The Numerical Class of a Surface on a Toric Manifold

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Volume 2012, Article ID 536475,9pages doi:10.1155/2012/536475

Research Article

The Numerical Class of

a Surface on a Toric Manifold

Hiroshi Sato

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

Correspondence should be addressed to Hiroshi Sato,[email protected]

Received 29 January 2012; Accepted 24 February 2012

Academic Editor: Harvinder S. Sidhu

Copyrightq2012 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.

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 Fanod-folds is an important and interesting problem. They are classified ford 3 by1,2, ford 4 by3,4, and ford 5 by5. In Øbro’s recent excellent paper6, an algorithm which classifies all the smooth toric Fanod-folds for any given natural number dwas 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 in7 seeDefinition 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 manifoldsseeSection 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 ρX 2 and for the case of dimX ≤ 4. We remark that Nobili classified smooth toric 2-Fano 4-folds in8 by using a Maple program.

The contents of this paper are as follows. InSection 2, we define the basic notation such as nef 2-cocycle and 2-Mori cone for our theory. InSection 3, we define a polynomial

IY/X for a torus invariant subvariety Y ⊂ X. This polynomial has all the information of

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ofY. For a some special surfaceS,IS/Xhas a good property to calculate intersection numbers

see Theorems3.4and3.5. As applications, we classify toric 2-Fano manifolds under some assumptions inSection 4.

Notation. We will work over an algebraically closed fieldkthroughout this paper. We denote a projective toricd-fold byXXΣ, whereΣis the associated fan inN:Zd.GΣ⊂Nis the set of the primitive generators for the 1-dimensional cones inΣ.

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. See9–11for the details.

Let X be a smooth projective toric d-fold. Put Z2X to be the free Z-module of 2-cocycles on X and Z2X the free Z-module of 2-cycles on X. We define the numerical

equivalence “≡” on Z2Xand Z2X. A 2-cocycleE ∈ Z2X is numerically equivalent to 0;

that is,E≡0 if the intersection numberE·S 0 for any 2-cycleS∈Z2X, while a 2-cycle S∈Z2Xis numerically equivalent to 0; that is,S≡0 if the intersection numberE·S 0 for

any 2-cocycleE∈Z2X. We define N2X: Z2_X_/_≡_⊗_R_{and N}

2X: Z2X/≡⊗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∈Z2X.

Definition 2.2. For a projective toric manifoldX, let NE2X⊂N2Xbe the cone of eﬀective

2-cycles; namely,

NE2X:

i

aiSi

∈N2X|ai≥0

. 2.1

One calls NE2X⊂N2Xthe 2-Mori cone ofX.

We should remark thatNl_X_,_N

lX, and NElXcan be defined for any 1 ≤ l ≤ d

similarly.

The following is an immediate consequence of the projectivity ofX.

Proposition 2.3. NE2Xis a strongly convex cone.

Proof. LetDbe an ample divisor onX. Then, for anyS∈NE2X\ {0}, we haveD2·S>0;

namely, NE2Xis strongly convex.

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

Proposition 2.4. LetXbe a smooth projective toricd-fold. Then, NE2Xis a polyhedral cone.

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We end this section by giving the following simple examples.

Example 2.5. 1IfXPd_{, then}

NE2X R≥0S, 2.2

whereSis a plane inX. 2IfXP1_×_P3_{, then}

NE2X R≥0a point ×P2

R≥0

P1_×_P1_. _2.3

3IfXP2_×_P2_{, then}

NE2X R≥0a point ×P2

R≥0P1_×_P1 _R≥0_P2_×_{a point} _. _2.4

3. Combinatorial Descriptions

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

LetY Yσ ⊂Xbe a torus invariant subvariety of dimY lassociated to a coneσ∈Σ

and GΣ {x1, . . . , xm}. Put

IY/X IY/XX1, . . . , Xm:

1≤i1,...,il≤m

Dxi₁· · ·Dx_il ·Y

Xi1· · ·Xil

∈ZX1, . . . , Xm,

3.1

where Dxi is the torus invariant prime divisor corresponding toxi, while Xi is defined to

be the independent variable corresponding toxi. We will use this notation throughout this

paper.

Remark 3.1. IY/Xhas all the informations of intersection numbers ofY onX. So, we can

considerIY/X as the numerical class ofY ∈NlX.

Example 3.2. LetC Cτ ⊂ X be a torus invariant curve, whereτ is ad−1-dimensional

cone, that is, a wall inΣ. In this case,

IC/X

i

Di·CXi 3.2

is a polynomial of degree 1. On the other hand,

i

Di·Cxi0 3.3

is the so-called Reid’s wall relation associated to the wallτsee12; namely,IC/Xis calculated

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Example 3.3. WhenY X,IX/X sometimes becomes a simple shape as follows.

1Projective spaces. LetXbe thed-dimensional projective spacePd_{and GΣ}_{_x1 _:

e1, . . . , xd :ed, xd 1:−e1 · · · ed}. Then,

IX/X X1 · · · Xd 1d. 3.4

2Hirzebruch surfaces. Let X be the Hirzebruch surface Fα of degree αand GΣ

{x1:e1, x2 :e2, x3:−e1 αe2, x4−e2}. Then,

IX/XαX2 X42 2X2 X4X1 X3−αX2. 3.5

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,}_I

S/X IC/X2.

Proof. LetτR≥0x1 · · · R≥0xd−2∈Σbe thed−2-dimensional cone associated toSSτ,

whereτ∩GΣ {x1, . . . , xd−2}. Then, there exist exactly three maximal conesτ R≥0y1, τ R≥0y2, andτ R≥0y3∈Σwhich containτ. Put

y1 y2 y3 a1x1 · · · ad−2xd−20 3.6

to be the wall relation corresponding toC. For the proof, it is suﬃcient to show that

DzDwSazaw, 3.7

for anyz, w∈GΣ, whereDzis the prime torus invariant divisor corresponding toz, while

azis the coeﬃcient ofzin the above wall relation.

Suppose thatzorw /∈ {x1, . . . , xd−2, y1, y2, y3}; namely,az 0 oraw 0. In this case,

trivially,DzS0 orDwS0. So,DzDwSazaw0.

For any 1≤i, j≤3,

DyiDyjS

Dyi|S Dyj|S

C2 1. 3.8

So, the remaining case is z or w ∈ {x1, . . . , xd−2}. By calculating the rational functions

associated to aZ-basis{x1, . . . , xd−2, y1, y2}forN, we have the relations

Dx1−a1Dy3 E10, . . . , Dxd−2−ad−2Dy3 Ed−20,

Dy1−Dy3 Ed−10, Dy1−Dy3 Ed0

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in PicX, whereE1, . . . , Edare torus invariant divisors such that SuppEi∩S∅for any 1≤i≤

d. Therefore, we have

Dx1Sa1Dy3S, . . . , Dxd−2Sad−2Dy3S. 3.10

By these relations, the equalityDzDwSazawis obvious.

Theorem 3.5. SupposeS ∼ Fα, that is, a Hirzebruch surface of degreeα. LetCfib ⊂Sbe a fiber of

the projectionSFα → P1, while letCnegbe the negative section ofS. Then,IS/X αICfib/X 2

2ICfib/XICneg/X.

Proof. LetτR≥0x1 · · · R≥0xd−2∈Σbe thed−2-dimensional cone associated toSSτ,

whereτ∩GΣ {x1, . . . , xd−2}. Then, there exist exactly four maximal conesτ R≥0y1, τ

R≥0y2, τ R≥0y3, and τ R≥0y4∈Σwhich containτ. Put

y1 y3−αy2 a1x1 · · · ad−2xd−20 3.11

to be the wall relation corresponding toCneg, while

y2 y4 b1x1 · · · bd−2xd−20 3.12

to be the wall relation corresponding toCfib. As in the proof ofTheorem 3.4, by calculating the rational functions associated to aZ-basis{x1, . . . , xd−2, y1, y2}forN, we have the relations

Dx1Sa1Dy3S b1Dy4S, . . . , Dxd−2Sad−2Dy3S bd−2Dy4S,

Dy1SDy3S, Dy2 −αDy3S Dy4S.

3.13

First, we remark that, for any 1≤i, j≤4,

DyiDyjS

Dyi|S Dyj|S

3.14

onS. So, these intersection numbers can be recovered fromIS/SseeExample 3.3.

The above relations say that, for any 1≤i, j≤d−2,

DxiDxjSαbibj aibj ajbi, 3.15

while for any 1≤i≤d−2,

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On the other hand, put f1 f1X1, . . . , Xd−2 : a1X1 · · · ad−2Xd−2 and f2

f2X1, . . . , Xd−2:b1X1 · · · bd−2Xd−2. Then,

αICfib/X

2 ₂

ICfib/XICneg/Xα

Y2 Y4 f1 2 2Y2 Y4 f1 Y1 Y3−αY2 f2

IS/SY1, Y2, Y3, Y4 αf₂2 2f1f2

2Y1f2 2Y2f1 2Y3f2 Y42f1 2αf2 .

3.17

This coincides withIS/Xby the above calculations.

4. 2-Fano Manifolds

As an application ofSection 3, we study on toric 2-Fano manifolds in this section. The notion of 2-Fano manifolds was introduced in7.

Definition 4.1. A smooth projecive algebraic variety X is a Fano manifold if its first Chern classc1X −KX is an ample divisor.

Definition 4.2see7. A Fano manifoldXis a 2-Fano manifold if its second Chern character ch2X 1/2c1X2−2c2Xis 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 Øbro6for any dimension.

For a projective toric manifoldX, one can easily see that ch2X 1/2mi1D2i, where

D1, . . . , Dmare the torus invariant prime divisors. So, the following is immediate.

Proposition 4.4. For a torus invariant surfaceS⊂X, putIS/X:

i,jaijXiXj. Then,ch2X·S

1/2mi1aii.

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

Theorem 4.5see13. Every complete toric manifold of the Picard number 2 is a projective space

bundle over a projective space.

ByTheorem 4.5, we can put

XXΣPPn−1O ⊕ Oa1⊕ · · · ⊕ Oa_m₋₁, 4.1

wherea1≥ · · · ≥am−1≥0,m n−2d:dimX. Let

x1 · · · xm0, 4.2

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be the wall relations ofΣwhich correspond to the extremal rays of NEX, where

GΣ x1, . . . , xm, y1, . . . , yn

. 4.4

LetC1andC2be the extremal torus invariant curves corresponding to the wall relations4.2 and4.3, respectively.

First, we determine the extremal rays of NE2X. By calculating the rational functions

for aZ-basis{x1, . . . , xm−1, y1, . . . , yn−1}, we have the relations

D1−Dm a1En0, . . . , Dm−1−Dm am−1En0,

E1−En0, . . . , En−1−En0

4.5

in N1X, whereD1, . . . , Dm, E1, . . . , Enare torus invariant prime divisors corresponding to

x1, . . . , xm, y1, . . . , yn. Therefore, for 1≤i, j ≤m−1,

DjDi

ai−aj En,

E1E2· · ·En.

4.6

On the other hand, everyd−2-dimensional coneτ ∈Σis expressed as

τ R≥0xi1 · · · R≥0xik R≥0yj1 · · · R≥0yjl, 4.7

for some 1≤i1<· · ·< ik≤m, 1≤j1<· · ·< jl≤nsuch thatk < m, l < n, andk ld−2. So,

the corresponding torus invariant surfaceSτis expressed as

Sτ Di1· · ·DikEj1· · ·Ejl ∈N2X. 4.8

By using4.6, anySτ is expressed as a linear combination of 2-cycles:

D1· · ·DpEq

p≤m−1, q≤n−1, p qd−2 , 4.9

whose coeﬃcients are nonnegative, because i < j implies ai −aj ≥ 0. Moreover, since

D1· · ·Dm E1· · ·En 0 by wall relations4.2and4.3, the possibilities for the generators

of NE2Xare

S1:D1· · ·Dm−3En−1, S2:D1· · ·Dm−2En−2, or

S3:D1· · ·Dm−1En−3.

4.10

In fact, the following hold:

NE2X R≥0S1 R≥0S2 R≥0S3 ifm≥3, n≥3.

NE2X R≥0S2 R≥0S3 if m2, n≥3.

NE2X R≥0S1 R≥0S2 if m≥3, n2.

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For each case, dim N2X 3, dim N2X 2, and dim N2X 2, respectively. So, NE2X

is a simplicial cone for each case, andS1,S2, andS3are extremal surfaces. Next, we will check whenXbecomes a 2-Fano manifold.

So, letC2be the torus invariant curve which generates the extremal ray corresponding to the wall relation4.3. Then,

−KX·C2 n−a1 · · · am−1. 4.12

Therefore,Xis a Fano manifold if and only if

n−a1 · · · am−1>0. 4.13

SinceS1∼S3∼P2_,_ch

2X·S1≥0 andch2X·S3≥0 are trivial byTheorem 3.4.

On the other hand, we can easily check thatS2∼Fam−1. ByTheorem 3.5, we have

IS2 am−1IC1

2 ₂_I

C1IC2am−1X1 · · · Xm 2

2X1 · · · XmY1 · · · Yn−a1X1 · · · am−1Xm−1. 4.14

So, we obtain

ch2X·S2 mam−1−2a1 · · · am−1. 4.15

In4.15, suppose thatm≥3 andch2X·S2≥0. Then,

ch2X·S2 m−2am−1−2a1 · · · am−2. 4.16

The assumptiona1≥ · · · ≥am−1 ≥0 says thata1 · · ·am−10; that is,X ∼Pm−1×Pn−1. On

the other hand, suppose thatm2 in4.15. Then,ch2X·S2 0; that is, ch2Xis nef.

By4.13, we can summarize as follows.

Theorem 4.6. IfXis a toric 2-Fano manifold of the Picard number 2, thenXis one of the following:

1a direct product of projective spaces,

2PPd−1O ⊕ Oa 1≤a≤d−1.

Remark 4.7. This calculation shows that there exist infinitely many projective toric manifolds

of fixed dimension d whose second Chern character is nef.

Next, we consider the classification of toric 2-Fano manifolds of a fixed dimensiond.

Ford≤4, fortunately, these classifications can be done by only Theorems3.4and3.5.Table 1 is the classification listsee8for the detail.

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Table 1

d 1 2 3 4

# of toric Fano 1 5 18 124

# of toric 2-Fano 1 3 8 25

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

c₁4X−2c2₁Xc2X≥0. 4.17

For any smooth toric Fano 4-foldX,c4₁Xandc2₁Xc2Xare calculated in3. One can see that for 52 smooth toric Fano 4-folds, they are not 2-Fano manifolds byLemma 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 ResearchCno. 23540062 from the JSPS. This paper is dedicated to Professor Shihoko Ishii on her sixtieth birthday.

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