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Toric 2-Fano manifolds and extremal contractions.

1. Introduction. We assume all algebraic varieties are defined over C. A Fano manifold X is a smooth projective algebraic variety with the ample first Chern class [c.sub.1](X) = - [K.sub.X]. In order to study rational surfaces on a Fano manifold, the following special class of Fano manifolds was introduced in de Jong-Starr [4]:

Definition 1.1. A Fano manifold X is a 2-Fano (resp. weakly 2-Fano) manifold if the second Chern character

[ch.sub.2](X) = 1/2 [([(c.sub.1](X)).sup.2] - 2[c.sub.2](X))

is ample (resp. nef). Here, an algebraic cycle F of codimension 2 on X is ample (resp. nef) if the intersection number (F x S) is positive (resp. (F x S) [greater than or equal to] 0) for any surface S in X.

The classification is an important problem for 2-Fano manifolds, and Schrack [9] shows the following results:

Theorem 1.2 (Schrack [9]). Let X be a 2-Fano manifold of dim X = 4. Then, the following hold:

(a) If there exists a Fano contraction [[phi].sub.R] : X [right arrow] [bar.X], then [bar.X] is a point.

(b) If there exists a divisorial contraction [[phi].sub.R] : X [right arrow] [bar.X] with the exceptional divisor E, then [[phi].sub.R](E) is not a point.

The purpose of this paper is to generalize Theorem 1.2 for any dimension d when X is a toric variety. Namely, we obtain the following

Theorem 1.3. Let X be a toric 2-Fano manifold of dimension d = dim X [greater than or equal to] 2. Then, the following hold:

(a) If there exists a Fano contraction [[phi].sub.R] : X [right arrow] [bar.X], then X is a point.

(b) Suppose that d [greater than or equal to] 3. If there exists a divisorial contraction [[phi].sub.R] : X [right arrow] [bar.X] with the exceptional divisor E, then [[phi].sub.R] (E) is not a point.

We will prove (a) of Theorem 1.3 in Section 3 without the assumption that X is Fano. Moreover, we determine the structure of a projective toric manifold X with the nef second Chern character [ch.sub.2](X) and a Fano contraction. For (b), we need the assumption that X is Fano. Namely, we use the classification of toric Fano manifolds. The assertion (b) is generalized for toric weakly 2-Fano manifolds.

2. Preliminaries. In this section, for the proof of the main theorem, we explain the way to calculate intersection numbers for toric cycles, and the notion of primitive collections and primitive relations. For the fundamental properties of toric manifolds, we refer to Cox-Little-Schenck [2], Fulton [3] and Oda [6].

Let X = [X.sub.[SIGMA]] be a smooth complete toric d-fold associated to a fan [SIGMA] in [Z.sup.d]. We put G([SIGMA]) as the set of the primitive generators for the 1-dimensional cones in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. be the torus invariant divisor corresponding to [x.sub.i]. For a torus invariant subvariety Y [subset] X of codimension l, we define the polynomial [I.sub.Y/X] = [I.sub.Y/X]([X.sub.1], ..., [X.sub.m]) [member of] [R.sub.x] := Z[[X.sub.1], ..., [X.sub.m]], by introducing the independent elements [X.sub.1], ..., [X.sub.m] associated to [x.sub.1], ..., [x.sub.m], respectively, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can say that [I.sub.Y/X] has all the numerical information of Y on X.

Example 2.1. For a torus invariant curve C = [C.sub.[sigma]] C X corresponding to a (d - 1)-dimensional cone [sigma] [member of] [SIGMA], let

[y.sub.1] + [y.sub.2] + [a.sub.1][x.sub.i] + ... + [a.sub.d-1][X.sub.d-1] = 0

be the wall relation associated to [sigma], where [y.sub.1], [y.sub.2], [x.sub.1], ..., [x.sub.d-1] [member of] G([SIGMA]) and [a.sub.1], ..., [a.sub.d-1] [member of] Z.

Then, [I.sub.C/X] = [Y.sub.1] + [Y.sub.2] + [a.sub.1][X.sub.i] + ... + [a.sub.d- 1] [X.sub.d-1] [member of] [R.sub.x], where [X.sub.1], ..., [X.sub.d-1], [Y.sub.1], [Y.sub.2] are the independent elements in Rx corresponding to [x.sub.1], ..., [x.sub.d-1], [y.sub.1], [y.sub.2], respectively.

So, we can easily calculate [I.sub.C/X] for a torus invariant curve C. For the case of surfaces, the following holds:

Theorem 2.2 (Sato [8]). Let S [subset] X be a torus invariant surface. Then the following hold:

(a) If S [congruent to] [P.sup.2], then [I.sub.S/X] = [([I.sub.C/X]).sup.2] for a torus invariant curve C [subset] S.

(b) If S is isomorphic to the Hirzebruch surface [F.sub.a] of degree a [greater than or equal to] 0, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [C.sub.fib] is the fiber of S [right arrow] [P.sup.1], while [C.sub.neg] is the negative section of S.

For a toric variety X, it is well known that

[ch.sub.2](X) = is ([D.sup.2.sub.1] + ... + [D.sup.2.sub.m]).

So, for a torus invariant surface S [subset] X, we can calculate ([ch.sub.2](X) x S) from [I.sub.S/X] easily. Therefore, Theorem 2.2 is crucial in our proof. We note that for the toric case, in order to check the ampleness or nefness for [ch.sub.2](X), it suffices to consider the intersection numbers with torus invariant surfaces. Namely, the following holds:

Proposition 2.3 (Nobili [5]). For a projective toric manifold X, [ch.sub.2](X) is ample (resp. nef) if and only if ([ch.sub.2](X) S) > 0 (resp. ([ch.sub.2](X) S) [greater than or equal to] 0) for any torus invariant surface S in X.

Next, we briefly introduce the notion of primitive collections and primitive relations. We will use these concepts to describe the fans for certain toric Fano manifolds in Section 4.

Definition 2.4. Let X = XE be the smooth complete toric d-fold associated to a fan E. A subset P [subset] G([SIGMA]) is called a primitive collection if it does not generate a cone in [SIGMA], while any proper subset generates a cone in [SIGMA].

For a primitive collection P = {[x.sub.1], ... [x.sub.s]} [subset] G([SIGMA]), there exists the unique element [sigma](P) [member of] [SIGMA] such that [x.sub.1] + ... + [x.sub.s] is contained in the relative interior of [sigma](P). So, we have a linear relation

[x.sub.1] + ... + [x.sub.s] = [a.sub.1] [y.sub.1] + ... + [a.sub.t][y.sub.t];

where {[y.sub.1], ..., [y.sub.t]} is the set of generators for [sigma](P) and [a.sub.1], ..., [a.sub.t] are positive integers. We call it the primitive relation corresponding to P.

We can recover the fan [SIGMA] from the data of all the primitive relations (see Proposition 3.6 in Sato [7]). So, we can describe a fan by giving all the primitive relations. We also remark that the primitive collections and primitive relations are convenient to deal with the toric Mori theory.

3. Fano contractions. The following theorem is an assertion for not necessarily Fano toric varieties.

Theorem 3.1. Let X be a smooth projective toric d-fold, and suppose that there exists a Fano contraction [[phi].sub.R]: X [right arrow] [bar.X]. Then, the following hold:

(a) If [ch.sub.2](X) is ample, then [bar.X] is a point, that is, X is isomorphic to [P.sup.d].

(b) If [ch.sub.2](X) is nef but not ample, then [[phi].sub.R] gives either a [P.sup.1]-bundle structure or a direct product structure.

Proof. Since X = [X.sub.[SIGMA]] is a smooth toric variety, [[phi].sub.R] simply gives a projective space bundle structure. So, let s - 1 be the dimension of a fiber of [[phi].sub.R]. There exists the primitive relation [x.sub.1] + ... + [x.sub.s] = 0 associated to [[phi].sub.R], where {[x.sub.1], ..., [x.sub.s]} [subset] G([SIGMA]). Suppose that s - 1 < d, that is, dim [bar.X] > 0. Then, we can take a (d - 1)-dimensional cone in [SIGMA] generated by {[x.sub.1], ..., [x.sub.s-1], [z.sub.1], ..., [z.sub.d-s]} [subset] G([SIGMA]). Let

[y.sub.1] + [y.sub.2] + [a.sub.1] [x.sub.1] + ... + [a.sub.s-1] [x.sub.s-1]

+ [b.sub.1] [z.sub.1] + ... + [b.sub.d-s][z.sub.d-s] = 0

be the associated wall relation, where both {[y.sub.1]; [x.sub.1], ... [x.sub.s-1] [z.sub.1], ..., [z.sub.d-s]} and ([y.sub.2], [x.sub.1], ..., [x.sub.s-1], [z.sub.1], ..., [z.sub.d-s]} generate maximal cones in [SIGMA], for [y.sub.1] [not equal to] [y.sub.2] [member of] G([SIGMA]) and [a.sub.1], ..., [a.sub.s-1], [b.sub.1], ..., [b.sub.d-s] [member of] Z. If max{[a.sub.1], ..., [a.sub.s-1]} = [a.sub.i] > 0, then by the equality [x.sub.i] = -([x.sub.1] + ... + [[??].sub.i] + ... + [x.sub.s]), the above wall relation becomes

[y.sub.1] + [y.sub.2] + ([a.sub.1] - [a.sub.1])[x.sub.1] + ... + [[??].sub.i] + ... + ([a.sub.s-1] - [a.sub.i])[x.sub.s-1] + (-[a.sub.i])[x.sub.s] + [b.sub.1][z.sub.1] + ... + [b.sub.d-s] [z.sub.d-s] = 0.

Therefore, by reordering [x.sub.1], ..., [x.sub.s], we can assume that [a.sub.1] [less than or equal to] ... [less than or equal to] [a.sub.s-1] [less than or equal to] 0. Put [tau] as the (d - 2)-dimensional cone in [SIGMA] whose generators are [x.sub.1], ..., [x.sub.s-2], [z.sub.1], ..., [z.sub.d-s]. Since [x.sub.s-1] + [x.sub.s] = 0 and [y.sub.1] + [y.sub.2] = (-[a.sub.s-1])[x.sub.s-1] in [R.sup.d]/Span[tau], the torus invariant subsurface S - [S.sub.[tau]] [subset] X associated to [tau] is isomorphic to the Hirzebruch surface [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So, Theorem 2.2 says that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [X.sub.1], ..., [X.sub.s], [Y.sub.1], [Y.sub.2], [Z.sub.1], ..., [Z.sub.d- s] are the independent elements in Rx corresponding to [x.sub.1], ..., [x.sub.s], [y.sub.1], [y.sub.2], [z.sub.1], ..., [z.sub.d-s], respectively. Therefore, we have

2([ch.sub.2](X) x S) = -s[a.sub.s-1] + 2([a.sub.1] + ... + [a.sub.s-1]) = ([a.sub.1] - [a.sub.s-1]) + ... + ([a.sub.s-1] - [a.sub.s-1]) + [a.sub.1] + ... + [a.sub.s-2] [less than or equal to] 0.

In particular, [ch.sub.2](X) is not ample.

If [[phi].sub.R] is a Pi-bundle structure, that is, s = 2, then ([ch.sub.2](X) x S) = 0. If s > 2, then the above equality says that [a.sub.1] = ... = [a.sub.s-1] = 0. In this case, X becomes a direct product of [bar.X] and a fiber of [[phi].sub.R].

By assuming X to be a Fano manifold, we have the following. The former is (a) in Theorem 1.3:

Corollary 3.2. Let X be a smooth toric 2-Fano d-fold, and suppose that there exists a Fano contraction [[phi].sub.R] : X [right arrow] [bar.X]. Then, dim [bar.X] = 0, that is, X is isomorphic to the d-dimensional projective space [P.sup.d].

Corollary 3.3. Let X be a smooth toric weakly 2-Fano d-fold, and suppose that there exists a Fano contraction [[phi].sub.R] : X [right arrow] [bar.X] such that dim [bar.X] > 0. Then, X is either a projective line bundle over [bar.X] or the direct product of [bar.X] and a fiber of [[phi].sub.R].

4. Divisorial contractions. In this section, we give the proof of (b) of Theorem 1.3. First, we suppose d [greater than or equal to] 3. The case d = 2 will be studied later. Let X be a toric Fano manifold equipped with a divisorial contraction [[phi].sub.R] : X [right arrow] [bar.X] such that dim [[phi].sub.R] (E) = 0, where E is the exceptional divisor. In this case, we need the condition where X is Fano for our proof. Toric Fano manifolds with such contractions are completely classified by Bonavero [1]. There exist the following two cases:

(b1) X is a [P.sup.1]-bundle over [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(b2) The Picard number of X - [X.sub.[SIGMA]] is 3 and the primitive relations of [SIGMA] are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where G(X) = {[x.sub.1], ..., [x.sub.d+3]} and i [less than or equal to] [alpha] [less than or equal to] d - 1.

In the case (b1), X has a Fano contraction. So, we can use the results of Section 3. Therefore, it suffices to consider the case (b2).

Put [tau] [member of] [SIGMA] as the (d - 2)-dimensional cone generated by G([tau]) = {[x.sub.2], ..., [x.sub.d-2], [x.sub.d+3]}. By the above primitive relations, we see that there exist exactly 4 maximal cones in [SIGMA] which contain [tau], i.e., the cones generated by G([tau])[union]{[x.sub.1], [x.sub.d-1]}, G([tau]) [union] {[x.sub.1], [x.sub.d]}, G([tau])[union]{[x.sub.d-1], [x.sub.d+2]} and G([tau])[union] {[x.sub.d], [x.sub.d+2]}, respectively. So, the associated torus invariant subsurface S = [S.sub.[tau]] [subset] X is isomorphic to a Hirzebruch surface. Since [x.sub.1] + [x.sub.d+2] = 0 and [x.sub.d-1] + [x.sub.d] = ([alpha] - 1)[x.sub.1] in [R.sup.d]/Span[tau], its degree is [alpha] - 1. Obviously, the wall relation corresponding to the fiber [C.sub.fib] [subset] X of S [right arrow] [P.sup.1] is

[x.sub.1] + [x.sub.d+2] - [x.sub.d+3] = 0.

On the other hand, the wall corresponding to the negative section [C.sub.neg] [subset] X of S is generated by {[x.sub.1], ..., [x.sub.d-2], [x.sub.d+3]}. Therefore, its wall relation is

-([alpha] - 1)[x.sub.1] + [x.sub.2] + ... + [x.sub.d] + [alpha][x.sub.d+3] = 0.

So, by Theorem 2.2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [X.sub.1], ..., [X.sub.d+3] are the independent elements in Rx corresponding to [x.sub.1], ..., [x.sub.d+3], respectively, and

([ch.sub.2](X) x S) = 3([alpha] - 1) - 2([alpha] - 1) - 2[alpha] = -([alpha] + 1) < 0.

Namely, X is not a weakly 2-Fano manifold. So, we obtain the following result which implies (b) of Theorem 1.3:

Theorem 4.1. Let X be a toric weakly 2-Fano manifold. If there exists a divisorial contraction [[phi].sub.R] : X [right arrow] [bar.X], then dim [phi](E) > 0 for the exceptional divisor E of [[phi].sub.R].

Remark 4.2. There exists a toric variety of Picard number 3 determined by the primitive relations in (b2) for every [alpha] [greater than or equal to] 1. The argument above shows that the second Chern character of the toric variety is not ample for every [alpha].

For a smooth projective toric surface X = [X.sub.[SIGMA]], its second Chern character is calculated as [ch.sub.2](X) = (12 - 3m)/2, where m is the number of 1-dimensional cones in S. Thus, the following is obvious:

Proposition 4.3. Let X be a smooth projective toric surface. If [ch.sub.2](X) is nef but not ample, then X is isomorphic to a Hirzebruch surface.

Remark 4.4. Smooth toric weakly 2-Fano d-folds are completely classified for d [less than or equal to] 4 by Nobili [5] and Sato [8].

doi: 10.3792/pjaa.92.121

Acknowledgment. The author was partially supported by the Grant-in-Aid for Scientific Research (C)]23540062 from JSPS.

References

[1] L. Bonavero, Toric varieties whose blow-up at a point is Fano, Tohoku Math. J. (2) 54 (2002), no. 4, 593 597.

[2] D. A. Cox, J. B. Little and H. K. Schenck, Toric varieties, Graduate Studies in Mathematics, 124, Amer. Math. Soc., Providence, RI, 2011.

[3] W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton Univ. Press, Princeton, NJ, 1993.

[4] A. J. de Jong and J. Starr, Higher Fano manifolds and rational surfaces, Duke Math. J. 139 (2007), no. 1, 173 183.

[5] E. E. Nobili, Classification of toric 2-Fano 4-folds, Bull. Braz. Math. Soc. (N.S.) 42 (2011), no. 3, 399 414.

[6] T. Oda, Convex bodies and algebraic geometry, translated from the Japanese, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 15, Springer, Berlin, 1988.

[7] H. Sato, Toward the classification of higher-dimensional toric Fano varieties, Tohoku Math. J. (2) 52 (2000), no. 3, 383 413.

[8] H. Sato, The numerical class of a surface on a toric manifold, Int. J. Math. Math. Sci. 2012, Art. ID 536475.

[9] F. Schrack, Extremal contractions of 2-Fano four-folds, arXiv:1406.3180.

Hiroshi SATO

Department of Applied Mathematics, Faculty of Sciences, Fukuoka University, 8-19-1, Nanakuma, Jonan-ku, Fukuoka 814-0180, Japan

(Communicated by Shigefumi MORI, M.J.A., Nov. 14, 2016)

2010 Mathematics Subject Classification. Primary 14M25; Secondary 14J45, 14E30.
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