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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 536475, 9 pages
http://dx.doi.org/10.1155/2012/536475
Research Article

The Numerical Class of a Surface on a Toric Manifold

Faculty of Economics and Information, Gifu Shotoku Gakuen University, 1-38 Nakauzura, Gifu 500-8288, Japan

Received 29 January 2012; Accepted 24 February 2012

Academic Editor: Harvinder S. Sidhu

Copyright © 2012 Hiroshi Sato. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we give a method to describe the numerical class of a torus invariant surface on a projective toric manifold. As applications, we can classify toric 2-Fano manifolds of the Picard number 2 or of dimension at most 4.

1. Introduction

The classification of smooth toric Fano -folds is an important and interesting problem. They are classified for by [1, 2], for by [3, 4], and for by [5]. In Øbro's recent excellent paper [6], an algorithm which classifies all the smooth toric Fano -folds for any given natural number was constructed. So, we can say that the classification of smooth toric Fano varieties is completed.

On the other hand, de Jong and Starr defined a special class of Fano manifolds called 2-Fano manifolds in [7] (see Definition 4.2). So, we consider the problem of the classification of toric 2-Fano manifolds as a next step. For this classification, we give a method to describe the numerical class of a 2-cycle on projective toric manifolds (see Section 3). This method makes calculations of intersection numbers much easier. As results, we obtain the classification of toric 2-Fano manifolds for the case of the Picard number and for the case of . We remark that Nobili classified smooth toric 2-Fano 4-folds in [8] by using a Maple program.

The contents of this paper are as follows. In Section 2, we define the basic notation such as nef 2-cocycle and 2-Mori cone for our theory. In Section 3, we define a polynomial for a torus invariant subvariety . This polynomial has all the information of intersection numbers of on . So, we can consider this polynomial as the numerical class of . For a some special surface , has a good property to calculate intersection numbers (see Theorems 3.4 and 3.5). As applications, we classify toric 2-Fano manifolds under some assumptions in Section 4.

Notation 1. We will work over an algebraically closed field throughout this paper. We denote a projective toric -fold by , where is the associated fan in . is the set of the primitive generators for the 1-dimensional cones in .

2. Preliminaries

In this section, we explain the notation and some basic facts of the toric geometry and the birational geometry used in this paper. See [911] for the details.

Let be a smooth projective toric -fold. Put to be the free -module of 2-cocycles on and the free -module of 2-cycles on . We define the numerical equivalence” on and . A 2-cocycle is numerically equivalent to 0; that is, if the intersection number for any 2-cycle , while a 2-cycle is numerically equivalent to 0; that is, if the intersection number for any 2-cocycle . We define and .

The following definitions are similar to the case of divisors and curves.

Definition 2.1. A 2-cocycleis a nef 2-cocycle if for any effective 2-cycle .

Definition 2.2. For a projective toric manifold , let be the cone of effective 2-cycles; namely, One calls the 2-Mori cone of .

We should remark that , , and can be defined for any similarly.

The following is an immediate consequence of the projectivity of .

Proposition 2.3. is a strongly convex cone.

Proof. Let be an ample divisor on . Then, for any , we have ; namely, is strongly convex.

On the other hand, for the toric case, the following is obvious.

Proposition 2.4. Let be a smooth projective toric -fold. Then, is a polyhedral cone.

Thus, is a strongly convex polyhedral rational cone similarly as .

We end this section by giving the following simple examples.

Example 2.5. If , then where is a plane in .
If , then
If , then

3. Combinatorial Descriptions

In this section, we establish a method to describe the numerical class of a torus invariant subvariety. We assume that is a smooth projective toric variety.

Let be a torus invariant subvariety of associated to a cone and . Put where is the torus invariant prime divisor corresponding to , while is defined to be the independent variable corresponding to . We will use this notation throughout this paper.

Remark 3.1.  has all the informations of intersection numbers of    on  . So, we can consider    as the numerical class of  .

Example 3.2. Let    be a torus invariant curve, where   is a -dimensional cone, that is, a wall in  . In this case, is a polynomial of degree 1. On the other hand, is the so-called Reid's wall relation associated to the wall (see [12]); namely, is calculated from the wall relation immediately.

Example 3.3. When ,   sometimes becomes a simple shape as follows.Projective spaces. Let be the -dimensional projective space and . Then, Hirzebruch surfaces. Let be the Hirzebruch surface of degree and . Then,

Let be a smooth projective toric variety and a torus invariant surface. For some special cases, is simply calculated as follows. These are the main theorems of this paper.

Theorem 3.4. Suppose . Let be a torus invariant curve. Then, .

Proof. Let be the -dimensional cone associated to , where . Then, there exist exactly three maximal cones which contain . Put to be the wall relation corresponding to . For the proof, it is sufficient to show that for any , where is the prime torus invariant divisor corresponding to , while is the coefficient of in the above wall relation.
Suppose that or ; namely, or . In this case, trivially, or . So, .
For any , So, the remaining case is or . By calculating the rational functions associated to a -basis for , we have the relations in , where are torus invariant divisors such that Supp for any . Therefore, we have By these relations, the equality is obvious.

Theorem 3.5. Suppose , that is, a Hirzebruch surface of degree . Let be a fiber of the projection , while let be the negative section of . Then, .

Proof. Let be the -dimensional cone associated to , where . Then, there exist exactly four maximal cones which contain . Put to be the wall relation corresponding to , while to be the wall relation corresponding to . As in the proof of Theorem 3.4, by calculating the rational functions associated to a -basis for , we have the relations
First, we remark that, for any , on . So, these intersection numbers can be recovered from (see Example 3.3).
The above relations say that, for any , while for any ,
On the other hand, put and . Then, This coincides with by the above calculations.

4. 2-Fano Manifolds

As an application of Section 3, we study on toric 2-Fano manifolds in this section. The notion of 2-Fano manifolds was introduced in [7].

Definition 4.1. A smooth projecive algebraic variety   is a Fano manifold if its first Chern class  is an ample divisor.

Definition 4.2 (see [7]). A Fano manifold   is a 2-Fano manifold if its second Chern character is a nef 2-cocycle.

Remark 4.3. Since a 2-Fano manifold is a Fano manifold by the definition, for the classification of toric 2-Fano manifolds, all we have to do is to check the list of toric Fano manifolds. The classification of toric Fano manifolds can be done by the algorithm of Øbro [6] for any dimension.

For a projective toric manifold , one can easily see that , where are the torus invariant prime divisors. So, the following is immediate.

Proposition 4.4. For a torus invariant surface , put . Then, .

First of all, we classify toric 2-Fano manifolds of Picard number 2. So, let be a complete toric manifold of . In this case, the structure of is very simple as follows.

Theorem 4.5 (see [13]). Every complete toric manifold of the Picard number 2 is a projective space bundle over a projective space.

By Theorem 4.5, we can put where , . Let be the wall relations of which correspond to the extremal rays of , where Let and be the extremal torus invariant curves corresponding to the wall relations (4.2) and (4.3), respectively.

First, we determine the extremal rays of . By calculating the rational functions for a -basis , we have the relations in , where are torus invariant prime divisors corresponding to . Therefore, for , On the other hand, every -dimensional cone is expressed as for some , such that , and . So, the corresponding torus invariant surface is expressed as By using (4.6), any is expressed as a linear combination of 2-cycles: whose coefficients are nonnegative, because implies . Moreover, since by wall relations (4.2) and (4.3), the possibilities for the generators of are In fact, the following hold: For each case, , and , respectively. So, is a simplicial cone for each case, and , , and are extremal surfaces.

Next, we will check when becomes a 2-Fano manifold.

So, let be the torus invariant curve which generates the extremal ray corresponding to the wall relation (4.3). Then, Therefore, is a Fano manifold if and only if

Since , and are trivial by Theorem 3.4.

On the other hand, we can easily check that . By Theorem 3.5, we have So, we obtain In (4.15), suppose that and . Then, The assumption says that ; that is, . On the other hand, suppose that in (4.15). Then, ; that is, is nef.

By (4.13), we can summarize as follows.

Theorem 4.6. If is a toric 2-Fano manifold of the Picard number 2, then is one of the following: a direct product of projective spaces, .

Remark 4.7. This calculation shows that there exist infinitely many projective toric manifolds of fixed dimension whose second Chern character is nef.

Next, we consider the classification of toric 2-Fano manifolds of a fixed dimension . For , fortunately, these classifications can be done by only Theorems 3.4 and 3.5. Table 1 is the classification list (see [8] for the detail).

tab1
Table 1

Since there exist 124 smooth toric Fano 4-folds, it is hard to check all the smooth toric Fano 4-folds. However, by using the following trivial Lemma 4.8, we can omit a large part of the calculations.

Lemma 4.8. Let be a 4-dimensional toric 2-Fano manifold. Then,

For any smooth toric Fano 4-fold , and are calculated in [3]. One can see that for 52 smooth toric Fano 4-folds, they are not 2-Fano manifolds by Lemma 4.8.

Acknowledgments

The author would like to thank Professor Osamu Fujino for advice and encouragement. He was partially supported by the Grant-in-Aid for Scientific Research (C) no. 23540062 from the JSPS. This paper is dedicated to Professor Shihoko Ishii on her sixtieth birthday.

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