# The Numerical Class of a Surface on a Toric Manifold

Authors: Hiroshi Sato
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## 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.


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## Body

## 1. Introduction

The classification of smooth toric Fanod-folds is an important and interesting problem. They are classified for d=3 by [1, 2], for d=4 by [3, 4], and for d=5 by [5]. In Øbro's recent excellent paper [6], an algorithm which classifies 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 [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 ρ(X)=2 and for the case of dim⁡(X)≤4. 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 Section2, we define the basic notation such as  nef 2-cocycle and  2-Mori cone for our theory. In Section 3, we define a polynomial IY/X for a torus invariant subvariety Y⊂X. This polynomial has all the information 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, IS/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.Notation 1.
We will work over an algebraically closed fieldk throughout this paper. We denote a projective toric d-fold by X=XΣ, where Σ is the associated fan in N:=ℤd. G(Σ)⊂N is the set of the primitive generators for the 1-dimensional cones in Σ.


## 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 [9–11] for the details.LetX be a smooth projective toric d-fold. Put Z2(X) to be the free ℤ-module of 2-cocycles on X and Z2(X) the free ℤ-module of 2-cycles on X. We define thenumerical equivalence “≡” on Z2(X) and Z2(X). A 2-cocycle E∈Z2(X) is numerically equivalent to0; that is, E≡0 if the intersection number (E·S)=0 for any 2-cycle S∈Z2(X), while a 2-cycle S∈Z2(X) is numerically equivalent to 0; that is, S≡0 if the intersection number (E·S)=0 for any 2-cocycle E∈Z2(X). We define N2(X):=(Z2(X)/≡)⊗ℝ and N2(X):=(Z2(X)/≡)⊗ℝ.The following definitions are similar to the case of divisors and curves.Definition 2.1.
A 2-cocycleE∈Z2(X)is a  nef 2-cocycle if (E·S)≥0 for any effective 2-cycle S∈Z2(X).Definition 2.2.
For a projective toric manifoldX, let NE2(X)⊂N2(X) be the cone of effective 2-cycles; namely,
(2.1)NE2(X):={[∑iaiSi]∈N2(X)∣ai≥0}.
One calls NE2(X)⊂N2(X) the  2-Mori cone of X.We should remark thatNl(X), Nl(X), and NEl(X) can be defined for any 1≤l≤d similarly.The following is an immediate consequence of theprojectivity of X.Proposition 2.3.
NE2(X) is a strongly convex cone.Proof.
LetD be an ample divisor on X. Then, for any S∈NE2(X)∖{0}, we have (D2·S)>0; namely, NE2(X) is strongly convex.On the other hand, for the toric case, the following is obvious.Proposition 2.4.
LetX be a smooth projective toric  d-fold. Then,  NE2(X) is a polyhedral cone.Thus,NE2(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=ℙd, then
(2.2)NE2(X)=R≥0[S],
where S is a plane in X.
(2) If X=ℙ1×ℙ3, then
(2.3)NE2(X)=R≥0[(apoint)×P2]+R≥0[P1×P1].
(3) If X=ℙ2×ℙ2, then
(2.4)NE2(X)=R≥0[(apoint)×P2]+R≥0[P1×P1]+R≥0[P2×(apoint)].


## 3. Combinatorial Descriptions

In this section, we establish a method to describe the numerical class of a torus invariant subvariety. We assume thatX=XΣ is a smooth projective toric variety.LetY=Yσ⊂X be a torus invariant subvariety of dim⁡Y=l associated to a cone σ∈Σ and G(Σ)={x1,…,xm}. Put (3.1)IY/X=IY/X(X1,…,Xm):=∑1≤i1,…,il≤m(Dxi1⋯Dxil⋅Y)Xi1⋯Xil∈Z[X1,…,Xm],
where Dxi is the torus invariant prime divisor corresponding to xi, while Xi is defined to be the independent variable corresponding to xi. We will use this notation throughout this paper.Remark 3.1.
IY/X has all the informations of intersection numbers of  Y  on  X. So, we can consider  IY/X  as the numerical class of  Y∈Nl(X).Example 3.2.
LetC=Cτ⊂X  be a torus invariant curve, where  τ is a (d-1)-dimensional cone, that is, a  wall in  Σ. In this case,
(3.2)IC/X=∑i(Di⋅C)Xi
is a polynomial of degree 1. On the other hand,
(3.3)∑i(Di⋅C)xi=0
is the so-called Reid's wall relation associated to the wall τ (see [12]); namely, IC/X is calculated from the wall relation immediately.Example 3.3.
WhenY=X,IX/X  sometimes becomes a simple shape as follows.(1)
Projective spaces. Let X be the d-dimensional projective space ℙd and G(Σ)={x1:=e1,…,xd:=ed,xd+1:=-(e1+⋯+ed)}. Then,
(3.4)IX/X=(X1+⋯+Xd+1)d.(2)
Hirzebruch surfaces. Let X be the Hirzebruch surface Fα of degree α and G(Σ)={x1:=e1,x2:=e2,x3:=-e1+αe2,x4=-e2}. Then,
(3.5)IX/X=α(X2+X4)2+2(X2+X4)(X1+X3-αX2).LetX be a smooth projective toric variety and S⊂X a torus invariant surface. For some special cases, IS/X is simply calculated as follows. These are the main theorems of this paper.Theorem 3.4.
SupposeS≅ℙ2. Let  C⊂S be a torus invariant curve. Then,  IS/X=(IC/X)2.Proof.
Letτ=ℝ≥0x1+⋯+ℝ≥0xd-2∈Σ be the (d-2)-dimensional cone associated to S=Sτ, where τ∩G(Σ)={x1,…,xd-2}. Then, there exist exactly three maximal cones τ+ℝ≥0y1,τ+ℝ≥0y2,andτ+ℝ≥0y3∈Σ which contain τ. Put
(3.6)y1+y2+y3+a1x1+⋯+ad-2xd-2=0
to be the wall relation corresponding to C. For the proof, it is sufficient to show that
(3.7)DzDwS=azaw,
for any z,w∈G(Σ), where Dz is the prime torus invariant divisor corresponding to z, while az is the coefficient of z in the above wall relation.
Suppose thatz or w∉{x1,…,xd-2,y1,y2,y3}; namely, az=0 or aw=0. In this case, trivially, DzS=0 or DwS=0. So, DzDwS=azaw=0.
For any1≤i,j≤3,
(3.8)DyiDyjS=(Dyi∣S)(Dyj∣S)=C2=1.
So, the remaining case is z or w∈{x1,…,xd-2}. By calculating the rational functions associated to a ℤ-basis {x1,…,xd-2,y1,y2} for N, we have the relations
(3.9)Dx1-a1Dy3+E1=0,…,Dxd-2-ad-2Dy3+Ed-2=0,Dy1-Dy3+Ed-1=0,Dy1-Dy3+Ed=0
in PicX, where E1,…,Ed are torus invariant divisors such that SuppEi∩S=∅ for any 1≤i≤d. Therefore, we have
(3.10)Dx1S=a1Dy3S,…,Dxd-2S=ad-2Dy3S.
By these relations, the equality DzDwS=azaw is obvious.Theorem 3.5.
SupposeS≅Fα, that is, a Hirzebruch surface of degree  α. Let  Cfib⊂S be a fiber of the projection  S=Fα→ℙ1, while let  Cneg be the negative section of  S. Then,  IS/X=α(ICfib/X)2+2ICfib/XICneg/X.Proof.
Letτ=ℝ≥0x1+⋯+ℝ≥0xd-2∈Σ be the (d-2)-dimensional cone associated to S=Sτ, where τ∩G(Σ)={x1,…,xd-2}. Then, there exist exactly four maximal cones τ+ℝ≥0y1,τ+ℝ≥0y2,τ+ℝ≥0y3,andτ+ℝ≥0y4∈Σ which contain τ. Put
(3.11)y1+y3-αy2+a1x1+⋯+ad-2xd-2=0
to be the wall relation corresponding to Cneg, while
(3.12)y2+y4+b1x1+⋯+bd-2xd-2=0
to be the wall relation corresponding to Cfib. As in the proof of Theorem 3.4, by calculating the rational functions associated to a ℤ-basis {x1,…,xd-2,y1,y2} for N, we have the relations
(3.13)Dx1S=a1Dy3S+b1Dy4S,…,Dxd-2S=ad-2Dy3S+bd-2Dy4S,Dy1S=Dy3S,Dy2=-αDy3S+Dy4S.
First, we remark that, for any1≤i,j≤4,
(3.14)DyiDyjS=(Dyi∣S)(Dyj∣S)
on S. So, these intersection numbers can be recovered from IS/S (see Example 3.3).
The above relations say that, for any1≤i,j≤d-2,
(3.15)DxiDxjS=αbibj+aibj+ajbi,
while for any 1≤i≤d-2,
(3.16)Dy1Dxi=bi,Dy2Dxi=ai,Dy3Dxi=bi,Dy4Dxi=ai+αbi.
On the other hand, putf1=f1(X1,…,Xd-2):=a1X1+⋯+ad-2Xd-2 and f2=f2(X1,…,Xd-2):=b1X1+⋯+bd-2Xd-2. Then,
(3.17)α(ICfib/X)2+2ICfib/XICneg/X=α(Y2+Y4+f1)2+2(Y2+Y4+f1)(Y1+Y3-αY2+f2)=IS/S(Y1,Y2,Y3,Y4)+αf22+2f1f2+2Y1f2+2Y2f1+2Y3f2+Y4(2f1+2αf2).
This coincides with IS/X by the above calculations.


## 4. 2-Fano Manifolds

As an application of Section3, we study on toric2-Fano manifolds in this section. The notion of 2-Fano manifolds was introduced in [7].Definition 4.1.
A smooth projecive algebraic varietyX  is aFano manifoldif its first Chern classc1(X)=-KX  is an ample divisor.Definition 4.2 (see [7]).
A Fano manifoldX is a 2-Fano manifoldif its second Chern  character ch2(X)=(1/2)(c1(X)2-2c2(X)) 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 manifoldX, one can easily see that ch2(X)=(1/2)∑i=1mDi2, where D1,…,Dm are the torus invariant prime divisors. So, the following is immediate.Proposition 4.4.
For a torus invariant surfaceS⊂X, put  IS/X:=∑i,jaijXiXj. Then,  (ch2(X)·S)=(1/2)∑i=1maii.First of all, we classify toric 2-Fano manifolds of Picard number 2. So, letX be a complete toric manifold of ρ(X)=2.In this case, the structure of X 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 Theorem4.5, we can put (4.1)X=XΣ=PPn-1(O⊕O(a1)⊕⋯⊕O(am-1)),
where a1≥⋯≥am-1≥0, m+n-2=d:=dim⁡X. Let (4.2)x1+⋯+xm=0,(4.3)y1+⋯+yn=a1x1+⋯+am-1xm-1
be the wall relations of Σ which correspond to the extremal rays of NE(X), where (4.4)G(Σ)={x1,…,xm,y1,…,yn}.
Let C1 and C2 be the extremal torus invariant curves corresponding to the wall relations (4.2) and (4.3), respectively.First, we determine the extremal rays ofNE2(X). By calculating the rational functions for a ℤ-basis {x1,…,xm-1,y1,…,yn-1}, we have the relations (4.5)D1-Dm+a1En=0,…,Dm-1-Dm+am-1En=0,E1-En=0,…,En-1-En=0
in N1(X), where D1,…,Dm,E1,…,En are torus invariant prime divisors corresponding to x1,…,xm,y1,…,yn. Therefore, for 1≤i,j≤m-1, (4.6)Dj=Di+(ai-aj)En,E1=E2=⋯=En.
On the other hand, every (d-2)-dimensional cone τ∈Σ is expressed as (4.7)τ=R≥0xi1+⋯+R≥0xik+R≥0yj1+⋯+R≥0yjl,
for some 1≤i1<⋯<ik≤m, 1≤j1<⋯<jl≤n such that k<m,l<n, and k+l=d-2. So, the corresponding torus invariant surface Sτ is expressed as (4.8)Sτ=Di1⋯DikEj1⋯Ejl∈N2(X).
By using (4.6), any Sτ is expressed as a linear combination of 2-cycles: (4.9)D1⋯DpEq(p≤m-1,q≤n-1,p+q=d-2),
whose coefficients are nonnegative, because i<j implies ai-aj≥0. Moreover, since D1⋯Dm=E1⋯En=0 by wall relations (4.2) and (4.3), the possibilities for the generators of NE2(X) are (4.10)S1:=D1⋯Dm-3En-1,S2:=D1⋯Dm-2En-2,orS3:=D1⋯Dm-1En-3.
In fact, the following hold: (4.11)NE2(X)=R≥0S1+R≥0S2+R≥0S3ifm≥3,n≥3.NE2(X)=R≥0S2+R≥0S3ifm=2,n≥3.NE2(X)=R≥0S1+R≥0S2ifm≥3,n=2.
For each case, dim⁡N2(X)=3,dim⁡N2(X)=2, and dim⁡N2(X)=2, respectively. So, NE2(X) is a simplicial cone for each case, and S1, S2, and S3 are extremal surfaces.Next, we will check whenX becomes a 2-Fano manifold.So, letC2 be the torus invariant curve which generates the extremal ray corresponding to the wall relation (4.3). Then, (4.12)(-KX⋅C2)=n-(a1+⋯+am-1).
Therefore, X is a Fano manifold if and only if (4.13)n-(a1+⋯+am-1)>0.SinceS1≅S3≅ℙ2, (ch2(X)·S1)≥0 and (ch2(X)·S3)≥0 are trivial by Theorem 3.4.On the other hand, we can easily check thatS2≅Fam-1. By Theorem 3.5, we have (4.14)IS2=am-1(IC1)2+2IC1IC2=am-1(X1+⋯+Xm)2+2(X1+⋯+Xm)(Y1+⋯+Yn-(a1X1+⋯+am-1Xm-1)).
So, we obtain (4.15)(ch2(X)⋅S2)=mam-1-2(a1+⋯+am-1).
In (4.15), suppose that m≥3 and (ch2(X)·S2)≥0. Then, (4.16)(ch2(X)⋅S2)=(m-2)am-1-2(a1+⋯+am-2).
The assumption a1≥⋯≥am-1≥0 says that a1=⋯=am-1=0; that is, X≅ℙm-1×ℙn-1. On the other hand, suppose that m=2 in (4.15). Then, (ch2(X)·S2)=0; that is, ch2(X) is nef.By (4.13), we can summarize as follows.Theorem 4.6.
IfX is a toric  2-Fano manifold of the Picard number  2, then  X is one of the following:   (1)
a direct product of projective spaces,(2)
ℙℙd-1(𝒪⊕𝒪(a))(1≤a≤d-1).Remark 4.7.
This calculation shows that there exist  infinitely many projective toric manifolds of fixed dimensiond whose second Chern character is nef.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. Table 1 is the classification list (see [8] for the detail).Table 1d1234# of toric Fano1518124# of toric 2-Fano13825Since 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 Lemma4.8, we can omit a large part of the calculations.Lemma 4.8.
LetX be a  4-dimensional toric  2-Fano manifold. Then,
(4.17)c14(X)-2c12(X)c2(X)≥0.For any smooth toric Fano 4-foldX, c14(X) and c12(X)c2(X) 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.



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