# Some remarks on log surfaces.

1. Introduction. We work over an algebraically closed field of
arbitrary characteristic throughout this paper. We will also follow the
language and notational conventions of the book [KM98] unless stated
otherwise.

Let (X, [DELTA]) be a log surface. Remember that a pair (X, [DELTA]) is called log surface if X is a normal algebraic surface and [DELTA] is a boundary R-divisor on X such that [K.sub.X] + [DELTA] is R-Cartier (See [Fjn12, Definition 3.1]). To complete Fujita's results [Fjt84] on the semi-ampleness of semi-positive parts of Zariski decompositions of log canonical divisors and the finite generation of log canonical rings for smooth projective log surfaces, Fujino [Fjn12] developed the log minimal model program for projective log surfaces in characteristic 0. It is generalized to characteristic p > 0 by Tanaka in his paper [Tnk14]. One of their main results is the following Theorem 1.1 ([Fjn12, Theorem 3.3], [Tnk14, Theorem 1.1]). Let (X, [DELTA]) be a log surface which is not necessarily log canonical, and let [pi] : X [right arrow] S be a projective morphism onto an algebraic variety S. Assume that X is Q-factorial. Then we can run the log minimal model program over S with respect to [K.sub.X] + [DELTA] and get a sequence of at most [rho](X/S) - 1 contractions

(X, [DELTA]) = ([X.sub.0], [[DELTA].sub.0]) [right arrow] ([X.sub.1], [[DELTA].sub.1]) [right arrow] *** [right arrow] ([X.sub.k], [[DELTA].sub.k]) = ([X.sup.*], [[DELTA].sup.*])

over S such that one of the following holds:

(1) (Minimal model) [mathematical expression not reproducible] is nef over S. In this case, ([X.sup.*], [[DELTA].sup.*]) is called a minimal model of (X, [DELTA]).

(2) (Mori fiber space) There is a morphism g : [X.sup.*] [right arrow] C over S such that [mathematical expression not reproducible] is g-ample, dim C < 2, C is projective over S and [rho]([X.sup.*]/C) = 1. We sometimes call g : ([X.sup.*], [[DELTA].sup.*]) [right arrow] C a Mori fiber space.

Note that [X.sub.i] is Q-factorial for every i. Furthermore, if [K.sub.X] + [DELTA] is big, then on the minimal model [mathematical expression not reproducible] is nef and big over S.

First, we try to clarify that, given such a log surface (X, [DELTA]) where X is smooth, what every intermediate surface [X.sub.i] would look like after running this log minimal model program. Note that the final log surface ([X.sup.*], [[DELTA].sup.*]) could be a minimal model or a Mori fiber space g : ([X.sup.*], [[DELTA].sup.*]) [right arrow] C. The following theorem is our main result in this paper to achieve this aim.

Theorem 1.2 (Theorem 3.1). Notations are as in Theorem 1.1. Let e be a real number such that 0 [less than or equal to] [epsilon] < 1. If X is smooth and the coefficients of [DELTA] are [less than or equal to] 1 - [epsilon], then [X.sub.i] is [epsilon]-log terminal for every i. In particular, [X.sup.*] is [epsilon]-log terminal.

Next, a natural question is that, given a log surface (X, [DELTA]) where X is not smooth, what every intermediate surface [X.sub.i] would look like after running log minimal model program (Theorem 1.1).

Proposition 1.3. In Theorem 1.1, [X.sub.i] is not necessarily log canonical even if X is log canonical.

Moreover, we have:

Proposition 1.4. In Theorem 1.1, [X.sub.i] is not necessarily log canonical even if X is [epsilon]-log canonical and the coefficients of [DELTA] are [less than or equal to] 1 - [epsilon] for some 0 < [epsilon] < 1.

In Section 4 we construct some examples to show that Propositions 1.3 and 1.4 are true. Furthermore, we show that [X.sub.i] could not even be MR log canonical if X is not smooth. In fact this shows that Fujino and Tanaka's minimal model program on log surfaces is more general than Alexeev's minimal model program which is running mainly on MR log canonical surfaces in [Alex94, Section 10] (see Definition 2.2 for the definition of MR log canonical).

2. Preliminaries. Let (X, [DELTA]) be a log surface. If X is smooth, then it is Q-factorial. Choose a set I [subset] [0,1 - [epsilon]] where [epsilon] [member of] [0, 1] is a fixed real number. Assume that the coefficients of [DELTA] are in I. We say that I satisfies the descending chain condition or I is a DCC set for short, if it does not contain any infinite strictly decreasing sequence. Finally, recall that the volume of an R-divisor D on a normal projective variety X of dimension n is defined as

[mathematical expression not reproducible]

We recall some kinds of singularities and MR singularities following the same way of Alexeev.

Definition 2.1 ([Alex94, Definition 1.5]). Let (X, [DELTA]) be a log surface. Let [epsilon] be a real number such that 0 [less than or equal to] [epsilon] < 1. It is called:

1. [epsilon]-log canonical, if the total discrepancies [greater than or equal to]

-1 + [epsilon],

2. [epsilon]-log terminal, if the total discrepancies >

-1 + [epsilon]

for every resolution f : Y [right arrow] X in both cases. Simply, we call it [epsilon]-lc or [epsilon]-lt instead. Note that when [epsilon] is not zero, we can replace [epsilon] by a smaller positive [epsilon]', and assume that [epsilon]-log canonical is [epsilon]'-log terminal.

Definition 2.2 ([Alex94, Definition 1.7]). We call a log surface (X, [DELTA]) MR log canonical, MR [epsilon]-log canonical, MR [epsilon]-log terminal etc. if we require the previous inequalities in Definition 2.1 to hold not for all resolutions f : Y [right arrow] X but only for a minimal desingularization.

A strange but trivial example of MR log canonical log surface is the following

Example 2.3. Given a log surface (X, [DELTA]), where X is smooth and [DELTA] is a boundary. (X, [DELTA]) is not necessarily log canonical in the usual sense. But id : X [right arrow] X is the minimal desingularization, therefore (X, [DELTA]) is MR log canonical.

3. Main results. Now we go to the proof of Theorem 1.2. Note that e in this theorem could be zero:

Theorem 3.1. Notations are as in Theorem 1.1. Let [epsilon] be a real number such that 0 [less than or equal to] [epsilon] < 1. If X is smooth and the coefficients of [DELTA] are [less than or equal to] 1 - [epsilon], then [X.sub.i] is [epsilon]-log terminal for every i. In particular, [X.sup.*] is [epsilon]-log terminal.

Proof. Step 1. Run log minimal model program (Theorem 1.1) for [K.sub.X] + [DELTA] as in Theorem 1.1:

(X, [DELTA]) = ([X.sub.0], [[DELTA].sub.0]) [right arrow] ([X.sub.1], [[DELTA].sub.1]) [right arrow] *** [right arrow] ([X.sub.k], [[DELTA].sub.k]) = ([X.sup.*], [[DELTA].sup.*])

where ([X.sup.*], [[DELTA].sup.*]) is a minimal model or a Mori fiber space. In the following proof, we consider everything over [X.sub.j] for a fixed j. Put [X.sup.[dagger]] = [X.sub.j] for this fixed j. Then take [X.sup.[dagger]] as a base (that is, replace S by [X.sup.[dagger]] and hence we are reduced to the case where S = [X.sup.*] = [X.sub.j]. Therefore, if needed, shrink [X.sup.[dagger]] to be affine since [epsilon]-log terminal or not is a local property) and run ([K.sub.X] + A)-LMMP for the relative morphism f : X [right arrow] [X.sup.[dagger]], which ends up again on [X.sup.[dagger]] and [mathematical expression not reproducible] is nef over [X.sup.[dagger]]. In each step we have a relative morphism [X.sub.i] [right arrow] [X.sup.[dagger]] (i [less than or equal to] j) and denote it by [X.sub.i]/[X.sup.[dagger]]. We use [f.sub.i] and [h.sub.i] to denote the morphisms ([X.sub.i], [[DELTA].sub.i])/[X.sup.[dagger]] [right arrow] ([X.sup.[dagger]], [[DELTA].sup.[dagger]])/[X.sup.[dagger]] and (X, [DELTA])/[X.sup.[dagger]] [right arrow] ([X.sub.i], [[DELTA].sub.i])/[X.sup.[dagger]] with that [h.sub.j] = [f.sub.0] = f. By [Fjn12, Section 3] and [Tnk14, Section 3],

[mathematical expression not reproducible],

where [E.sub.i] is all effective over [X.sup.[dagger]] for every 0 [less than or equal to] i < j. In particular, [mathematical expression not reproducible]. Furthermore, every curve in Exc(f) = Supp([E.sub.0]) is a smooth rational curve by [Fjn12, Proposition 3.8] and [Tnk14, Theorem 3.19].

Step 2. Now we may assume that there is no (-1)-curve in Exc(f). Indeed, if there is some (-1)-curve, say C, in Exc(f), then by Castelnuovo's theorem, contracting this (-1)-curve in X/[X.sup.[dagger]] leads to a new smooth surface X'/[X.sup.[dagger]]. Therefore we can run another ([K.sub.X'] + [DELTA]')-LMMP over [X.sup.[dagger]] until reaching to a final log surface [mathematical expression not reproducible], where [DELTA]' is the image of [DELTA]. Every assumption of (X, [DELTA]) is obviously keeping if we replace (X, [DELTA]) by (X', [DELTA]') except that we need to prove [mathematical expression not reproducible]. We have three morphisms over [X.sup.[dagger]]: [pi]: X [right arrow] X', g : X' [right arrow] [??] and [rho] : [??][right arrow] [X.sup.[dagger]] such that

[mathematical expression not reproducible]

where [pi] : X [right arrow] X' is the Castelnuovo's contraction, [rho] is not necessarily the identity and [mathematical expression not reproducible] is nef over [X.sup.[dagger]]. Then by negativity lemma (see [KM98, Lemma 3.39 and Lemma 3.40]), we have that -D [greater than or equal to] 0, since [mathematical expression not reproducible] is [rho]-exceptional. Remember that [mathematical expression not reproducible]. That is, [E.sub.0] [~.sub.f] [[pi].sup.*] [g.sup.*] D + [[pi].sup.*] [E.sub.0] + aC. By negativity lemma again, D > 0 since [E.sub.0] is effective and both sides have the same support. Therefore we get a contradiction unless p is the identity. That is, [mathematical expression not reproducible]. Then, by contracting (-1)-curves finitely many times, we may assume that Exc(f) contains no (-1)-curve from now on.

Step 3. Assume that [C.sub.i] is the contracted curve in step i of the log minimal model program, then [mathematical expression not reproducible]. Therefore

[mathematical expression not reproducible].

Note that [([h.sup.*.sub.i] ([C.sub.i])).sup.2] = [([C.sub.i]).sup.2] < 0 by the negativity lemma. Then [K.sub.X] - [h.sup.*.sub.i] ([C.sub.i]) [greater than or equal to] 0 since [h.sup.*.sub.i] ([C.sub.i]) is effective and its support contains no (-1)-curve. Indeed, if [K.sub.X] - [h.sup.*.sub.i] ([C.sub.i]) < 0, there must be a curve, say E, in Supp[h.sup.*.sub.i] ([C.sub.i]) such that [K.sub.X] x E < 0. But [E.sup.2] < 0 since E is in Exc(f). Thus it is a (-1)-curve which contradicts our assumption. Therefore [DELTA] - [h.sup.*.sub.i] ([C.sub.i]) < 0. Then

[mathematical expression not reproducible].

That is, [C.sub.i] is in Supp [[DELTA].sub.i], and its strict transform is in Supp[DELTA]. Therefore all those curves in Exc(f) must be such a strict transform of [C.sub.i] under the assumption of the above step.

Step 4. Next, we need to prove that, for the resolution f : X [right arrow] [X.sup.[dagger]] where [mathematical expression not reproducible], we have that [a.sub.i] > -1 + [epsilon]. Note that [mathematical expression not reproducible] where [E.sub.0] is effective in Exc(f) and [F.sub.i] is in Supp[DELTA] by the above steps. Furthermore, let [DELTA] = [SIGMA] [[delta].sub.i] [F.sub.i] + [DELTA]' where [SIGMA] [F.sub.i] and [DELTA]' have no common components. Therefore, [f.sub.*] [DELTA]' = [[DELTA].sup.[dagger]]. Then

[mathematical expression not reproducible].

That is,

[SIGMA]([a.sub.i] + [[delta].sub.i])[F.sub.i] = [f.sup.*] [[DELTA].sup.[dagger]] - [DELTA]' + [E.sub.0]

in which both sides are supported in Exc(f) and the right hand side is effective since [f.sup.*] [[DELTA].sup.[dagger]] - [DELTA]' = [f.sup.*] [f.sub.*] [DELTA]' - [DELTA]' and [DELTA]' is effective. Note that Supp[E.sub.0] = Supp(Exc(f)) by our log minimal model program (Theorem 1.1). Thus comparing both sides, [a.sub.i] + [[delta].sub.i] > 0. That is, [a.sub.i] > - [[delta].sub.i] [greater than or equal to] -1 + [epsilon] since the coefficients of [DELTA] are [less than or equal to] 1 - [epsilon].

Step 5. We claim that, the resolution f : X [right arrow] [X.sup.[dagger]] is a log resolution. That is, the reduced [SIGMA] [F.sub.i] must be a simple normal crossing curve. Remember that [F.sub.i] is all smooth extremal rational curves since [X.sup.[dagger]] has rational singularities by [FT12, Theorem 6.2] for any characteristic. Furthermore, the dual graph of [SIGMA] [F.sub.i] must be a tree. This shows that the reduced [SIGMA] [F.sub.i] must be a simple normal crossing curve. We get what we want.

Remark 3.2. It is pointed out by Tanaka that, our claim in Step 5 can be proved by [KM98, Theorem 4.7].

From the above theorem, we know that when X is smooth, those contracting curves in log minimal model program consist of some images of (-1)-curves and some components of Supp[DELTA]. Several direct but important implications of Theorem 3.1 are the following. When [K.sub.X] + [DELTA] is big, [mathematical expression not reproducible] is nef and big on the minimal model. What we have done in the proof of Theorem 3.1 is in fact showing that f : X' [right arrow] [X.sup.*] is exactly the minimal desingularization and ([X.sup.*], [[DELTA].sup.*]) is MR [epsilon]-log terminal. Then the following corollaries are just simple consequences of [Alex94, Theorem 7.6, Theorem 7.7, Theorem 8.2]. It is another way to see that Fujino and Tanaka's minimal model program on log surfaces cover Alexeev's minimal model program stated in [Alex94, Section 10].

Corollary 3.3. Let (X, [DELTA]) be a projective log surface where X is smooth and [K.sub.X] + [DELTA] is big. Fixing [epsilon] > 0, let I [subset] [0, 1 - [epsilon]] be a DCC set and the coefficients of [DELTA] be in I. If there is a positive integer M such that [mathematical expression not reproducible] where ([X.sup.*], [[DELTA].sup.*]) is a minimal model of (X, [DELTA]), then these ([X.sup.*], Supp[[DELTA].sup.*]) belong to a bounded family.

Corollary 3.4. Let (X, [DELTA]) be a projective log surface where X is smooth and [K.sub.X] + [DELTA] is big. Fixing [epsilon] [greater than or equal to] 0, let I [subset] [0, 1 - [epsilon]] be a DCC set and the coefficients of [DELTA] be in I. Then [mathematical expression not reproducible] is a DCC set. In particular the volume vol ([K.sub.X] + [DELTA]) is bounded from below away from 0.

Proof. Since [mathematical expression not reproducible] by Theorem 3.1, this corollary is a direct consequence of [Alex94, Theorem 8.2].

Remark 3.5. Note that in Corollary 3.3, the [epsilon] is smaller, the bounded family of ([X.sup.*], Supp[[DELTA].sup.*]) is bigger. When e goes to 0, all those [X.sup.*] may not be in a bounded family, so may not be ([X.sup.*], Supp[[DELTA].sup.*]). See [Lin03, Remark 1.5] for the example showing that [X.sup.*] could be Q-Fano and not in a bounded family. Note also that Corollary 3.4 is an answer of the question coming from the first version of Di Cerbo's paper which has been confirmed by his second version [Dic, Corollary 4.3].

4. Examples. By [Alex94, Section 10], we easily see that if the log surface (X, [DELTA]) is MR [epsilon]-log canonical, then so is every ([X.sub.i], [[DELTA].sub.i]) in the step of log minimal model program; by Grothendieck spectral sequence and [FT12, Theorem 6.2], it is also easy to see that if X has only rational singularities, then so has every [X.sub.i]. Now it is natural to generalize Theorem 3.1 and ask that if X is [epsilon]-log canonical, is so every [X.sub.i] or not. But unfortunately we have the following example:

Example 4.1. Let us first recall an example of log canonical surface which is rational but not log terminal. Blowing up at a point of [P.sup.2], we get a (-1)-curve [E.sub.0]; find three points at [E.sub.0] and blow up several times (at these three points and some points at the exceptional curves over them), we can easily get a smooth and projective surface Y and four smooth rational curves [E.sub.0], [E.sub.1], [E.sub.2], [E.sub.3] on it such that [n.sub.0] = -[E.sup.2.sub.0] [greater than or equal to] 3, [n.sub.1] = - [E.sup.2.sub.1] = 2, [n.sub.2] = - [E.sup.2.sub.2] = 3, [n.sub.3] = -[E.sup.2.sub.3] = 6 where by abusing of notations, we still use [E.sub.0] to denote its strict transform on Y. By construction, [E.sub.i] x [E.sub.0] = 1, [E.sub.i] x [E.sub.j] = 0 for 1 [less than or equal to] i < j [less than or equal to] 3. Let E = [E.sub.0] + [E.sub.1] + [E.sub.2] + [E.sub.3], then its dual graph is a triple fork. See also that its intersection matrix is negative definite. Therefore by Artin's criterion [Art62, Theorem 2.3], we can contract E and finally get a surface X with a singular point. Now we have f : Y [right arrow] X with [K.sub.Y] = [f.sup.*] [K.sub.X] + [SIGMA] [a.sub.i] [E.sub.i]. Using adjunction, we have that:

[mathematical expression not reproducible].

Solving these equations we have [a.sub.0] = -1, [a.sub.1] = - 1/2, [a.sub.2] = - 2/3, [a.sub.3] = - 5/6. These show that the singularity of X is exactly log canonical but not log terminal.

Keeping this example in mind, we construct an example as follows:

Similar to the above blowing-up method, we can easily construct a surface Y and five smooth rational curves D, [E.sub.0], [E.sub.1], [E.sub.2], [E.sub.3] on it such that n = -[D.sup.2] is as big as we want, [n.sub.0] = -[E.sup.2.sub.0] [greater than or equal to] 3, [n.sub.1] = -[E.sup.2.sub.1] = 2, [n.sub.2] = -[E.sup.2.sub.2] = 3, [n.sub.3] = -[E.sup.2.sub.3] = 6 and [E.sub.i] x [E.sub.0] = D x [E.sub.0] = 1, [E.sub.i] x [E.sub.i] = D x [E.sub.i] = 0 for 1 [less than or equal to] i < j [less than or equal to] 3. Let E = [E.sub.0] + [E.sub.1] + [E.sub.2] + [E.sub.3] and F = E + D. Then E is a triple fork and F is a quadruple fork in dual graph. Note that both of the intersection matrices of E and F are negative definite which are exercises of diagonalization of matrix. By contracting E on Y we get a morphism f from Y to a log canonical surface X which is rational but not log terminal as above. Now consider the log surface (X, D') where D0 is the image of the smooth rational curve D. Note that D and D' are isomorphic outside the point E [intersection] D and its image. Note also that ([K.sub.X] + D') x D' < 0. Indeed, let [f.sup.*] D' = D + [SIGMA] [c.sub.i] [E.sub.i]. Then by [E.sub.i] x [f.sup.*] D' = 0,

[mathematical expression not reproducible].

That is, [c.sub.0] = 1/[n.sub.0] - 1, [c.sub.1] = [c.sub.0]/2, [c.sub.2] = [c.sub.0]/3, [c.sub.3] = [c.sub.0]/6. Then

[mathematical expression not reproducible]

since [c.sub.0] = 1/[n.sub.0] - 1 < 1. Therefore, by [Tnk14, (1) of Theorem 3.19], D' is a smooth rational curve. Moreover, contracting D' on X is indeed a step of log minimal model program (Theorem 1.1). Finally, we get a log surface ([X.sup.*], 0) where [X.sup.*] is no longer log canonical since the dual graph of F is a quadruple fork which is not in the classification of dual graph of log canonical singularities in [KM98, Theorem 4.7]. Furthermore, it is not even MR log canonical by calculating the discrepancy of [E.sub.0] (which is 2 - [nn.sub.0]/[nn.sub.0] - n < -1). But remember that [X.sup.*] still has rational singularities by [FT12, Theorem 6.2]. This example proves Proposition 1.3.

Example 4.2. We just gave an example for Proposition 1.3 where [epsilon] = 0. In fact, by a similar construction as above, we can get some examples where [epsilon] > 0. [DELTA] sketch of construction is the following. As Example 4.1, we can easily construct a smooth and projective surface Y and five smooth rational curves D, [E.sub.0], [E.sub.1], [E.sub.2], [E.sub.3] on it with n = -[D.sup.2], [n.sup.i] = -[E.sup.2.sub.i] such that n = 3, [n.sub.0] = 5, [n.sub.1] = [n.sub.2] = [n.sub.3] = 2, [E.sub.i] - [E.sub.0] = D x [E.sub.0] = 1, [E.sub.i] x [E.sub.j] = D x [E.sub.i] = 0 for 1 [less than or equal to] i < j [less than or equal to] 3. Let E = [E.sub.0] + [E.sub.1] + [E.sub.2] + [E.sub.3] and F = E + D. Then E is a triple fork and F is a quadruple fork which is not in the classification of dual graph of log canonical singularities. Note that both of the intersection matrices of E and F are negative definite again by diagonalization of the intersection matrices. Choose an [epsilon] such that 0 < [epsilon] [less than or equal to] 1/7. The same calculation as Example 4.1 shows that by contracting E on Y we get a morphism f from Y to an [epsilon]-log canonical surface X. Now consider the log surface (X, bD') where D' is the image of D' and b is a non-negative real number. Note that D' is still a smooth rational curve by construction. Choose a proper real number b such that ([K.sub.X] + bD') x D' < 0. Indeed, by careful calculations as in Example 4.1, we can check that ([K.sub.X] + bD') x D' < 0 for b > 13/19. Therefore, ([K.sub.X] + (1 - [epsilon])D') x D' < 0 for 0 < [epsilon] [less than or equal to] 1/7. Now contracting D' on (X, bD') by log minimal model program, we get a log surface ([X.sup.*], 0) where [X.sup.*] is no longer log canonical. This gives an example to confirm Proposition 1.4.

Remark 4.3. The above two examples are based on one of the dual graphs of log canonical singularities in [KM98, Theorem 4.7]. In fact, we can construct similar examples based on the other dual graphs there and get a bunch of similar examples.

It will be interesting to ask the following question:

Question 4.4. In Theorem 1.1, if X is canonical, is X, log canonical?

doi: 10.3792/pjaa.93.115

Acknowledgements. The author would like to thank Prof. Osamu Fujino for so many inspirational suggestions and comments. The author would like to thank Dr. Hiromu Tanaka and the referees for their helpful comments too. He would also like to thank Dr. Chen Jiang for many discussions on MR log canonical when they were attending in the conference of Higher Dimensional Algebraic Geomentry (HDAG) held in Utah.

References

[Alex94] V. Alexeev, Boundedness and [K.sup.2] for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779 810.

[Art62] M. Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485 496.

[Dic] G. Di Cerbo, On Fujita's spectrum conjecture, arXiv:1603.09315v2.

[FT12] O. Fujino and H. Tanaka, On log surfaces, Proc. Japan Acad. Ser. [DELTA] Math. Sci. 88 (2012), no. 8, 109 114.

[Fjn12] O. Fujino, Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), no. 2, 339 371.

[Fjt84] T. Fujita, Fractionally logarithmic canonical rings of algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 685-696.

[KM98] J. Kollar and S. Mori, Birational geometry of algebraic varieties, translated from the 1998 Japanese original, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, 1998.

[Lin03] J. Lin, Birational unboundedness of Q-Fano threefolds, Int. Math. Res. Not. 2003, no. 6, 301 312.

[Tnk14] H. Tanaka, Minimal models and abundance for positive characteristic log surfaces, Nagoya Math. J. 216 (2014), 170.

By Haidong LIU

Department of Mathematics, Graduate School of Science, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan

(Communicated by Shigefumi MORI, M.J.A., Nov. 13, 2017)

2010 Mathematics Subject Classification. Primary 14E05; Secondary 14E30.

Let (X, [DELTA]) be a log surface. Remember that a pair (X, [DELTA]) is called log surface if X is a normal algebraic surface and [DELTA] is a boundary R-divisor on X such that [K.sub.X] + [DELTA] is R-Cartier (See [Fjn12, Definition 3.1]). To complete Fujita's results [Fjt84] on the semi-ampleness of semi-positive parts of Zariski decompositions of log canonical divisors and the finite generation of log canonical rings for smooth projective log surfaces, Fujino [Fjn12] developed the log minimal model program for projective log surfaces in characteristic 0. It is generalized to characteristic p > 0 by Tanaka in his paper [Tnk14]. One of their main results is the following Theorem 1.1 ([Fjn12, Theorem 3.3], [Tnk14, Theorem 1.1]). Let (X, [DELTA]) be a log surface which is not necessarily log canonical, and let [pi] : X [right arrow] S be a projective morphism onto an algebraic variety S. Assume that X is Q-factorial. Then we can run the log minimal model program over S with respect to [K.sub.X] + [DELTA] and get a sequence of at most [rho](X/S) - 1 contractions

(X, [DELTA]) = ([X.sub.0], [[DELTA].sub.0]) [right arrow] ([X.sub.1], [[DELTA].sub.1]) [right arrow] *** [right arrow] ([X.sub.k], [[DELTA].sub.k]) = ([X.sup.*], [[DELTA].sup.*])

over S such that one of the following holds:

(1) (Minimal model) [mathematical expression not reproducible] is nef over S. In this case, ([X.sup.*], [[DELTA].sup.*]) is called a minimal model of (X, [DELTA]).

(2) (Mori fiber space) There is a morphism g : [X.sup.*] [right arrow] C over S such that [mathematical expression not reproducible] is g-ample, dim C < 2, C is projective over S and [rho]([X.sup.*]/C) = 1. We sometimes call g : ([X.sup.*], [[DELTA].sup.*]) [right arrow] C a Mori fiber space.

Note that [X.sub.i] is Q-factorial for every i. Furthermore, if [K.sub.X] + [DELTA] is big, then on the minimal model [mathematical expression not reproducible] is nef and big over S.

First, we try to clarify that, given such a log surface (X, [DELTA]) where X is smooth, what every intermediate surface [X.sub.i] would look like after running this log minimal model program. Note that the final log surface ([X.sup.*], [[DELTA].sup.*]) could be a minimal model or a Mori fiber space g : ([X.sup.*], [[DELTA].sup.*]) [right arrow] C. The following theorem is our main result in this paper to achieve this aim.

Theorem 1.2 (Theorem 3.1). Notations are as in Theorem 1.1. Let e be a real number such that 0 [less than or equal to] [epsilon] < 1. If X is smooth and the coefficients of [DELTA] are [less than or equal to] 1 - [epsilon], then [X.sub.i] is [epsilon]-log terminal for every i. In particular, [X.sup.*] is [epsilon]-log terminal.

Next, a natural question is that, given a log surface (X, [DELTA]) where X is not smooth, what every intermediate surface [X.sub.i] would look like after running log minimal model program (Theorem 1.1).

Proposition 1.3. In Theorem 1.1, [X.sub.i] is not necessarily log canonical even if X is log canonical.

Moreover, we have:

Proposition 1.4. In Theorem 1.1, [X.sub.i] is not necessarily log canonical even if X is [epsilon]-log canonical and the coefficients of [DELTA] are [less than or equal to] 1 - [epsilon] for some 0 < [epsilon] < 1.

In Section 4 we construct some examples to show that Propositions 1.3 and 1.4 are true. Furthermore, we show that [X.sub.i] could not even be MR log canonical if X is not smooth. In fact this shows that Fujino and Tanaka's minimal model program on log surfaces is more general than Alexeev's minimal model program which is running mainly on MR log canonical surfaces in [Alex94, Section 10] (see Definition 2.2 for the definition of MR log canonical).

2. Preliminaries. Let (X, [DELTA]) be a log surface. If X is smooth, then it is Q-factorial. Choose a set I [subset] [0,1 - [epsilon]] where [epsilon] [member of] [0, 1] is a fixed real number. Assume that the coefficients of [DELTA] are in I. We say that I satisfies the descending chain condition or I is a DCC set for short, if it does not contain any infinite strictly decreasing sequence. Finally, recall that the volume of an R-divisor D on a normal projective variety X of dimension n is defined as

[mathematical expression not reproducible]

We recall some kinds of singularities and MR singularities following the same way of Alexeev.

Definition 2.1 ([Alex94, Definition 1.5]). Let (X, [DELTA]) be a log surface. Let [epsilon] be a real number such that 0 [less than or equal to] [epsilon] < 1. It is called:

1. [epsilon]-log canonical, if the total discrepancies [greater than or equal to]

-1 + [epsilon],

2. [epsilon]-log terminal, if the total discrepancies >

-1 + [epsilon]

for every resolution f : Y [right arrow] X in both cases. Simply, we call it [epsilon]-lc or [epsilon]-lt instead. Note that when [epsilon] is not zero, we can replace [epsilon] by a smaller positive [epsilon]', and assume that [epsilon]-log canonical is [epsilon]'-log terminal.

Definition 2.2 ([Alex94, Definition 1.7]). We call a log surface (X, [DELTA]) MR log canonical, MR [epsilon]-log canonical, MR [epsilon]-log terminal etc. if we require the previous inequalities in Definition 2.1 to hold not for all resolutions f : Y [right arrow] X but only for a minimal desingularization.

A strange but trivial example of MR log canonical log surface is the following

Example 2.3. Given a log surface (X, [DELTA]), where X is smooth and [DELTA] is a boundary. (X, [DELTA]) is not necessarily log canonical in the usual sense. But id : X [right arrow] X is the minimal desingularization, therefore (X, [DELTA]) is MR log canonical.

3. Main results. Now we go to the proof of Theorem 1.2. Note that e in this theorem could be zero:

Theorem 3.1. Notations are as in Theorem 1.1. Let [epsilon] be a real number such that 0 [less than or equal to] [epsilon] < 1. If X is smooth and the coefficients of [DELTA] are [less than or equal to] 1 - [epsilon], then [X.sub.i] is [epsilon]-log terminal for every i. In particular, [X.sup.*] is [epsilon]-log terminal.

Proof. Step 1. Run log minimal model program (Theorem 1.1) for [K.sub.X] + [DELTA] as in Theorem 1.1:

(X, [DELTA]) = ([X.sub.0], [[DELTA].sub.0]) [right arrow] ([X.sub.1], [[DELTA].sub.1]) [right arrow] *** [right arrow] ([X.sub.k], [[DELTA].sub.k]) = ([X.sup.*], [[DELTA].sup.*])

where ([X.sup.*], [[DELTA].sup.*]) is a minimal model or a Mori fiber space. In the following proof, we consider everything over [X.sub.j] for a fixed j. Put [X.sup.[dagger]] = [X.sub.j] for this fixed j. Then take [X.sup.[dagger]] as a base (that is, replace S by [X.sup.[dagger]] and hence we are reduced to the case where S = [X.sup.*] = [X.sub.j]. Therefore, if needed, shrink [X.sup.[dagger]] to be affine since [epsilon]-log terminal or not is a local property) and run ([K.sub.X] + A)-LMMP for the relative morphism f : X [right arrow] [X.sup.[dagger]], which ends up again on [X.sup.[dagger]] and [mathematical expression not reproducible] is nef over [X.sup.[dagger]]. In each step we have a relative morphism [X.sub.i] [right arrow] [X.sup.[dagger]] (i [less than or equal to] j) and denote it by [X.sub.i]/[X.sup.[dagger]]. We use [f.sub.i] and [h.sub.i] to denote the morphisms ([X.sub.i], [[DELTA].sub.i])/[X.sup.[dagger]] [right arrow] ([X.sup.[dagger]], [[DELTA].sup.[dagger]])/[X.sup.[dagger]] and (X, [DELTA])/[X.sup.[dagger]] [right arrow] ([X.sub.i], [[DELTA].sub.i])/[X.sup.[dagger]] with that [h.sub.j] = [f.sub.0] = f. By [Fjn12, Section 3] and [Tnk14, Section 3],

[mathematical expression not reproducible],

where [E.sub.i] is all effective over [X.sup.[dagger]] for every 0 [less than or equal to] i < j. In particular, [mathematical expression not reproducible]. Furthermore, every curve in Exc(f) = Supp([E.sub.0]) is a smooth rational curve by [Fjn12, Proposition 3.8] and [Tnk14, Theorem 3.19].

Step 2. Now we may assume that there is no (-1)-curve in Exc(f). Indeed, if there is some (-1)-curve, say C, in Exc(f), then by Castelnuovo's theorem, contracting this (-1)-curve in X/[X.sup.[dagger]] leads to a new smooth surface X'/[X.sup.[dagger]]. Therefore we can run another ([K.sub.X'] + [DELTA]')-LMMP over [X.sup.[dagger]] until reaching to a final log surface [mathematical expression not reproducible], where [DELTA]' is the image of [DELTA]. Every assumption of (X, [DELTA]) is obviously keeping if we replace (X, [DELTA]) by (X', [DELTA]') except that we need to prove [mathematical expression not reproducible]. We have three morphisms over [X.sup.[dagger]]: [pi]: X [right arrow] X', g : X' [right arrow] [??] and [rho] : [??][right arrow] [X.sup.[dagger]] such that

[mathematical expression not reproducible]

where [pi] : X [right arrow] X' is the Castelnuovo's contraction, [rho] is not necessarily the identity and [mathematical expression not reproducible] is nef over [X.sup.[dagger]]. Then by negativity lemma (see [KM98, Lemma 3.39 and Lemma 3.40]), we have that -D [greater than or equal to] 0, since [mathematical expression not reproducible] is [rho]-exceptional. Remember that [mathematical expression not reproducible]. That is, [E.sub.0] [~.sub.f] [[pi].sup.*] [g.sup.*] D + [[pi].sup.*] [E.sub.0] + aC. By negativity lemma again, D > 0 since [E.sub.0] is effective and both sides have the same support. Therefore we get a contradiction unless p is the identity. That is, [mathematical expression not reproducible]. Then, by contracting (-1)-curves finitely many times, we may assume that Exc(f) contains no (-1)-curve from now on.

Step 3. Assume that [C.sub.i] is the contracted curve in step i of the log minimal model program, then [mathematical expression not reproducible]. Therefore

[mathematical expression not reproducible].

Note that [([h.sup.*.sub.i] ([C.sub.i])).sup.2] = [([C.sub.i]).sup.2] < 0 by the negativity lemma. Then [K.sub.X] - [h.sup.*.sub.i] ([C.sub.i]) [greater than or equal to] 0 since [h.sup.*.sub.i] ([C.sub.i]) is effective and its support contains no (-1)-curve. Indeed, if [K.sub.X] - [h.sup.*.sub.i] ([C.sub.i]) < 0, there must be a curve, say E, in Supp[h.sup.*.sub.i] ([C.sub.i]) such that [K.sub.X] x E < 0. But [E.sup.2] < 0 since E is in Exc(f). Thus it is a (-1)-curve which contradicts our assumption. Therefore [DELTA] - [h.sup.*.sub.i] ([C.sub.i]) < 0. Then

[mathematical expression not reproducible].

That is, [C.sub.i] is in Supp [[DELTA].sub.i], and its strict transform is in Supp[DELTA]. Therefore all those curves in Exc(f) must be such a strict transform of [C.sub.i] under the assumption of the above step.

Step 4. Next, we need to prove that, for the resolution f : X [right arrow] [X.sup.[dagger]] where [mathematical expression not reproducible], we have that [a.sub.i] > -1 + [epsilon]. Note that [mathematical expression not reproducible] where [E.sub.0] is effective in Exc(f) and [F.sub.i] is in Supp[DELTA] by the above steps. Furthermore, let [DELTA] = [SIGMA] [[delta].sub.i] [F.sub.i] + [DELTA]' where [SIGMA] [F.sub.i] and [DELTA]' have no common components. Therefore, [f.sub.*] [DELTA]' = [[DELTA].sup.[dagger]]. Then

[mathematical expression not reproducible].

That is,

[SIGMA]([a.sub.i] + [[delta].sub.i])[F.sub.i] = [f.sup.*] [[DELTA].sup.[dagger]] - [DELTA]' + [E.sub.0]

in which both sides are supported in Exc(f) and the right hand side is effective since [f.sup.*] [[DELTA].sup.[dagger]] - [DELTA]' = [f.sup.*] [f.sub.*] [DELTA]' - [DELTA]' and [DELTA]' is effective. Note that Supp[E.sub.0] = Supp(Exc(f)) by our log minimal model program (Theorem 1.1). Thus comparing both sides, [a.sub.i] + [[delta].sub.i] > 0. That is, [a.sub.i] > - [[delta].sub.i] [greater than or equal to] -1 + [epsilon] since the coefficients of [DELTA] are [less than or equal to] 1 - [epsilon].

Step 5. We claim that, the resolution f : X [right arrow] [X.sup.[dagger]] is a log resolution. That is, the reduced [SIGMA] [F.sub.i] must be a simple normal crossing curve. Remember that [F.sub.i] is all smooth extremal rational curves since [X.sup.[dagger]] has rational singularities by [FT12, Theorem 6.2] for any characteristic. Furthermore, the dual graph of [SIGMA] [F.sub.i] must be a tree. This shows that the reduced [SIGMA] [F.sub.i] must be a simple normal crossing curve. We get what we want.

Remark 3.2. It is pointed out by Tanaka that, our claim in Step 5 can be proved by [KM98, Theorem 4.7].

From the above theorem, we know that when X is smooth, those contracting curves in log minimal model program consist of some images of (-1)-curves and some components of Supp[DELTA]. Several direct but important implications of Theorem 3.1 are the following. When [K.sub.X] + [DELTA] is big, [mathematical expression not reproducible] is nef and big on the minimal model. What we have done in the proof of Theorem 3.1 is in fact showing that f : X' [right arrow] [X.sup.*] is exactly the minimal desingularization and ([X.sup.*], [[DELTA].sup.*]) is MR [epsilon]-log terminal. Then the following corollaries are just simple consequences of [Alex94, Theorem 7.6, Theorem 7.7, Theorem 8.2]. It is another way to see that Fujino and Tanaka's minimal model program on log surfaces cover Alexeev's minimal model program stated in [Alex94, Section 10].

Corollary 3.3. Let (X, [DELTA]) be a projective log surface where X is smooth and [K.sub.X] + [DELTA] is big. Fixing [epsilon] > 0, let I [subset] [0, 1 - [epsilon]] be a DCC set and the coefficients of [DELTA] be in I. If there is a positive integer M such that [mathematical expression not reproducible] where ([X.sup.*], [[DELTA].sup.*]) is a minimal model of (X, [DELTA]), then these ([X.sup.*], Supp[[DELTA].sup.*]) belong to a bounded family.

Corollary 3.4. Let (X, [DELTA]) be a projective log surface where X is smooth and [K.sub.X] + [DELTA] is big. Fixing [epsilon] [greater than or equal to] 0, let I [subset] [0, 1 - [epsilon]] be a DCC set and the coefficients of [DELTA] be in I. Then [mathematical expression not reproducible] is a DCC set. In particular the volume vol ([K.sub.X] + [DELTA]) is bounded from below away from 0.

Proof. Since [mathematical expression not reproducible] by Theorem 3.1, this corollary is a direct consequence of [Alex94, Theorem 8.2].

Remark 3.5. Note that in Corollary 3.3, the [epsilon] is smaller, the bounded family of ([X.sup.*], Supp[[DELTA].sup.*]) is bigger. When e goes to 0, all those [X.sup.*] may not be in a bounded family, so may not be ([X.sup.*], Supp[[DELTA].sup.*]). See [Lin03, Remark 1.5] for the example showing that [X.sup.*] could be Q-Fano and not in a bounded family. Note also that Corollary 3.4 is an answer of the question coming from the first version of Di Cerbo's paper which has been confirmed by his second version [Dic, Corollary 4.3].

4. Examples. By [Alex94, Section 10], we easily see that if the log surface (X, [DELTA]) is MR [epsilon]-log canonical, then so is every ([X.sub.i], [[DELTA].sub.i]) in the step of log minimal model program; by Grothendieck spectral sequence and [FT12, Theorem 6.2], it is also easy to see that if X has only rational singularities, then so has every [X.sub.i]. Now it is natural to generalize Theorem 3.1 and ask that if X is [epsilon]-log canonical, is so every [X.sub.i] or not. But unfortunately we have the following example:

Example 4.1. Let us first recall an example of log canonical surface which is rational but not log terminal. Blowing up at a point of [P.sup.2], we get a (-1)-curve [E.sub.0]; find three points at [E.sub.0] and blow up several times (at these three points and some points at the exceptional curves over them), we can easily get a smooth and projective surface Y and four smooth rational curves [E.sub.0], [E.sub.1], [E.sub.2], [E.sub.3] on it such that [n.sub.0] = -[E.sup.2.sub.0] [greater than or equal to] 3, [n.sub.1] = - [E.sup.2.sub.1] = 2, [n.sub.2] = - [E.sup.2.sub.2] = 3, [n.sub.3] = -[E.sup.2.sub.3] = 6 where by abusing of notations, we still use [E.sub.0] to denote its strict transform on Y. By construction, [E.sub.i] x [E.sub.0] = 1, [E.sub.i] x [E.sub.j] = 0 for 1 [less than or equal to] i < j [less than or equal to] 3. Let E = [E.sub.0] + [E.sub.1] + [E.sub.2] + [E.sub.3], then its dual graph is a triple fork. See also that its intersection matrix is negative definite. Therefore by Artin's criterion [Art62, Theorem 2.3], we can contract E and finally get a surface X with a singular point. Now we have f : Y [right arrow] X with [K.sub.Y] = [f.sup.*] [K.sub.X] + [SIGMA] [a.sub.i] [E.sub.i]. Using adjunction, we have that:

[mathematical expression not reproducible].

Solving these equations we have [a.sub.0] = -1, [a.sub.1] = - 1/2, [a.sub.2] = - 2/3, [a.sub.3] = - 5/6. These show that the singularity of X is exactly log canonical but not log terminal.

Keeping this example in mind, we construct an example as follows:

Similar to the above blowing-up method, we can easily construct a surface Y and five smooth rational curves D, [E.sub.0], [E.sub.1], [E.sub.2], [E.sub.3] on it such that n = -[D.sup.2] is as big as we want, [n.sub.0] = -[E.sup.2.sub.0] [greater than or equal to] 3, [n.sub.1] = -[E.sup.2.sub.1] = 2, [n.sub.2] = -[E.sup.2.sub.2] = 3, [n.sub.3] = -[E.sup.2.sub.3] = 6 and [E.sub.i] x [E.sub.0] = D x [E.sub.0] = 1, [E.sub.i] x [E.sub.i] = D x [E.sub.i] = 0 for 1 [less than or equal to] i < j [less than or equal to] 3. Let E = [E.sub.0] + [E.sub.1] + [E.sub.2] + [E.sub.3] and F = E + D. Then E is a triple fork and F is a quadruple fork in dual graph. Note that both of the intersection matrices of E and F are negative definite which are exercises of diagonalization of matrix. By contracting E on Y we get a morphism f from Y to a log canonical surface X which is rational but not log terminal as above. Now consider the log surface (X, D') where D0 is the image of the smooth rational curve D. Note that D and D' are isomorphic outside the point E [intersection] D and its image. Note also that ([K.sub.X] + D') x D' < 0. Indeed, let [f.sup.*] D' = D + [SIGMA] [c.sub.i] [E.sub.i]. Then by [E.sub.i] x [f.sup.*] D' = 0,

[mathematical expression not reproducible].

That is, [c.sub.0] = 1/[n.sub.0] - 1, [c.sub.1] = [c.sub.0]/2, [c.sub.2] = [c.sub.0]/3, [c.sub.3] = [c.sub.0]/6. Then

[mathematical expression not reproducible]

since [c.sub.0] = 1/[n.sub.0] - 1 < 1. Therefore, by [Tnk14, (1) of Theorem 3.19], D' is a smooth rational curve. Moreover, contracting D' on X is indeed a step of log minimal model program (Theorem 1.1). Finally, we get a log surface ([X.sup.*], 0) where [X.sup.*] is no longer log canonical since the dual graph of F is a quadruple fork which is not in the classification of dual graph of log canonical singularities in [KM98, Theorem 4.7]. Furthermore, it is not even MR log canonical by calculating the discrepancy of [E.sub.0] (which is 2 - [nn.sub.0]/[nn.sub.0] - n < -1). But remember that [X.sup.*] still has rational singularities by [FT12, Theorem 6.2]. This example proves Proposition 1.3.

Example 4.2. We just gave an example for Proposition 1.3 where [epsilon] = 0. In fact, by a similar construction as above, we can get some examples where [epsilon] > 0. [DELTA] sketch of construction is the following. As Example 4.1, we can easily construct a smooth and projective surface Y and five smooth rational curves D, [E.sub.0], [E.sub.1], [E.sub.2], [E.sub.3] on it with n = -[D.sup.2], [n.sup.i] = -[E.sup.2.sub.i] such that n = 3, [n.sub.0] = 5, [n.sub.1] = [n.sub.2] = [n.sub.3] = 2, [E.sub.i] - [E.sub.0] = D x [E.sub.0] = 1, [E.sub.i] x [E.sub.j] = D x [E.sub.i] = 0 for 1 [less than or equal to] i < j [less than or equal to] 3. Let E = [E.sub.0] + [E.sub.1] + [E.sub.2] + [E.sub.3] and F = E + D. Then E is a triple fork and F is a quadruple fork which is not in the classification of dual graph of log canonical singularities. Note that both of the intersection matrices of E and F are negative definite again by diagonalization of the intersection matrices. Choose an [epsilon] such that 0 < [epsilon] [less than or equal to] 1/7. The same calculation as Example 4.1 shows that by contracting E on Y we get a morphism f from Y to an [epsilon]-log canonical surface X. Now consider the log surface (X, bD') where D' is the image of D' and b is a non-negative real number. Note that D' is still a smooth rational curve by construction. Choose a proper real number b such that ([K.sub.X] + bD') x D' < 0. Indeed, by careful calculations as in Example 4.1, we can check that ([K.sub.X] + bD') x D' < 0 for b > 13/19. Therefore, ([K.sub.X] + (1 - [epsilon])D') x D' < 0 for 0 < [epsilon] [less than or equal to] 1/7. Now contracting D' on (X, bD') by log minimal model program, we get a log surface ([X.sup.*], 0) where [X.sup.*] is no longer log canonical. This gives an example to confirm Proposition 1.4.

Remark 4.3. The above two examples are based on one of the dual graphs of log canonical singularities in [KM98, Theorem 4.7]. In fact, we can construct similar examples based on the other dual graphs there and get a bunch of similar examples.

It will be interesting to ask the following question:

Question 4.4. In Theorem 1.1, if X is canonical, is X, log canonical?

doi: 10.3792/pjaa.93.115

Acknowledgements. The author would like to thank Prof. Osamu Fujino for so many inspirational suggestions and comments. The author would like to thank Dr. Hiromu Tanaka and the referees for their helpful comments too. He would also like to thank Dr. Chen Jiang for many discussions on MR log canonical when they were attending in the conference of Higher Dimensional Algebraic Geomentry (HDAG) held in Utah.

References

[Alex94] V. Alexeev, Boundedness and [K.sup.2] for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779 810.

[Art62] M. Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485 496.

[Dic] G. Di Cerbo, On Fujita's spectrum conjecture, arXiv:1603.09315v2.

[FT12] O. Fujino and H. Tanaka, On log surfaces, Proc. Japan Acad. Ser. [DELTA] Math. Sci. 88 (2012), no. 8, 109 114.

[Fjn12] O. Fujino, Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), no. 2, 339 371.

[Fjt84] T. Fujita, Fractionally logarithmic canonical rings of algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 685-696.

[KM98] J. Kollar and S. Mori, Birational geometry of algebraic varieties, translated from the 1998 Japanese original, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, 1998.

[Lin03] J. Lin, Birational unboundedness of Q-Fano threefolds, Int. Math. Res. Not. 2003, no. 6, 301 312.

[Tnk14] H. Tanaka, Minimal models and abundance for positive characteristic log surfaces, Nagoya Math. J. 216 (2014), 170.

By Haidong LIU

Department of Mathematics, Graduate School of Science, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan

(Communicated by Shigefumi MORI, M.J.A., Nov. 13, 2017)

2010 Mathematics Subject Classification. Primary 14E05; Secondary 14E30.

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Author: | Liu, Haidong |
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Publication: | Japan Academy Proceedings Series A: Mathematical Sciences |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Oct 1, 2017 |

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