_i, n≥2,
  b_1=1, b_2>0, and B_1∩ B_2∅.


  Y can be taken to be the minimal resolution of a dlt
  blowup of (X,B).




A set ⊂ is called a DCC (resp. ACC) set if it satisfies
the descending (resp. ascending) chain condition: a strictly
decreasing (resp. increasing) subsequence of  terminates.


  Let ⊂ (0,1]. The derivative set
  is defined as follows:
  
    ' = {
          1 -  1- ∑ n_j b_j/m|
          b_j∈,  m,n_j∈,
           1- ∑ n_jb_j ≥0
        }∪{1}.

    It is easy to see that if  is a DCC set then so is
    '. Note that _0'=_1'=_2'=_2. 


One standard way the derivative set appears is the adjunction formula
<cit.>. Let (Y,  E+Δ)∈()
be a log canonical  surface

with E a reduced curve. 
In particular, K_Y+E+Δ is
an -Cartier and the curve E is at worst nodal. Then the
restriction of K_Y+E+Δ to E is K_E + _E(Δ), where
_E(Δ) = ∑ b_k' Q_k, called the different, is an
effective divisor with coefficients b'_k∈'.  
The explicit formula of the coefficients b'_k is as follows.


  Let (Y, E + ∑ b_jΔ_j) be a log canonical surface as above.  Then
  for each k
  
  
  * either b'_k=1,
  
  * or Q_k∈ Y is smooth and m=1, n_j= (E·Δ_j)_Q_k
    in equation (<ref>),

  * or Q_k∈ Y is a singularity and the exceptional locus of its
  minimal resolution g Z→ Y is a chain of curves
  F_1, …, F_r with F_i^2=-p_i≤ 2, and EF_1=1, EF_i=0
  for i>1. Then
  
  
  * m= (p_1,…, p_r) is the determinant of the matrix M
    corresponding to the resolution, with p_i on the main diagonal and
    -1 next to the main diagonal. The number m is the index of Q_k∈ Y.
  
  * n_j = ∑_i=1^r (f_*Δ_j· F_i) (p_i+1, …,
    p_r). Here, =1 if i=r. 
  

  




  Assume that Q_k∈ Y is singular and that b'_k<1. By the
  classification of log canonical surface singularities with
  coefficients in {1}  (<cit.>),
  the resolution graph of Q_k∈ Y is a chain. The log discrepancies
  (1+f_i) of K_Y+E+Δ along F_i are solutions to a
  r× r system of linear equations with the matrix M. The
  formula for M in terms of cofactors gives the stated formula.
  The number (p_i+1, …, p_r) is the minor of M obtained
  by removing the first column and i-th row.


In particular, we have the following well known fact:

  
  For the set _2, if b_k' = 1-1/n<1 then n=mn', where
  m=(p_1, …, p_r) is the index of Q_k∈ Y, and 
  
  
  * either g_*Δ intersects only the last curve
    F_r, once, and the corresponding coefficient of g_*Δ
    equals 1-1/n';
  
  * or g_*Δ is disjoint from ∪ F_i, and
    m=n, n'=1.
  
  In particular, n(K_Y+E+Δ) is integral and Cartier in a
  neighborhood of Q_k.




We now introduce an auxiliary ACC set. 


  Let ⊂(0,1] is a DCC set and m∈. Then the following
  set
  
    T_m() =  {1-b/{mb}| b∈, {mb}0
           }∪{1}

  is an ACC set and thus attains the maximum t_m()∈_≥1.
  For any b∈ with { mb }0 one has
    b + t_m {mb}≥ 1. Here, {x} denotes the fractional part of x.


  For =_0, _1 or _2, one has t_m=1 for all m.


The proofs of these lemmas are straightforward, and are skipped.
  









§ CONSTRUCTION OF ACCUMULATION POINTS


In this section, we prove one direction of Theorem <ref>: if
there exists a log canonical surface (X,B)∈(∪{1}) with ample
K_X+B satisfying the conditions (ii) and (iii) then (K_X+B)^2 is
an accumulation point of ().



 §.§ Accumulation points of volumes due to accumulating coefficients



Let ⊂ (0,1] be a DCC set. Let (X, B)∈() such that K_X+B big and nef, and at least one coefficient of B is an accumulation point of . Then (K_X+B)^2 is an accumulation point of (). 


  Write B=∑_j∈ J b_j B_j. Let J_∞ be the set of the
  indices j such that b_j is an accumulation point of . Then
  J_∞ is nonempty by assumption. Since bigness is an open
  condition on the N^1(X)_, the space of numerical classes of
  -divisors on X, there is a strictly increasing sequence
  b_j^(s)∈ converging to b_j for each j∈ J_∞ such
  that K_X+B^(s) is big, where

    B^(s):=∑_j∈ J_∞ b_j^(s)B_j +∑_j∈ J-J_∞ b_j B_j.

Since (X, B) is log canonical and 0≤ B^(s)< B for any s,
the log surfaces (X, B^(s)) are all log canonical. By construction,
(X, B^(s))∈().
Since K_X + B is nef, one has (K_X+B^(s)) < (K_X+B). 
Finally, since the volume is a continuous function on N^1(X)_, we have

    lim_s→∞(K_X+B^(s)) = (K_X+ B) =(K_X+B)^2.






 §.§ Accumulation points of volumes due to nklt loci




  We denote by _B
  the set of coefficients appearing in B.



  Let (X, B) be a log canonical surface with ample K_X+B and
  nonempty accessible nklt locus.  For any ϵ>0, there is a
  log canonical surface (X', B')∈(_B) with ample K_X'+B'
  such that


  * X' birational to X and B' is the strict transform of B, and

  * (K_X+B)-ϵ < (K_X'+B') <(K_X+B).
 



The construction appeared in <cit.> when the coefficient set is _1.


  Consider a resolution as in Lemma <ref>
  and K_Y+B_Y = f^*(K_ X+B).  By blowing up a point of
  intersection of B_1 and B_2 several times, if necessary, we can
  assume that f is effective, and B_1,B_2 meet transversally at a
  single point p∈ Y.

  In our notation, b_1=1 and b_2>0.
  Let us blow up p∈ Y and then
  its preimage on the strict transforms of B_2 on the blown-up
  surfaces. Let h Y^(s)→ Y be the resulting morphism
  after s blow-ups. 
  The inverse image of B_1+B_2 on Y^(s) has the
  following dual graph:


  
    
           [inner sep=0pt](a0) at (0,0) ⊗;
                      [inner sep=0pt](a5) at (5,0) ⊗;
    
      [draw, inner sep=1pt,fill=white,circle] (a1) at (1,0) ;
      [draw, inner sep=1pt,fill=white,circle] (a2) at (2,0) ;
            [draw, inner sep=1pt,fill=white,circle] (a3) at (3,0) ;
      [draw, inner sep=1pt,fill=white,circle] (a4) at (4,0) ;

      [below of=a0, node distance=1em] B_1^(s);
           [below of=a5, node distance=1em] B_2^(s);

      [above of=a1, node distance=1em](1);
      [below of=a1, node distance=1em]E_s;
      [above of=a2, node distance=1em](2);
      [below of=a2, node distance=1em]E_s-1;
      
    
     
     
      [above of=a3,node distance=1em](2);
      [below of=a3,node distance=1em]E_2;
      [above of=a4,node distance=1em](2);
      [below of=a4,node distance=1em]E_1;
      (a0)–(a1)–(a2);
      (a3)–(a4)–(a5);
      [dashed] (a2)–(a3);
            
  
  

  where the B_j^(s)'s are the strict transforms of the B_j's
  (j=1,2), the E_i's (1≤ i≤ n) are the exceptional curves,
  and the numbers above the nodes are the negatives of the
  self-intersections of the corresponding curves.

Let K_Y^(s) + B_Y^(s)= h^*(K_Y+B_Y).  If we write
B_Y^(s) = h^-1_*B_Y + ∑_1≤ i≤ n e_iE_i as the sum of
distinct components, then e_1 = e_2=…=e_n=b_2>0.
Now let

    B_Y^(s)'=B_Y^(s)-b_2∑_1≤ i≤ si/s E_i=  h^-1_*B_Y +b_2∑_1≤ i ≤ ss-i/sE_i.

Since K_Y^(s)+B_Y^(s) is big and nef and B_Y^(s)'< B_Y^(s), one has 

    (K_Y^(s)+B_Y^(s)')<(K_Y^(s)+B_Y^(s)).

On the other hand, one computes

    (K_Y^(s)+B_Y^(s)')^2=(K_Y^(s)+B_Y^(s))^2 -b_2^2/s=(K_X+B)^2-b_2^2/s

Take s>max{b_2^2/ϵ, b_2^2/(K_X+B)^2}. Then

    max{0,  (K_X+B)^2-ϵ}< (K_Y^(s)+B_Y^(s)')^2 < (K_X+B)^2.

Since h_*(K_Y^(s)+B_Y^(s)')= K_Y+B_Y is big, we infer that,
K_Y^(s)+B_Y^(s)'
is big.  Let (X^(s), B^(s)) be its log canonical model.
 
One computes easily that (K_Y^(s)+B_Y^(s)')E_i=0 for
1≤ i≤ s-1.  Thus, all the exceptional curves F E_s of Y^(s)→ X
satisfy (K_Y^(s)+B_Y^(s)')F≤ 0. Obviously, they
are all contracted by the morphism g Y^(s)→ X^(s).
 Since the coefficient of E_s in B'_Y^(s) is zero, one has
 (X^(s), B_X^(s))∈(_B). Moreover,

    (K_X^(s)+B^(s))^2≥(K_Y^(s)+B_Y^(s)')^2=
    (K_X+B)^2-b_2^2/s>(K_X+B)^2 - ϵ.
 We take
(X', B')=(X^(s), B^(s)).



  Let (X, B) be a log canonical surface with ample K_X+B and
  nonempty accessible nklt locus. For any ϵ>0, there is a
  log canonical surface (X', B')∈(_B) with ample K_X'+B'
  such that


  * X' is birational to X and B' is the strict transform of B;

  * (X', B') has no accessible nklt locus;

  * (K_X+B)-ϵ < (K_X'+B') <(K_X+B).



  By Theorem <ref> there is a log surface
  (X^(1), B^(1))∈(_B) with ample K_X^(1)+B^(1)
  such that X^(1) is birational to X and B^(1) is the strict
  transform of B, and
  
    (K_X+B)-ϵ/2 < (K_X^(1)+B^(1)) <(K_X+B).

  If (X^(1),B^(1)) has nonempty accessible nklt locus, then we can apply
  Theorem <ref> again to (X^(1), B^(1)) to  get
  (X^(2), B^(2)).  By induction, if we
  have constructed (X^(s), B^(s)) in (_B), and if
  (X^(s), B^(s)) has nonempty accessible nklt locus, then we
  can apply Theorem <ref> to (X^(s), B^(s)) to obtain
  a log surface (X^(s+1), B^(s+1))∈(_B) with ample
  K_X^(s+1)+B^(s+1) such
  that
  
    (K_X^(s)+B^(s))-ϵ/2^s+1 <
      (K_X^(s+1)+B^(s+1)) <(K_X^(s)+B^(s)).


  Since (_B) is a DCC set, this process must stop after, say,
  N-1 steps.
  Then (X^(N), B^(N)) has
  ample K_X^(N)+B^(N) and empty accessible nklt locus, and
  it is birational to X and B^(N) is the strict transform of
  B. Moreover,

    (K_X+B)^2-ϵ < (K_X+B)^2-∑_s=1^N 1/2^sϵ<(K_X^(N)+ B^(N))^2 <(K_X+B)^2.

Now we take (X', B')=(X^(N), B^(N)).



  Let (X, B) be a log canonical surface with ample K_X+B and
  nonempty accessible nklt locus. Then there is an infinite
  sequence of log canonical surfaces (X^(s), B^(s))∈(_B) with ample K_X^(s)+ B^(s)
  such that


  * X^(s) is birational to X and B^(s) is the strict transform of B;

  * (X^(s), B^(s)) has no accessible nklt locus;

  * the volumes (K_X^(s)+B^(s)) are strictly increasing and 

    (K_X+B)=lim_s→∞(K_X^(s)+B^(s)).







 §.§ The if part of Theorem <ref>


  First we assume that 1∈, so
  ∪{1}=. If (X, B) satisfies
  (ii.a), then there is a birational morphism f Y→ X
  extracting a single curve, with codiscrepancy contained in
  (). As usual, we write K_B+B_Y = f^*(K_X+B). Note that B_Y is
  effective and the log surface (Y, B_Y) satisfies the conditions of
  Theorem <ref>, so we have
    
      (K_X+B)^2 = (K_Y+B_Y)^2∈(()).
    
  If (X,B) satisfies (ii.b), then by  Corollary <ref>
  
(K_X+B)∈((_B)).    
  
In this case, we can assume that (X,B) does not satisfy (ii.a), so
_B⊂ and hence (K_X+B)∈(()).

Now we assume that 1∉.
Then each irreducible component of ⌊ B⌋ has geometric genus at
most 1 by assumption (iii).  By Corollary <ref>, we get a
sequence (X^(s), B^(s))∈(∪{1})
with volumes converging to (K_X+B)^2
but with an additional
property: each component of ⌊ B^(s)⌋ has
geometric genus ≤1. But since there is no accessible nklt locus,
each component of ⌊ B^(s)⌋ must be a smooth curve
lying in the smooth locus of X^(s) and disjoint from the rest of
B^(s). By adjunction, it then must have genus ≥2. Thus,
⌊ B^(s)⌋=0 and (X^(s), B^(s))∈().




First consider the log canonical surface (^1×^1, B), where B is a reduced divisor consisting 3 horizontal lines L_i and n≥ 3 vertical lines V_j. Then K_^1×^1+B is ample and (K_^1×^1+B)^2= 2(n-2), thus exhausting all the positive even integers as n varies. By Theorem <ref>, each of the numbers 2(n-2) is an accumulation point of (_0).

Now let _1 is the Hirzebruch surface obtained by blowing up one
point of ^2 and let
B=Γ_0 + Γ_1+Γ_2 + ∑_1≤ j≤ n F_j
(n≥ 2), where Γ_0 is the section with Γ_0^2=-1,
Γ_1 and Γ_2 are two distinguished sections with
Γ_1^2=Γ_2^2=1, and the F_j's are distinct fibers 
  not passing through Γ_1∩Γ_2. Then the pair
(_1, B) is log canonical, has ample K__1+B and
(K__1+B)^2=2n-3, which exhausts all the positive odd integers
as n varies. By Theorem <ref>, each of 2n-3 with
n≥ 2 is an accumulation point of (_0).

As a result of the above construction, we obtain _>0⊂((_0)), which proves one part of <cit.>.








§ THE LIMIT SURFACE


  Let Z be a projective normal surface and B=∑_j b_jB_j an
  effective -Cartier divisor on Z with b_0<1. Then there are
  finitely many effective resolutions g_t Z_t→ Z,
  1≤ t≤ r such that, if g Y→ Z is an
  effective resolution, then there is an t such that the birational
  map g_t^-1∘ g Y Z_t is a morphism and
  it is an isomorphism over a Zariski open neighborhood of the strict
  transform g_t*^-1 B_0 of B_0.


  Since any resolution f Y→ Z dominates the minimal
  resolution of Z, we can replace Z by its minimal resolution and
  the log canonical divisor K_Z+B by its
  pull-back. In other words, we can assume that Z is smooth.

  We will take g_t Z_t→ Z to be effective
  resolutions (cf. (<ref>))
  that only blow up
  points on B_0 and its strict transforms. The lemma is equivalent to
  the statement that there are only finitely many such effective
  resolutions.

  Suppose that f Y→ Z is such a resolution. Then
  f cannot blow up any smooth point of B that lies in B_0,
  since b_0<1. So there are only finitely many choices for the first
  level of blow-ups at the points of B_0.

  Over every point p∈ B_0 that is blown up, there are finitely
  many blow-ups that make the support of the total transform of B
  becomes simple normal crossing over p.  Let p_1, …, p_k be
  the inverse images of p lying on the strict transform of
  B_0. Then the codiscrepancy of the exceptional divisor of the
  k-th blowup over p_i and at the strict transform of B_0 takes
  the form a_i - k(1-b_0), where a_i is the coefficient of the
  (unique) boundary component in the pull-back of K_Z+B that
  intersects the strict transform of B_0 at p_i. Under the
  condition that b_0<1 and a_i - k(1-b_0)≥ 0 there are only
  finitely many further blow-ups over p_i.


Let ⊂ (0,1]. We define the following subset of () 

    _() := {(K_X+B)^2| (X, B)∈() , K_X+B  ample }


Let ⊂(0,1] be a DCC set. Let v_∞∈(()). Then there are a smooth projective surface (Z, D) with D a reduced  divisor, and a sequence of smooth projective surfaces Y^(s) with a boundary divisor B_Y^(s) such that 


  * there is a birational morphism g_s Y^(s)→ Z which only blows up the nodes of D and of its the total transforms;

  * (Y^(s), B_Y^(s)) is log canonical, K_Y^(s)+B_Y^(s) is big and nef, g_s* B_Y^(s)≤ D;

  * (K_Y^(s)+ B_Y^(s)) form a strictly increasing sequence, converging to v_∞;

  * the log canonical model (X^(s), B^(s)) of (Y^(s), B_Y^(s)) lies in ();

  * there are no (-1)-curves E on Y^(s) such that (K_Y^(s)+B_Y^(s))E=0, that is, Y^(s)→ X^(s) is the minimal resolution;

  * if v_∞∈_() then, additionally, (X^(s), B^(s)) is klt and it holds

    g_t^*(K_Z+g_s* B_Y^(s)) ≤ K_Y^(t) + B_Y^(t)

for any s<t.




Since () is a DCC set (<cit.>), v_∞
is approached by a strictly increasing sequence (K_X^(s)+B^(s))^2
in (), where the (X^(s), B^(s))'s are surfaces in ()
whose log canonical divisors K_X^(s)+B^(s) are ample.

By <cit.> (see also <cit.>),
there is a diagram

    Y^(s)[d, "f_s"'] [r, "g_s"]    Z^(s)
    
    X^(s)

such that 


  (a) f_s Y^(s)→ X^(s) is the minimal resolution of
   (X^(s), B^(s)),  

  (b) g_s Y^(s)→ Z^(s) is birational, and, defining
  K_Y^(s)+B_Y^(s) = f_s^*(K_X^(s)+B^(s)),
  D^(s)=g_s( B_Y^(s)∪(f_s)), the log surfaces (Z^(s), D^(s)) form a
  bounded class.

Since the log surfaces (Z^(s), D^(s)) form a bounded class, it suffices
to assume that (Z^(s), D^(s)) are fixed, to be denoted by
(Z, D) (<cit.>). By
<cit.>, we can assume
that Z is smooth and g_s Y^(s)→ Z blows up only
nodes of D and of its total transforms.

Thus we have found (Z, D) and (Y^(s), B_Y^(s)) satisfying
(i)-(v). The last very important property (vi) is
<cit.> (see also
<cit.>).



Now we are ready to complete the proof of Theorem <ref>.

Let v_∞∈(()). Let (Z, D), (Y^(s), B_Y^(s)) and (X^(s), B^(s)) be as in Lemma <ref>.



klt case. First we assume that v_∞∈(_()). Then by Lemma <ref>, (X^(s), B^(s)) have klt
singularities for all s and for every s<t,

    g_t^*(K_Z+B_Z^(s)) ≤ K_Y^(t) + B_Y^(t)

where the divisor B_Z^(s)=g_s*(B_Y^(s)). In particular, it follows
that B_Z^(s)≤ B_Z^(t) for any s<t.

Let B_Z=lim_s→∞ B_Z^(s). Since
K_Z+B_Z^(s) is nef, so is their limit K_Z+B_Z. Moreover, the
bigness of K_Z+B_Z^(s) implies that of K_Z+B since
K_Z+B_Z^(s)≤ K_Z+B. By (<ref>) one has
(K_Y^(s) + B_Y^(s))^2≤ (K_Z+B_Z^(s))^2≤ (K_Y^(t) + B_Y^(t))^2,
one sees that

    (K_Z+B_Z)^2=lim_s→∞ (K_Z+B_Z^(s))^2 = lim_s→∞(K_X^(s)+B^(s))^2=v_∞.
 

We need to show that (Z, B_Z) is log canonical. Obviously, the
divisor B_Z has coefficients in (0,1]. Let E be a divisor over
Z. By (<ref>) we have, for every s<t,

    b(E, Z, B_Z^(s))≤ b(E, Y^(t), B_Y^(t))≤ 1,
 where b denotes
the codiscrepancies of the log surfaces. It follows that

    b(E, Z, B_Z)=lim_s→∞ b(E, Z, B_Z^(s))≤ b(E, Y^(t),B_Y^(t))≤ 1.

Hence (Z, B_Z) is log canonical.

However, it is possible that the coefficient set _B_Z is not contained in ∪{1}, as required. We need to contract all the components of B_Z that have coefficients not in ∪{1}. For this, we consider (Z, B_Z)→ (Z_, B_)=:(X, B), the contraction onto the log canonical model. We will show that (X, B)∈(∪{1}) and it satisfies the properties (i), (ii), (iii) of Theorem <ref>.



Claim. (X, B)∈(∪{1}).
 
Being the log canonical model of (Z, B_Z), the surface (X, B) is log canonical. We need to show that _B⊂∪{1}, or equivalently, any component B_j of B_Z with coefficient b_j∉∪{1} is
contracted by the morphism Z→ X.

Since b_j<1, up to
taking a subsequence, there is a resolution h Z'→ Z such that for each s, the birational map g_s'=h^-1∘ g_s Y^(s) Z' is a morphism and is an isomorphism over a Zariski open neighborhood of h_*^-1 B_j by Lemma <ref>. 

 By taking a subsequence, we can assume that,
for any s, b(B_j, K_Y^(s), B_Y^(s))∉, which implies that
B_j is contracted by the morphism Y^(s)→ X^(s) and hence
(K_Y^(s)+ B_Y^(s))B_j, Y^(s)=0, where B_j, Y^(s) is the strict
transform of B_j on Y^(s). 

We compute (K_Z+B_Z)B_j on Z'. Note that, for any fixed curve E on Z', the codiscrepancies b(E, Y^(s), B_Y^(s)) converge to b(E, Z, B_Z). For the computation of (K_Z+B_Z)B_j, only the codiscrepancies of those finitely many curves in h^-1(D) that intersect B_j are relevant. Thus, we have

    (K_Z+B_Z)B_j = h^*(K_Z+B_Z) h_*^-1 B_j 
    = lim_s→∞ g_s*'(K_Y^(s)+ B_Y^(s)) h_*^-1 B_j =
      lim_s→∞ (K_Y^(s)+ B_Y^(s))B_j, Y^(s)=0.





Claim.  (X, B) satisfies condition (ii) of Theorem <ref>.


Suppose that (X, B) does not satisfy condition (ii.b). We need to show that  (X, B) satisfies (ii.a). 

Since (X, B) does not satisfy condition (ii.b), B_Z is
smooth along (Z, B_Z) and so g_s Y^(s)→ Z
blow up only at the klt points of (Z, B_Z) by Lemma <ref>(i). By Lemma <ref> there are only finitely many
possible blow-ups f Y→ Z such that the pull-back
f^*(K_Z+B_Z)=K_Y+B_Y is still a log canonical divisor with an
effective boundary B_Y. Since B_Y^(s) is effective and
g_s*(B_Y^(s))≤ B_Z, the morphisms Y^(s)→ Z
must be among these finitely many blow-ups. Thus, up to taking a
subsequence, we can assume that Y^(s)=Y and B_Y^(s)
are the same for all s.

Since (K_Y^(s)+B_Y^(s))^2 is strictly increasing, up to taking a subsequence, the boundary divisors B_Y^(s) form a strictly
increasing sequence. There must be a component of B_Y^(s), say B_0,
that is not contracted by f_s Y^(s)→ X^(s) for any s, and
b(B_0, Y^(s), B_Y^(s)) is strictly increasing and converging to
b(B_0, Z, B_Z), the coefficient of B_0 in B_Z. Since B_0 is
not contracted by f_s Y^(s)→ X^(s) and _B^(s)⊂, its coefficient in B_Y^(s) is
in . This means b(B_0, Z, B_Z) is an accumulation point of
, and condition (ii,a) of Theorem <ref> is verified.




Claim. If 1∉, then (X, B) satisfies  (iii) of Theorem <ref>.


Suppose ⌊ B⌋≠ 0 and B_0 is a component of ⌊ B⌋. Let B_0,Y^(s) be the strict transform of B_0 on Y^(s). Then 

    lim_s→∞ b(B_0, Y^(s), Y^(s), B_Y^(s)) = 1

Since 1∉, for sufficiently large s, b(B_0, Y^(s), Y^(s), B_Y^(s)) does not lie in . On the other hand, the log canonical model (X^(s), B_s) lies in (). It can only happen that B_0, Y^(s) is contracted by f_s Y^(s)→ X^(s) to an lc singularity, and hence the geometric genus of B_0 (and B_0, Y^(s)) is at most one.


In conclusion, the theorem is proved in the case when  v_∞∈(_()). 



nklt case.  Suppose that v_∞∈(())-(_()). Then there are at most finitely many s
such that (X^(s), B_s) is klt. By taking an
infinite subsequence, we can assume that
(X^(s), B^(s))≠∅ for all s. 

For each s, let
μ_sX̅^(s)→ X^(s) be a -factorial dlt
blowup, extracting only divisors with discrepancy -1. Let
K_X̅^(s)+B_X̅^(s) = μ_s^*(K_X^(s)+B^(s)), and write
B_X̅^(s)=B_X̅^(s)^(1)+B_X̅^(s)^(2) where
B_X̅^(s)^(1) is the fractional part of B_X̅^(s) and
B_X̅^(s)^(2) the (nonzero) reduced part.

We can choose a strictly decreasing sequence ϵ_s∈ (0,1]
with lim_s→∞ϵ_s=0 such that


  * 1-ϵ_s∉;

  * K_X̅^(s)+B_X̅^(s)' is lc and big,
  where
  B_X̅^(s)'= B_X̅^(s)^(1) + (1-ϵ_s) B_X̅^(s)^(2);

  * (K_X̅^(s)+B_X̅^(s)')^2 is strictly increasing and
  lim_s→∞ (K_X̅^(s)+B_X̅^(s)')^2=v_∞.

Let =∪{1-ϵ_s}_s.
Then  is a DCC set and
(X̅^(s), B_X̅^(s)')∈() are klt. As in the klt case,
we can then construct a log canonical surface (Z, B_Z) with big and nef
K_Z+B_Z whose log canonical model is in
(∪{1}) .

We need to show that the log canonical model (X, B) of (Z, B_Z) is
actually in (∪{1}). By construction, each
coefficient b of the divisor B is either in ()∪{1}
or it appears in the divisors B_X̅^(s)' 
infinitely many times. Since
()=()∪{1} and each of the coefficients
1-ϵ_s appears only once, we have
b∈∪{1}.



Similar arguments as in the klt case show that (X, B) satisfies the
conditions (ii) and (iii) of Theorem <ref>.






§ LOWER BOUNDS OF ACCUMULATION POINTS FOR (_2)


Let ⊂ (0,1] be a DCC subset. By
<cit.> the set () and so also
(()) are DCC sets. The paper
<cit.> gives an effectively computable
lower bound for () but, as we mentioned, it is way too small
to be useful. However, we have the following quite efficient bound:



    v_1()=min{(K_X+B)^2| (X, B)∈(∪{1}), K_X+B
     ample and ⌊ B⌋≠ 0}.
  



  
  One has v_1(_2) = 1/1764 =
  1/42^2. 


By Corollary <ref>,
min((_2))= min_(_2), where for the latter we
only consider log surfaces (X,B) with (X,B)∅. If
⌊ B⌋0 then we are done by (<ref>). Below, we
deal with the remaining case: ⌊ B⌋ = ∅ but
(X,B) has an isolated nklt center, a point p∈ X. The number
t_m() in the theorem below was defined in
Section <ref>. 


  Let (X,B=∑ b_jB_j) be a log canonical surface with
  coefficients in a DCC set . Suppose that K_X+B is ample
  and that 
  H^0(X, ⌊ m(K_X+B) ⌋) 0. Then
  
    (K_X+B) ≥v_1()/(1+mt_m())^2.
  



  Let C∼⌊ m(K_X+B) ⌋ be an effective curve. Hence,
  we have the relation m(K_X+B) ∼_ C + {mB}.  Now define
  
    D = C + max(B,⌈{mB} ⌉).    
  

  If B=∑ b_jB_j then the coefficient of B_j in
  max(B, ⌈{mB}⌉) is b_j if {mb}=0 and 1 if
  {mb}0.
  Thus, K_X+D≥ K_X + B, and the integral part of D satisfies
  ⌊ D⌋≥ m(K_X+B), so ⌊ D⌋ supports a big divisor.
  Note also that K_X+D is -Cartier because B does not pass through
  the non -factorial singularities of X since (X,B) is lc.
  
  We want to bound the divisor K_X + D from above in terms of
  K_X+B.  We search for a number t such that
  
    (1 + mt)(K_X + B) ≥ K_X + D 
        B + t( C + {mB} ) ≥ C + max(B, ⌈{mB}⌉)

  This is true for as long as t≥1 and for every b∈ with
  {mb}0 one has b + t {mb}≥ 1. By
  Lemma <ref> this holds if we set t=t_m().  This
  gives us
  
    (1+mt_m)^2  (K_X+B) ≥ (K_X+D).


  The log surface (X,D) may not be log canonical.
  Let f Y→ X be a dlt blowup as in Definition
  <ref>. Let K_Y+D_Y = f^*(K_X+D).
  If D_Y = ∑ d_jD_j + ∑ e_iE_i then one has e_i≥1. Define 
  D'_Y = ∑min(1,d_j)D_j + ∑min(1,e_i)E_i. Then we have  
     
   f^*(K_X + D)  ≥K_Y + D'_Y ≥f^*(K_X+B), 
  
  the latter being true because K_X+D ≥ K_X + B and (X,B) is
  log canonical. 

  This implies that the divisor K_Y+D'_Y is big. Its reduced part
  ⌊ D'_Y⌋ = f^-1 (⌊ D⌋) supports a big
  divisor. Thus, on the log canonical model (Z, D_Z) of (Y, D'_Y) the
  reduced part ⌊ D_Z⌋0. We have
  
    (K_Y + D'_Y) = (K_Z + D_Z) 
  
  and by Theorem <ref> (K_Z + D_Z) ≥ v_1().
  All together, 
  
    (K_X+D) = (   f^*(K_X + D)) ≥(K_Y+D'_Y) ≥v_1()
  
  Together with equation (<ref>) this proves the statement.



Next, we find a section  of a multiple of K_X+B provided that
(X,B) contains an isolated point.  


  Let (X,B) be in (_2) such that K_X+B is ample.
Let p be an isolated nklt center of
  (X,B). Assume that m(K_X+B) is integral and Cartier near the
  point p. Then


  * If m≥ 2 then 
H^0(X, ⌊ (m(K_X+B)⌋ )≥ 1.

  * If m=1 then H^0(X, ⌊ (2(K_X+B)⌋ )≥ 1.



  Part (ii) is obtained from (i) by using 2(K_X+B), so we only need to prove (i).
  Let f Y→ X be a log resolution. Write
  K_Y+B_Y=f^*(K_X+B) and E:=B_Y^=1. By assumption, the divisor
  E is reduced, nonempty, and has a connected component contained
  in f(p).  Since (m-1)(K_Y+B_Y) is big and nef, we
  have 
    H^1(Y, K_Y+⌈ (m-1)(K_Y+B_Y⌉))=0
 by the
  Kawamata–Viehweg vanishing theorem. The standard exact sequence
  0→_Y(-E)→_Y→_E→ 0 together with the above
  vanishing gives
  
    H^0(Y, K_Y+⌈(m-1)(K_Y+B_Y)⌉+ E)
    H^0(E, (K_Y+⌈(m-1)(K_Y+B_Y)⌉+ E )E)
  
  Let E_0 be a connected component of E contained in
  f(p). The coefficients of the divisors B_j in B_Y which
  intersect E_0 satisfy 0≤ b_j<1; they appear in the
  different. 
  Since m(K_X+B) is integral and Cartier at p, it
  follows that in a neighborhood of E_0 one has
  
    K_Y+⌈(m-1)(K_Y+B_Y)⌉+ E = m(K_Y+B_Y)
  
  and is trivial. Therefore, the above two H^0 groups are
  nonzero. Thus, the divisor 
  
    f_*(K_Y+⌈ (m-1)(K_Y+B_Y)⌉ + E)=
        K_X + ⌈ (m-1)(K_X+B)⌉ + B^=1

  is effective. Now it is easy to see that for a number of the form
  b=1-1/n, n∈ and for any m∈ one has
  ⌈ (m-1)b⌉ = ⌊ mb⌋. Thus, the last divisor equals
  
    K_X + ⌈(m-1)(K_X+B)⌉+ B^=1 =
    ⌊m(K_X+B) ⌋.
  



Let (X,B)∈(_2) with ample K_X+B and  k distinct isolated nklt centers p_i, 1≤ i≤ k. Assume that m(K_X+B) is integral and Cartier near each point p_i.  Then the same argument of Proposition <ref> gives:


  * If m≥ 2 then 
H^0(X, ⌊ (m(K_X+B)⌋ )≥ k.

  * If m=1 then H^0(X, ⌊ (2(K_X+B)⌋ )≥ k.





  For each DCC set ⊂(0,1]∩, there exists 
  m()∈ such that for any  log surface (X,B) in () with ample
  K_X+B and a point p∈ X that is an isolated nklt
  center of (X,B), the divisor m()(K_X+B) is integral and
  Cartier near p. 


  Let f Y→ X be a dlt blowup, as in 
  (<ref>).
  Let B_Y=E+Δ be the decomposition into
  the integral and fractional parts. We have E 0 by assumption;
  write E=∑ E_i. 
  By Shokurov's connectedness theorem
  <cit.>, E is connected.
  As in Section <ref>, the adjunction to E gives
  
    0 ≡(K_Y+ E + Δ) |_E = K_E + _E(Δ),
        _E(Δ) = ∑b_k' Q_k,  b_k'∈'.
  
  It follows that
  
  
  * Either p_a(E)=1, E is a smooth elliptic curve or a cycle
    of ^1s, and p∉ B. Then K_X+B is Cartier near p.
  
  * Or E is a ^1 or a chain of ^1's and Δ
    intersects only the end curves. 
  

  Restricting to an end curve gives an identity ∑ b'_k =1 or 2,
  with b'_k<1, b'_k∈'. Since ' is a DCC set, this identity has
  only finitely many solutions. 
  For a fixed b_k'<1 we have b_k'=1 -  1- ∑ n_j b_j/n and
  there are only finitely many solutions for n, n_j, b_j. By
  Lemma <ref>, n is the index of the singularity Q_k∈ Y.

  The end result is that there are only finitely many possibilities
  for b_j and the Cartier indices of K_Y+B_Y at the singular points over
  p. Since all b_j∈ by assumption, there exists a fixed
  multiple m depending only on  such that m(K_Y+B_Y) is
  integral and Cartier. If p is a point as in (i) above then near
  p the divisor K_X+B=K_X is already Cartier. Otherwise X is klt
  at p. By the Base Point Free Theorem for lc surfaces it follows
  that m(K_X+B) is Cartier.



  Let (X,B) be a log surface in (_2)  and let p∈ X be a point
  that is an isolated nklt center of (X,B). Then the divisor
  m(K_X+B) is integral and Cartier near p for m=1,2,3,4 or
  6.


  As in the proof of the above proposition, for a dlt blowup
  Y→ X on an exceptional divisor E=∑ E_i there are several
  singularities of index m_j and an identity of the form
  ∑ (1-1/n_j)=1 or 2.  As it is well known, the only
  solutions (n_j) to this identity are (2,2), (3,3,3),
  (2,4,4), (2,3,6), (2,2,2,2). By (<ref>), the
  indices m_j divide n_j: n_j=m_jn'_j and the respective
  coefficients of B are b_j=1-1/n'_j.  If m is the GCD of
  these numbers (i.e.  m=1,2,3,4 or 6) then m(K_Y+B_Y) is
  integral and Cartier. So is m(K_X+B) by the Base Point Free
  Theorem.


Putting this together we get:


  For any log surface (X,B)∈(_2) with ample
  K_X+B and (X,B) ∅ one has
  
    (K_X+B)^2 ≥1/7^2·42^2
  


  If ⌊ B⌋0 then (K_X+B)^2≥1/42^2 by
  Theorem <ref>. Otherwise, there is an isolated nklt
  center. Then we apply Theorem <ref> and use that
  t_m = 1 by Lemma <ref> and m≤6 by
  Lemma <ref>. 


We conclude by completing the proof of Theorem <ref>.


  (ii) The lower bound is proved in
  Theorem <ref>. The upper bound is obtained by
  applying Theorem <ref> to
  <cit.>.

  (iii) The lower bound follows from that for _2. The upper bound
  is obtained by applying Theorem <ref> to the example in
  <cit.> of a strictly log canonical surface
  X with ample K_X and K_X^2=1/462.

  (i) For _2, this follows from Corollary <ref>. For _0, and
  _1, using (<ref>), we have to rule out the possibility
  that the minimum is achieved on a log surface (X,B) with an
  inaccessible nklt locus, i.e. one that has 
  
  
  (a) either a simple elliptic singularity,
  
  (b) or a smooth component B_j of B lying in a smooth part.
  
  However, in the case (a) one has (K_X+B)^2≥1/143 by
  <cit.>. And in the case (b)
  (K_X+B)B_0∈, so by <cit.>
  
    (K_X+B)^2≥δ_1(_2)(K_X+B)B_0 ≥δ_1(_2),  
    
  where δ_1 = sup(t) such that K_X+B-t B_0 is big. One has
  δ_1(_2)=1/42 by
  <cit.>.  The existing limit point
  1/462 is smaller.






  * By Lemma <ref>, we can take m(_2) = 12. If
  (X, B)∈(_2) has ample K_X+B and at least two isolated
  nklt centers, then one has
  H^0(X, ⌊ 12(K_X+B)⌋)≥ 2 by Remark <ref>. In this case, one expects a better lower bound on
  (K_X+B), cf. <cit.>.

  * For any ϵ >0, the set of 
numbers in (_2) that are less than 1/86436-ϵ
is finite.

  * Kollár <cit.> gives a log canonical surface
  (X, B)∈(_2) such that K_X+B is ample and
  (K_X+B)^2=1/42^2· 43^2=1/3261636.
  On the other hand, the current record
  for the  smallest (positive) volume of 
  (X, B)∈(_0) and (_1)
  is 1/48983 (<cit.>).





§ ITERATED ACCUMULATION POINTS OF VOLUMES


  Let V⊂ be a non-empty subset. A point v∈ is
  a k-iterated accumulation point of V if
  v∈^k (V), where ^k (V)⊂ is defined
  inductively as the set of accumulation points of ^k-1(V), and
  ^0(V):=V.

Obviously, ^1(V)=(V) is exactly the set of accumulation
points of V; ^k(V)⊂^k-1(V) for k≥ 2.

  For any set ⊂ (0,1] and any k∈, one has
  ^k(^2())∅. 


  Consider (X,B)=(^2,∑_j=1^n L_j) with n≥ 4
where
  the L_j's are n distinct lines in general position. Then
  (X, B) is a log canonical surface with ample K_X+B and
  (K_X+B)^2=(n-3)^2. 
  Let p_j∈ L_j∩ L_n for 1≤ j≤ n-1 be the n-1 nodes of
  B on L_n. We apply the construction of Theorem <ref>
  to all of these points at once.

  Let h Z=Y^(s_1,…, s_n-1)→^2 be obtained by blowing
  up s_j times at the point p_j and its preimages on the the strict transforms of L_j,
  1≤ j≤ n-1. As in Theorem <ref>, we define the
  divisor B_Z by K_Z+B_Z = h^*(K_X+B) and the
  divisor B_Z' by 

  
    B_Z' 
    = B_Z - ∑_j=1^n-1 ∑_i=1^s_j i/s_j E_j,i
    = h_*B + ∑_j=1^n-1 ∑_i=1^s_j s_j-i/s_j E_j,i
  

  By construction, K_Z+B_Z' intersects the curves E_j,i
  non-negatively, and one has (K_Z+B_Z') h_* L_j = n-4 for
  1≤ j≤ n-1 and n-3-∑1/s_j for j=L_n. Thus, for
  s_j≫0 the divisor K_Z + B_Z' is nef of volume
  
    (K_Z+B_Z')^2 = (n-3)^2 - ∑_j=1^k 1/s_j
  
  and on the log canonical model of (Z,B_Z') the curve h_* L_n
  is not contracted and (Z,B_Z') has a nonempty accessible nklt locus.

  Each of the above numbers is an accumulation point of ^2(_0)
  by Theorem <ref>.  Considering the sequences with
  s_1≫ s_2≫… s_n-1≫0, we see that
  (n-3)^2 is in ^n(_0).  Now, for any n≥max(k,4) one has
  ^k()⊃^n(_0)∅.

amsalpha


'





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