The numerical class of a surface on a toric manifold

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 Picard number 2 or of dimension at most 4.


Introduction
The classification of smooth toric Fano d-folds is an important and interesting problem. They are classified for d = 3 by [B1] and [WW], for d = 4 by [B2] and [S], and for d = 5 by [KrNl]. In Øbro's recent excellent paper [Øb], an algorithm which classify all the smooth toric Fano d-folds for any given natural number d 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 [JS] (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 make calculations of intersection numbers much easier.
As results, we obtain the classification of toric 2-Fano manifolds for the case of Picard number ρ(X) = 2 and for the case of dim(X) ≤ 4. We remark that Nobili classified smooth toric 2-Fano 4-folds in [No] by using a Maple program.
The contents of this paper is 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 I Y /X for a torus invariant subvariety Y ⊂ X. This polynomial has all the informations of intersection numbers of Y on X. So, we can consider this polynomial as the numerical class of Y . For a some special surface S, I S/X 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.

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 [FjS], [Fl] and [Od] for the detail.
The following definitions are similar to the case of divisors and curves: Definition 2.1. A 2-cocycle E ∈ Z 2 (X) is a nef 2-cocycle if (E ·S) ≥ 0 for any 2-cycle S ∈ Z2 (X).
We should remark that N l (X), Nl (X) and NE l (X) can be defined for any 1 ≤ l ≤ d similarly.
The following is an immediate consequence of the projectivity of X: Proof. Let D be an ample divisor on X. Then, for any S ∈ NE 2 (X) \ {0}, we have (D 2 · S) > 0. Namely, NE 2 (X) is strongly convex.
On the other hand, for the toric case, the following is obvious: Proposition 2.4. Let X be a smooth projective toric d-fold. Then, NE 2 (X) is a polyhedral cone.
Thus, NE 2 (X) is a strongly convex polyhedral rational cone similarly as NE(X).
We end this section by giving the following simple examples: Example 2.5.
(1) If X = P d , then where S is a plane in X.

Combinatorial descriptions
In this section, we establish a method to describe the numerical class of a torus invariant subvariety.
Let Y = Y σ ⊂ X be a torus invariant subvariety of dim Y = l associated to a cone σ ∈ Σ and G (Σ) = {x 1 , . . . , x m }. Put where D x i is the torus invariant prime divisor corresponding to x i , while X i is defined to be the independent variable corresponding to x i . We will use this notation throughout this paper.
Remark 3.1. I Y /X has all the informations of intersection numbers of Y on X. So, we can consider I Y /X as the numerical class of Y ∈ Nl (X).
In this case, is the so-called Reid's wall relation associated to the wall τ (see [R]). Namely, I C/X is calculated from the wall relation immediately.
Example 3.3. When Y = X, I X/X sometimes becomes a simple shape as follows: (1) Projective spaces: Let X be the d-dimensional projective space (2) Hirzebruch surfaces: Let X be the Hirzebruch surface F α of degree α and G (Σ) = {x 1 := e 1 , x 2 := e 2 , x 3 := −e 1 + αe 2 , x 4 = −e 2 }. Then, Let X be a smooth projective toric variety and S ⊂ X be a torus invariant surface. For some special cases, I S/X is simply calculated as follows. These are the main theorems of this paper.
Then, there exist exactly three maximal cones τ + R≥0 y 1 , τ + R≥0 y 2 , τ + R≥0 y 3 ∈ Σ which contain τ . Put y 1 + y 2 + y 3 + a 1 x 1 + · · · + a d−2 x d−2 = 0 be the wall relation corresponding to C. For the proof, it is suffice to show that D z D w S = a z a w for any z, w ∈ G (Σ), where D z is the prime torus invariant divisor corresponding to z, while a z is the coefficient of z in the above wall relation. Suppose that z or w ∈ {x 1 , . . . , x d−2 , y 1 , y 2 , y 3 }. Namely, a z = 0 or a w = 0. In this case, trivially, D z S = 0 or D w S = 0. So, D z D w S = a z a w = 0.
For any 1 ≤ i, j ≤ 3, So, the remaining case is z or w ∈ {x 1 , . . . , x d−2 }. By calculating the rational functions associated to a Z -basis {x 1 , . . . , x d−2 , y 1 , y 2 } for N, we have the relations By these relations, the equality D z D w S = a z a w is obvious.
Theorem 3.5. Suppose S ∼ = F α , that is, a Hirzebruch surface of degree α. Let C fib ⊂ S be a fiber of the projection S = F α → P 1 , while let C neg be the negative section of S. Then, I S/X = α(I C fib /X ) 2 + 2I C fib /X I Cneg/X .

2-Fano manifolds
As an application of Section 3, we studies on toric 2-Fano manifolds in this section. The notion of 2-Fano manifolds was introduced in [JS].
Definition 4.1. A smooth projecive algebraic variety X is a Fano manifold if its first Chern class c 1 (X) = −K X is an ample divisor.
Definition 4.2. [JS] A Fano manifold X is a 2-Fano manifold if its second Chern character ch 2 (X) = 1 2 (c 1 (X) 2 −2c 2 (X)) is a nef 2-cocycle. Remark 4.3. Since a 2-Fano manifod is a Fano manifold by the definition, for the classification of toric 2-Fano manifolds, only 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 [Øb] for any dimension.
For a projective toric manifold X, one can easily see that ch 2 (X) = . . , D m are the torus invariant prime divisors. So, the following is immediate.
Proposition 4.4. For a torus invariant surface S ⊂ X, put I S/X := i,j a ij X i X j . Then, (ch 2 (X) · S) = 1 2 m i=1 a ii . 4.5. First of all, we classify toric 2-Fano manifolds of Picard number 2. So, let X be a complete toric manifold of ρ(X) = 2. In this case, the structure of X is very simple as follows: Theorem 4.6. [Kl] Every complete toric manifold of Picard number 2 is a projective space bundle over a projective space.
By (5), we can summarize as follows: Theorem 4.7. If X is a toric 2-Fano manifold of Picard number 2, then X is one of the following: (1) A direct product of projective spaces.
Remark 4.8. This calculation shows that there exist infinitely many projective toric manifolds of fixed dimension d whose second Chern character is nef.
4.9. Next, we consider the classification of toric 2-Fano manifolds of a fixed dimension d. For d ≤ 4, fortunately, these classifications can be done by only Theorems 3.4 and 3.5. The classification list is as follows (see [No] for the detail): 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 4.10, we can omit a large part of the calculations.