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Termination of extremal rays of divisorial type for the power of etale endomorphisms.

1. Introduction. The purpose of this note is to give a partial answer to the following question concerning termination of extremal rays of divisorial type on a smooth projective variety for the iteration of non-isomorphic etale endomorphisms.

Question. Let f: X [right arrow] X be a non-isomorphic etale endomorphism of a smooth projective variety X. Suppose that there exists an extremal ray R' [subset] [bar.NE](X) such that [K.sub.X] R' < 0 and the associated contraction morphism [phi] := [Cont.sub.R']: X [right arrow] Y is a fibration to a lower dimensional variety Y. Then is it true that for any KX-negative extremal ray R [subset] [bar.NE](X) of divisorial type, there exists a positive integer k such that [([f.sup.k]).sub.*](R) = R for the automorphism [([f.sup.k]).sub.*] : [N.sub.1](X) [equivalent] [N.sub.1](X) induced from the k-th power [f.sup.k] = f [??] ... [??] f?

If we apply the minimal model program (MMP, for short, cf. [6]) to the study of non-isomorphic surjective endomorphisms of projective varieties, we sometimes encounter serious troubles: Let f: X [right arrow] X be a non-isomorphic etale endomorphism. Then a [K.sub.X]-negative extremal ray R([subset] [bar.NE](X)) is not necessarily preserved under a suitable power [([f.sup.k]).sub.*] (k > 0) of the push-foward mapping [f.sub.*]: [N.sub.1](X) [equivalent] [N.sub.1](X) (cf. [3]). Hence it is not at all clear that we can apply the MMP working compatibly with etale endomorphisms. Thus it is an interesting problem to give a sufficient condition for a [K.sub.X]-negative extremal ray R to be preserved under a suitable power of f. A special case of the Question was studied in [3, Theorems 1.4, 8.6 and 9.6], which gives an affirmative answer to the Question in the case where dimX = 3 and dimY = 2. In this note, we shall give some generalization.

Theorem 1.1. The Question has an affirmative answer under the assumption that [rho](Y) = 2.

If we drop the assumption that [rho](Y) = 2, then the Question does not necessarily have an affirmative answer. In Section 4, we shall give an easy counterexample using a rational elliptic surface with infinitely many (-1)-curves.

2. Notations and preliminaries. In this paper, we work over the complex number field C. A projective variety is a complex variety embedded in a projective space. By an endomorphism f: X [right arrow] X, we mean a morphism from a projective variety X to itself.

The following symbols are used for a variety X.

[K.sub.X]: the canonical divisor of X.

Aut(X): the algebraic group of automorphisms of X.

[N.sub.1](X) := ({1-cycles on X}/[equivalent to]) [[cross product].sub.Z] R, where [equivalent to] means a numerical equivalence.

[N.sup.1](X) := ({Cartier divisors on X}/[equivalent to]) [[cross product].sub.Z] R, where [equivalent to] means a numerical equivalence.

NE(X): the smallest convex cone in [N.sub.1](X) containing all effective 1-cycles.

[bar.NE](X): the Kleiman-Mori cone of X, i.e., the closure of NE(X) in [N.sub.1](X) for the metric topology.

[rho](X) := [dim.sub.R] [N.sub.1](X), the Picard number of X.

[C]: the numerical equivalence class of a 1-cycle C.

cl(D): the numerical equivalence class of a Cartier divisor D.

[~.sub.Q]: the Q-linear equivalence of Q-divisors of X.

For an endomorphism f: X [right arrow] X and an integer k > 0, [f.sup.k] stands for the k-times composite f [??] ... [??] f of f.

Extremal rays. For a smooth projective variety X, an extremal ray R means a [K.sub.X]-negative extremal ray of [[bar.NE](X), i.e., a 1-dimensional face of [[bar.NE](X) with [K.sub.X]R < 0. An extremal ray R defines a proper surjective morphism with connected fibers [Cont.sub.R] : X [right arrow] Y such that, for an irreducible curve C [subset] X, [Cont.sub.R](C) is a point if and only if [C] [member of] R. This is called the contraction morphism associated to R.

We recall the following fundamental result.

Lemma 2.1 (cf. [1, Propositions 4.2 and 4.12]). Let f: Y [right arrow] X be a finite surjective morphism between smooth projective n-folds with [rho](X) = [rho](Y). Then the following hold.

(1) The push-forward map [f.sub.*]: [N.sub.1](Y) [right arrow] [N.sub.1](X) is an isomorphism and [f.sub.*] [[bar.NE](Y) = [[bar.NE](X).

(2) Let [f.sub.*]: [N.sup.1](Y) [right arrow] [N.sup.1](X) be the map induced from the push-forward map D [??] [f.sub.*]D of divisors. Then the dual map [f.sup.*]: [N.sub.1](X) [right arrow] [N.sub.1](Y) is an isomorphism and [f.sup.*] [[bar.NE](X) = [[bar.NE](Y).

(3) If f is etale and [K.sub.X] is not nef, then [f.sup.*] and [f.sub.*] above give a one-to-one correspondence between the set of extremal rays of X and Y.

(4) Under the same assumption as in (3), for an extremal ray R([subset] [[bar.NE](Y)), and for the contraction morphisms [Cont.sub.R]: Y [right arrow] Y' and [mathematical expression not reproducible], there exists a finite surjective morphism f': Y' [right arrow] X' such that [mathematical expression not reproducible].

The following theorem and some arguments used in its proof play another key role for proving our main Theorem 1.1.

Theorem 2.2 (cf. [7, Lemma 6.2]). Let X be a normal Q-factorial projective variety with at most log-canonical singularities and f: X [right arrow] X a surjective endomorphism. Let R([subset] [[bar.NE](X)) be an extremal ray and [infinity] := [Cont.sub.R]: X [right arrow] Y the contraction morphism associated to R. Suppose that E([subset] X) be a subvariety such that dim([pi](E)) < dim(E) and [f.sup.-1](E) = E. Then [([f.sup.k]).sub.*](R) = R for some positive integer k.

Remark 2.3. In [7, Lemma 6.2], the assumption that 'X is Q-factorial' is missing.

3. Proof of Theorem 1.1. Let f: X [right arrow] X be

a non-isomorphic etale endomorphism of a smooth projective variety X. Suppose that there exists an extremal ray R' [subset] [[bar.NE](X) such that for the contraction morphism [pi] := [Cont.sub.R]: X [right arrow] Y associated to R' is a Mori fiber space (i.e., dimY < dimX). Then by [5, Lemma 5-1-5], Y is Q-factorial. At first, we impose no restrictions on the Picard number [rho](Y) of Y. Then by [2], there exists a positive integer k such that [([f.sup.k]).sub.*] R' = R'. Hence replacing f by its power [f.sup.k], we may assume from the beginning that [f.sub.*] R' = R'. Hence there exists a surjective endomorphism g: Y [right arrow] Y such that [phi] [??] f = g [??] [phi]. Since [phi] is a Fano fibration, the general fiber of [phi] is simply-connected. Since f is etale, [mathematical expression not reproducible] gives a Stein factorization of [phi] [??] f: X [right arrow] Y and g is also a non-isomorphic etale endomorphism with deg(g) = deg(f). Thus we have the following Cartesian diagram:

We take an arbitrary extremal ray R([subset] [[bar.NE](X)) of divisorial type. We put [R.sub.n] := [([f.sup.n]).sub.*] R for n > 0 and [R.sub.0] := R. Then, by Lemma 2.1, Rn([subset] [[bar.NE](X)) is also an extremal ray for any n [greater than or equal to] 0 and let [mathematical expression not reproducible] be a divisorial contraction associated to [R.sub.n]. Then, by Lemma 2.1, there is induced a non-isomorphic finite morphism [g.sub.n] : [Z.sub.n] [right arrow] [Z.sub.n+1] which is etale in codimension one such that [[psi].sub.n+1] [??] f = [g.sub.n] [??] [[psi].sub.n] and deg([g.sub.n]) = deg(f) for any n [greater than or equal to] 0. In summary, there exists another commutative diagram below:

This diagram is Cartesian over a non-singular locus [Z.sup.0.sub.n+1] of [Z.sub.n+1]. Let [E.sub.n] := Exc([[psi].sub.n]) be the [[psi].sub.n]-exceptional divisor. Then [f.sup.-1]([E.sub.n+1]) = [E.sub.n] for any n [greater than or equal to] 0. Let [gamma]([subset] X) be an extremal rational curve whose numerical class [[gamma]] spans the extremal ray R'. Then [gamma] is contracted to a point by [psi]. Since each [E.sub.n] is effective, the restriction of [E.sub.n] to a general fiber of [phi] is also effective. Hence ([E.sub.n], [gamma]) [greater than or equal to] 0 for any n [greater than or equal to] 0.

Lemma 3.1. We have two cases;

(1) ([E.sub.n], [gamma]) = 0 for any n [greater than or equal to] 0, or

(2) ([E.sub.n], [gamma]) > 0 for any n [greater than or equal to] 0.

Proof. Since [f.sub.*]R' = R', we have [f.sub.*][gamma] = a[gamma] for a positive rational number a. If we set E := [E.sub.0], then E = [([f.sup.n]).sup.*] [E.sub.n]. Hence by the projection formula, we see that (E,[gamma]) = ([E.sub.n], [([f.sup.n]).sub.*] [gamma]) = [a.sup.n]([E.sub.n], [gamma]) for any n [greater than or equal to] 0, which shows the claim.

We set k := dimY. Let [l.sub.n] be an extremal rational curve whose numerical class [[l.sub.n]] spans the extremal ray [R.sub.n], i.e., [R.sub.n] = [R.sub.[greater than or equal to]0] [[l.sub.n]].

Case (1). First, we consider the case that ([E.sub.n], [gamma]) = 0 for any n [greater than or equal to] 0. Then by [4], we see that for any n [greater than or equal to] 0, there exists a Cartier divisor [D.sub.n] on Y such that [E.sub.n] ~ [[phi].sup.*] [D.sub.n]. Since [E.sub.n] = [f.sup.*][E.sub.n+1] and [phi] [??] f = g [??] [phi], we see that [E.sub.n] ~ [[phi].sup.*] [g.sup.*] [D.sub.n+1]. Hence [D.sub.n] ~ [g.sup.*] [D.sub.n+1] for any n [greater than or equal to] 0. We have the following easy lemma.

Lemma 3.2. [D.sub.n] is not nef and [([D.sub.n]).sup.k] = 0 for any n [greater than or equal to] 0.

Proof. Since [R.sub.n] [not equal to] R', [phi]([l.sub.n]) is an irreducible curve on Y. Hence, by the projection formula, we see that ([D.sub.n],[[phi].sub.*],[l.sub.n]) = ([[phi.sup.]*] [D.sub.n], [l.sub.n]) = ([E.sub.n], [l.sub.n]) < 0. Thus [D.sub.n] is not nef. For the latter assertion, we may assume that n = 0 without loss of generality. Since [D.sub.n] ~ [g.sup.*] [D.sub.n+1], we see that [([D.sub.n]).sup.k] = deg(g)[([D.sub.n+]1).sup.k] for any n [greater than or equal to] 0. Hence [([D.sub.0]).sup.k] = [(deg(g)).sup.n][([D.sub.n]).sup.k] for any n [greater than or equal to] 0. Suppose that [([D.sub.0]).sup.k] [not equal to] 0. Since deg(g) = deg(f) [greater than or equal to] 2, we see that 0 < [([D.sub.n]).sup.k] < 1 for a sufficiently positive integer n. This contradicts the fact that [([D.sub.n]).sup.k] [member of] Z. Thus [([D.sub.0]).sup.k] = 0.

Case (2). Next, we consider the case that ([E.sub.n], [gamma]) > 0 for any n [greater than or equal to] 0. We begin with an easy lemma.

Lemma 3.3. We have [f.sub.*][gamma] [equivalent] [gamma].

Proof. Since [f.sub.*]R' = R', we have [f.sub.*][gamma] [equivalent] a[gamma] for a positive rational number a. We set E := [E.sub.0]. Then by the projection formula, we have ([([f.sup.n]).sup.*]E, [gamma]) = (E, [([f.sup.n]).sub.*] [gamma]) = [a.sup.n](E, [gamma]) > 0 for any n > 0. Hence, if 0 < a < 1, then 0 < ([([f.sup.n]).sup.*]E, [gamma]) < 1 for a sufficiently positive integer n, which contradicts the fact that ([([f.sup.n]).sup.*] E, [gamma]) [member of] Z. Suppose that a > 1. Then by the same argument as in the proof of Lemma 3.1, we have (E, [gamma]) = [a.sup.n]([E.sub.n],[gamma]) for any n > 0. Thus 0 < ([E.sub.n], [gamma]) < 1 for a sufficiently positive integer n, which contradicts the fact that ([E.sub.n],[gamma]) [member of] Z. Hence a = 1.

Lemma 3.4. There exists a positive rational number b such that ([K.sub.X] + b[E.sub.n], [gamma]) = 0 for any n [greater than or equal to] 0.

Proof. Since ([K.sub.X], [gamma]) < 0 and ([E.sub.n], [gamma]) > 0, there exists a unique positive rational number [b.sub.n] such that ([K.sub.X] + [b.sub.n][E.sub.n], [gamma]) = 0 for any n [greater than or equal to] 0. Combining Lemma 3.3 with the fact that [E.sub.n-1] = [f.sup.*][E.sub.n], the projection formula shows that ([E.sub.n-1], [gamma]) = ([E.sub.n], [f.sub.*][gamma]) = ([E.sub.n], [gamma]). Since

[b.sub.n] = (-[K.sub.X], [gamma])/([E.sub.n], [gamma]),

we see that [b.sub.n-1] = [b.sub.n] for any n > 0. Thus [b.sub.n] [equivalent] b is indedependent of n.

Then by [4], for any n [greater than or equal to] 0, there exists a Q-Cartier Q-divisor [D'.sub.n] on Y such that [K.sub.X] + b[E.sub.n] [~.sub.Q] [[phi].sup.*] [D'.sub.n].

Lemma 3.5. Each [D'.sub.n] is not nef and [D'.sub.n-1] [~.sub.Q] [g.sup.*][D'.sub.n] for any n > 0. In particular, [D'.sub.n] is not numerically trivial.

Proof. Since ([K.sub.X], [l.sub.n]) < 0, ([E.sub.n], [l.sub.n]) < 0 and b > 0, we see by the projection formula that ([D'.sub.n], [[phi].sub.*] [l.sub.n]) = ([[phi].sup.*][D'.sub.n], [l.sub.n]) = ([K.sub.X], [l.sub.n])+ b ([E.sub.n], [l.sub.n]) < 0. Since [R.sub.n] [not equal to] R', [phi]([l.sub.n]) is an irreducible rational curve on Y and thus each [D'.sub.n] is not nef. Since [K.sub.X] ~ [f.sup.*][K.sub.X] and [f.sup.*][E.sub.n] = [E.sub.n-1], we see that

[K.sub.X] + b[E.sub.n-1] [~.sub.Q] [f.sup.*]([K.sub.X] + b[E.sub.n]) [~.sub.Q] [f.sup.*][[phi].sup.*][D'.sub.n] = [[phi].sup.*][g.sup.*][D'.sub.n].

Since [K.sub.X] + b[E.sub.n-1] [~.sub.Q] [[phi].sup.*][D'.sub.n-1], we have [D'.sub.n-1] [~.sub.Q] [g.sup.*][D'.sub.n] for any n > 0.

Lemma 3.6. For any n [greater than or equal to] 0, the self-intersection number of [D'.sub.n] is zero, i.e., [([D'.sub.n]).sup.k] = 0.

Proof. By Lemma 3.4,

b [equivalent] (-[K.sub.X], [gamma])/([E.sub.n], [gamma])

is independent of n. Hence a positive integer ([E.sub.n], [gamma]) is also independent of n. We set d := ([E.sub.n], [gamma]) and let [[GAMMA].sub.n] := d([K.sub.X] + b[E.sub.n]) be a Cartier divisor on X. Since ([[GAMMA].sub.n],[gamma]) = 0, by [4], the Cartier divisor [[GAMMA].sub.n] is linearly equivalent to the pullback of some Cartier divisor on Y by '. Hence for any n [greater than or equal to] 0, [[DELTA].sub.n] := d[D'.sub.n] is a Cartier divisor on Y and [[GAMMA].sub.n] ~ [[phi].sup.*][[DELTA].sub.n]. Since [[GAMMA].sub.n] ~ [f.sup.*] [[GAMMA].sub.n+1] ~ [f.sup.*][[phi].sup.*][[DELTA].sub.n+1] = [[phi].sup.*][g.sup.*][[DELTA].sub.n+1], we have [[DELTA].sub.n] ~ [g.sup.*][[DELTA].sub.n+1]. Since deg(g) > 1, applying the same method as in the proof of Lemma 3.3 to the Cartier divisor [[DELTA].sub.n] on Y, we see that [([[DELTA].sub.n]).sup.k] = 0. Hence [([D'.sub.n]).sup.k] = 0 for any n [greater than or equal to] 0.

Corollary 3.[gamma]. We have dimY [greater than or equal to] 2 and [rho](Y) [greater than or equal to] 2.

Proof. If dim(Y) [less than or equal to] 1, then by Lemma 3.6, [D'.sub.n] is numerically trivial, which contradicts Lemma 3.5. If [rho](Y) = 1, then by Lemma 3.5, - [D'.sub.n] is ample, which contradicts Lemma 3.6.

Proof of Theorem 1.1. Hereafter, we always assume that [rho](Y) = 2. Let [N.sup.1.sub.C](Y) := [N.sup.1](Y)[[cross product].sub.R] C be the complexification of the 2-dimensional R-vector space [N.sup.1](Y). We set H := {D [member of] [N.sup.1.sub.C](Y)|[D.sup.k] = 0}. Then H is an affine curve in [N.sup.1.sub.C](Y) [equivalent] [C.sup.2].

Case (1). First, we consider the case where ([E.sub.n], [gamma]) = 0 for any n [greater than or equal to] 0. For each n [greater than or equal to] 0, let [L.sub.n] := [<cl([D.sub.n])>.sub.C] be the 1-dimensional complex vector space spanned by cl([D.sub.n]). Then by Lemma 3.2, each affine line [L.sub.n] is an irreducible component of H. The pull-back [g.sup.*]: [N.sup.1.sub.C](Y) [equivalent] [N.sup.1.sub.C](Y) induces an automorphism of H. Since [g.sup.*][D.sub.n] ~ [D.sub.n-1], there is induced an isomorphism [mathematical expression not reproducible] for any n > 0. We have the following commutative diagram:

where j: H [??] [N.sup.1.sub.C](Y), [i.sub.n] : [L.sub.n] [??] H and [i.sub.n-1] : [L.sub.n-1] [??] H are all inclusions. The automorphism [g.sup.*][|.sub.H] : H [equivalent] H induces a permutation of a finite number of irreducible components of H. Hence, replacing g (resp. f) by its suitable power [g.sup.k](k > 0) (resp. [f.sup.k]), there exists some positive integer n such that [L.sub.n] = [g.sup.*][L.sub.n] = [L.sub.n-1]. Pulling back by [phi]: X [??] Y, we have [E.sub.n] [equivalent to] [lambda][E.sub.n-1] for some [lambda] [member of] [Q.sub.>0], since any [E.sub.n] is a non-zero effective divisor. Since ([l.sub.n-1], [E.sub.n]) = [lambda]([l.sub.n-1], [E.sub.n-1]) < 0, we see that [l.sub.n-1] [subset] [E.sub.n]. Since [l.sub.n-1] sweeps out [E.sub.n-1], we have [E.sub.n-1] [subset] [E.sub.n]. Thus [E.sub.n-1] = [E.sub.n], since [E.sub.n] is irreducible. Hence [f.sup.-1]([E.sub.n]) = [E.sub.n-1] = [E.sub.n] and we can apply Theorem 2.2 to the [[psi].sub.n]-exceptional divisor [E.sub.n]. There exists a positive integer p such that [([f.sup.p]).sub.*] [R.sub.n] = [R.sub.n]. Since [f.sub.*] : [N.sub.1](X) [equivalent] [N.sub.1](X) is an automorphism, we have [([f.sup.p]).sub.*]R = R.

Case (2). Next, we consider the case where ([E.sub.n], [gamma]) > 0 for any n [greater than or equal to] 0. For each n [greater than or equal to] 0, let [L'.sub.n] : = [<cl([D'.sub.n])>.sub.C] be the 1-dimensional complex vector space spanned by cl([D'.sub.n]). Then by Lemma 3.6, each affine line [L'.sub.n] is an irreducible component of the affine curve H. Since [f.sup.*]([K.sub.X] + b[E.sub.n]) [~.sub.Q] [K.sub.X] + b[E.sub.n-1], we see that [g.sup.*]([D'.sub.n]) [~.sub.Q] [D'.sub.n-1]. Thus the automorphism [g.sup.*] [member of] Aut([N.sup.1][(Y).sub.C]) induces an isomorphism [mathematical expression not reproducible] for each n > 0. Then we can apply the same argument as in the Case (1). After replacing g (resp. f) by its suitable power [g.sup.k] (k > 0) (resp. [f.sup.k]), there exists some positive integer n such that [L'.sub.n] = [g.sup.*] [L'.sub.n] = [L'.sub.n-1]. Pulling back by [phi]: X [right arrow] Y, we have [K.sub.X] + b[E.sub.n-1] = [lambda]([K.sub.X] + b[E.sub.n]) for some [lambda] [member of] Q. We have [lambda] [not equal to] 0, since [L'.sub.n-1] is not numericall trivial by Lemma 3.5. If [lambda] < 0, then we replace f by [f.sup.2]. Thus we may assume from the beginning that [lambda] > 0.

Lemma 3.8. We have [lambda] = 1.

Proof. Suppose that 0 < [lambda] < 1. Since [lambda]b[E.sub.n] [equivalent to] (1 - [lambda])[K.sub.X] + b[E.sub.n-1], we have

[lambda]b([E.sub.n], [l.sub.n-1]) = (1 - [lambda])([K.sub.X], [l.sub.n-1]) + b([E.sub.n-1], [l.sub.n-1]) < 0.

Hence ([E.sub.n], [l.sub.n-1]) < 0 and [l.sub.n-1] is contained in [E.sub.n]. Since [l.sub.n-1] sweeps out [E.sub.n-1], we have [E.sub.n-1] [subset] [E.sub.n]. Thus [E.sub.n-1] = [E.sub.n], since [E.sub.n] is irreducible. Then [K.sub.X] + b[E.sub.n] [equivalent to] [lambda]([K.sub.X] + b[E.sub.n]), which shows that [K.sub.X] + b[E.sub.n] [equivalent to] 0. This contradicts Lemma 3.5. Next, suppose that [lambda] > 1. Since b[E.sub.n-1] = ([lambda] - 1)[K.sub.X] + b[lambda][E.sub.n], we have

b([E.sub.n-1], [l.sub.n]) = ([lambda] - 1)([K.sub.X], [l.sub.n]) + b[lambda]([E.sub.n], [l.sub.n]) < 0.

Hence ([E.sub.n-1], [l.sub.n]) < 0. By the same argument as above, we have [E.sub.n-1] = [E.sub.n] and [K.sub.X] + b[E.sub.n] = 0, which again contradicts Lemma 3.5.

By Lemma 3.8, we have [E.sub.n-1] = [E.sub.n]. Then applying the same argument as in the Case (1), we see that [E.sub.n-1] = [E.sub.n]. Thus [E.sub.n] = [E.sub.n-1] = [f.sup.-1]([E.sub.n]). Hence we can apply Theorem 2.2 to the [[psi].sub.n]-exceptional divisor [E.sub.n]. By the same argument as in the Case (1), we have [([f.sup.p]).sub.*]R = R for some integer p > 0.

4. Counterexamples. The Question does not necessarily have an affirmative answer if we drop the assumption that [rho](Y) = 2. We shall construct such an example (cf. [2], Remark A.9). Let S be a rational elliptic surface with global sections whose Mordell-Weil rank is positive. It is obtained as 9-points blowing-up of [P.sup.2]. We regard S as an elliptic curve [C.sub.K] defined over the function field K of the base curve C. Since S is relatively minimal, the translation mapping [C.sub.K] [right arrow] [C.sub.K] given by the non-torsion section [gamma] induces a relative automorphism t: S [equivalent] S over C, which is of infinite order. Let X := S x E x [P.sup.1] be the product variety of S, an elliptic curve E and [P.sup.1]. We take a point o [member of] E and a point 0 [member of] [P.sup.1]. Since [gamma] is a (-1)-curve on S, the curve [gamma] x {o} x {0} on X spans the extremal ray R([subset] [bar.NE](X)) of divisorial type. If we denote by [[mu].sub.k]: E [right arrow] E multiplication by k > 1, then the product mapping f := t x [[mu].sub.k] x id [p.sup.1] : X [right arrow] X gives a non-isomorphic etale endomorphism of X. We set Y := S x E and let [p.sub.1,2] : X [right arrow] Y and p: Y [right arrow] S be natural projections. Then [p.sub.1,2] is a trivial [P.sup.1]-bundle, which is a Mori fiber space. Furthermore, the product mapping g := t x [[mu].sub.k]: Y [right arrow] Y also gives a non-isomorphic etale endomorphism of Y.

We have the following Cartesian diagram:

If [([f.sup.n]).sub.*]R = R for some n > 0, then [t.sup.n]([gamma]) = [gamma]. This is impossible, since t [member of] Aut (S/C) is of infinite order. Hence, [([f.sup.n]).sub.*]R [not equal to] R for any n > 0. In this case, [rho](S) = 10 and [rho](Y) [greater than or equal to] 10. Let [psi] := [Cont.sub.R]: X [right arrow] Z be the extremal contraction associated to R and [DELTA] := Exc([psi]) the [psi]-exceptional divisor. Then [DELTA] [equivalent] [gamma] x E x [P.sup.1] does not intersect with the general fiber of [p.sub.1,2]. Thus our counterexample corresponds to the Case (1) in Section 3.

doi: 10.3792/pjaa.95.47

Acknowledgement. The author wishes to express sincere thanks to Prof. Noboru Nakayama for many useful discussions and to the referee for the careful reading of the manuscript.

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By Yoshio FUJIMOTO

Department of Mathematics, Faculty of Liberal Arts and Sciences, Nara Medical University, 840, Shijo-cho, Kashihara, Nara 634-8521, Japan

(Communicated by Shigefumi MORI, M.J.A., April 12, 2019)

2010 Mathematics Subject Classification. Primary 14J15, 14J25, 14J30, 14J60; Secondary 32J17.
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Author:Fujimoto, Yoshio
Publication:Japan Academy Proceedings Series A: Mathematical Sciences
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Date:May 1, 2019
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