# A Note on Tsuji’s Criterion for Numerical Triviality

Authors: Shigetaka Fukuda
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this study, we give an alternative and elementary proof to Tsuji’s criterion for a Cartier divisor to be numerically trivial.


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## Body

## 1. Introduction

In this article, every algebraic variety is proper over the field of complex numbersC.In 1970s, Iitaka [1] initiated the classification theory of higher dimensional algebraic varieties by using the pluricanonical systems. In 1980s, Mori [2] deepened the Iitaka theory by cutting off the subvarieties of elliptic type.In [3], Tsuji gave an interesting and useful criterion for a Cartier divisor to be numerically trivial.Theorem 1 (Tsuji [[3], Lemma 5.1], cf. Bauer et al. [[4], Theorem 2.4]).
Letf:M⟶B a surjective morphism between complete varieties. Let L be a nef Cartier divisor on M and W some subvarieties of M, such that fW=B and B0 a subset of B which is a union of countably many proper Zariski-closed subsets. Assume that(1)
for someb∈B, L,C=0 for every curve C on f−1b(2)
L,C=0 for every irreducible curve C on W, such that fC⊈B0
Then,L is numerically trivial.Tsuji’s criterion for numerical triviality is one of the basic tools to decompose every algebraic variety into the varieties of elliptic type, of parabolic type, and of hyperbolic type by cutting off the varieties of parabolic type [5–7].Tsuji’s proof [3] is analytic and the proof [4] by Bauer et al. is algebraic.In this research note, we give an alternative and elementary proof to Tsuji’s criterion (Theorem1). The argument (Subcase 7), which uses the following corollary of the Hodge index theorem, due to Bauer et al. is essential.Lemma 1 (cf. [[4], Proposition 2.5]).
Letf:M⟶B be a surjective morphism from a complete surface M to a complete curve B. Let L be a nef Cartier divisor on M. Assume that(1)
For someb∈B, L,C=0 for every curve C on f−1b(2)
L,W=0 for some irreducible curve W on M, such that fW=B
Then,L is numerically trivial.Remark 1.
In the statement of Lemma1, condition (1) immediately implies that L is numerically trivial on every general fiber of the morphism f by considering the flattening. By the normalization, the Stein factorization, and the desingularization, the article ([4], Proposition 2.5), for an algebraically fibered surface, implies the assertion of Lemma 1.


## 2. Elementary Proof of Main Theorem1

Proof.
We prove the assertion by induction ondimM,dimB.
First, we take a commutative diagram as shown in Figure1 with the following properties:(1)
M′ and B′ are nonsingular projective varieties(2)
μ is a birational morphism(3)
ν is a generically finite morphism(4)
g is a morphism with only connected fibers
There exists some irreducible componentW′ of μ−1W, such that gW′=B′. We set L′≔μ∗L.
The locusg∪C′|C′isanirreduciblecurveonM′,gC′isapoint,andtheintersectionnumberL′,C′>0 is included in a union of at most countably many proper Zariski-closed subsets of B′ (Proposition 1). Thus, we obtain a union B0′⊇ν−1B0 of countably many proper Zariski-closed subsets of B′ with the following two properties:(1)
L′ is numerically trivial on every fiber of g over B′\B0′(2)
L′,CW′=0 for every irreducible curve CW′ on W′, such that gCW′⊈B0′
It suffices to prove thatL′,C′=0 for every irreducible curve C′ on M′. We fix an irreducible curve C′ on M′.Figure 1
The Stein factorization forf is as follows.Case 1.
gC′⊈B0′. This case divides into Subcases 1 and 2.Subcase 1.
gC′⊈B0′ and gC′ is a point.
We haveL′,C′=0 from (1).Subcase 2.
gC′⊈B0′ and gC′ is a curve. This subcase divides into Subcases 3 and 7.Subcase 3.
gC′⊈B0′, gC′ is a curve and dimB=1. This subcase divides into Subcases 4, 5, and 6.
We note thatgW1=gC′=B′ for some irreducible curve W1 on W′.Subcase 4.
gC′⊈B0′, gC′ is a curve, dimB=1, and dimM=1.
M′=W′=W1=C′. Thus, L′,C′=0.Subcase 5.
gC′⊈B0′, gC′ is a curve, dimB=1, and dimM=2.
BecauseL′,W1=0 from (2), Lemma 1 implies that L′ is numerically trivial, and thus, L′,C′=0.Subcase 6.
gC′⊈B0′, gC′ is a curve, dimB=1, and dimM≧3.
Because the codimensioncodimW1∪C′,M′≧2, we have an irreducible hyperplane section H of M′ that includes W1 and C′ (Proposition 2). Then, L′H is numerically trivial from the induction hypothesis. Consequently, L′,C′=0.Subcase 7 (cf. [[4], 2.1.2]).
gC′⊈B0′, gC′ is a curve, and dimB≧2.
LetS≔Si be the set of irreducible components of g−1gC′. We note that gW1=gC′ for some irreducible curve W1 on W′. Thus, L′S1 is numerically trivial for some S1∈S, such that S1⊇W1 from the property (2) and from the induction hypothesis.
Ifg∪m≠1Sm=gC′, then gS1∩∪m≠1Sm=gC′ from the connectedness of fibers of g, and therefore, gS1∩S2=gC′ for some S2∈S.
Thus,dimg∪m≠1Sm≦0 or gS1∩S2=gC′.
IfgS1∩S2=gC′ and g∪m≠1,2Sm=gC′, then gS1∪S2∩∪m≠1,2Sm=gC′ from the connectedness of fibers of g, and therefore, gS1∪S2∩S3=gC′ for some S3∈S. From this argument, we obtain the following properties:(1)
gS1=gS2=⋯=gSk=gC′(2)
gS1∪S2∪⋯∪Si−1∩Si=gC′ for all i with 1≦i≦k(3)
dimg∪i≠1,2,…kSi≦0
The fact thatL′S1 is numerically trivial and that gS1∩S2=gC′ implies that L′∣S2 is numerically trivial from the induction hypothesis. The fact that L′S1∪S2 is numerically trivial and that gS1∪S2∩S3=gC′ implies that L′S3 is numerically trivial from the induction hypothesis. This argument implies that L′∣S1∪S2∪⋯∪Sk is numerically trivial. Because dimg∪i≠1,2,…kSi≦0, we have that C′⊆S1∪S2∪⋯∪Sk. Consequently, L′,C′=0.Case 2.
gC′⊆B0′. This case divides into Subcases 8 and 11.Subcase 8.
gC′⊆B0′ and dimM=2. This subcase divides into Subcases 9 and 10.Subcase 9.
gC′⊆B0′, dimM=2, and dimB=1.
Lemma1 implies that L′ is numerically trivial, and thus, L′,C′=0.Subcase 10.
gC′⊆B0′, dimM=2, and dimB=2.
BecauseW′=M′, we have that L′,H=0 for an irreducible hyperplane section H of M′ from the property (2) of the divisor L′. The Hodge index theorem implies that L′ is numerically trivial. Thus, L′,C′=0.Subcase 11.
gC′⊆B0′ and dimM≧3.
Because the codimensioncodimC′,M′≧2, there exists an irreducible hyperplane section H of M′ that includes C′ (Proposition 2). We may assume that gH⊈B0′. Note that, from Case 1, L,C″=0 for every irreducible curve C″ on H, such that gC″⊈B0′. Thus, L′∣H is numerically trivial from the induction hypothesis. Consequently, L′,C′=0.□


## 3. Appendix

In this appendix, we state two elementary propositions and their proofs, which are well known to the experts, for the readers’ convenience.Proposition 1.
Letf:M⟶B be a surjective morphism between projective varieties and L a nef Cartier divisor on M.We assume that for someb∈B, the intersection number L,C=0 for every irreducible curve C on f−1b.Then, the locusf∪C|CisanirreduciblecurveonM,fCisapoint,andtheintersectionnumberL,C>0 is included in a union of at most countably many proper Zariski-closed subsets of B.Proof.
There exists some ample divisorA on B. Assume that C is an irreducible curve on M, such that f∗A,C=0 (i.e., fC is a point) and that L,C>0. There exists some irreducible component W of the universal scheme for the Hilbert scheme HilbM of M, such that W includes p2−1C, where C is the point ∈HilbM, which represents the subscheme C of M, and Figure 2 shows the projections p1 and p2 and the property that dimW=dimp2W+1. We set T:=p2W.
First, we consider the normalizationn1:Wn⟶W, n2:Tn⟶T, and n3:Wn⟶Tn of the morphism p2|W:W⟶T. Next, consider the Stein factorization Wn⟶s1T′⟶s2Tn of the morphism n3:Wn⟶Tn.
Last, consider the flatteningf1:W″⟶Wn, f2:T″⟶T′, and f3:W″⟶T″ of the morphism s1:Wn⟶T′, where the morphism f2 is birational and the variety T″ is nonsingular. We note that the morphism f3:W″⟶T″ is flat and with only connected fibers.
We puth≔n1f1.
Thus, we have the commutative diagram, as shown in Figure3.
From the flatness of the morphismf3:W″⟶T″, the intersection number h∗p1∗f∗A,F″>0 for every fiber F″ of the morphism f3:W″⟶T″ because p1∗f∗A,p2−1C=0. Thus, for every fiber F″ of the morphism f3:W″⟶T″, the morphism f contracts p1hF″ to one point from the connectedness of F″. In other words, p1hF″ is included in some fiber of f.
There exists some ample divisorA′ on W. Of course, A′,F>0 for every curve F on W. Because the morphism h is birational, we have that hF‴ is not a point (i.e., h∗A′,F‴>0) for some fiber F‴ of f3:W″⟶T″. From the flatness, h∗A′,F″>0 for every fiber F″ of f3:W″⟶T″. Thus, every fiber F″ of f3:W″⟶T″ cannot be contracted to a point by the morphism h.
There exists some fiberF1″ of f3:W″⟶T″, such that h−1p2−1C∩F1″≠∅. Then, F1″⊂h−1p2−1C because the morphism p2h maps F1″ to a point C∈T. Consequently, hF1″=p2−1C because F1″ does not contract to a point by the morphism h. Thus, h∗p1∗L,F1″>0. From the flatness of the morphism f3:W″⟶T″, the intersection number is h∗p1∗L,F″>0 for every fiber F″ of the morphism f3:W″⟶T″.
We note that every fiber off3:W″⟶T″ is mapped in some fiber of p2|W:W⟶T. In other words, every fiber of p2|W is swept out by fibers of f3.
So, for every fiberF of p2|W:W⟶T, the locus p1F is swept out by connected curves C′, such that fC′ is one point and that the intersection number L,C′>0. We note that we consider p1hF″ as C′ and that C′=p1hF″=Suppp1h∗F″ from the connectedness of F″. Thus, b∉fp1F. Consequently, b∉fp1W. In other words, p1W is disjoint with f−1b.
The countability of the irreducible components of the Hilbert schemeHilbM of M implies the assertion.Figure 2
The deformation ofC in M.Figure 3
The flattening of the deformation.Proposition 2.
LetM be a nonsingular projective variety and C a Zariski-closed subset with codimension codimC,M≥2. Then, there exists some irreducible hyperplane section H, such that H⊃C.Proof.
We take some ample divisorA on M. We have a birational morphism f:M′⟶M, such that M′ is a nonsingular projective variety, that f−1C is divisorial with only simple normal crossings and that there exists an effective divisor C0 with the property that SuppC0=f−1C and −C0 is f-ample. Then, mf∗A−C0 is ample for a sufficiently large integer m. For a sufficiently large and divisible integer l, the divisor lmA is very ample, and there exists a member H0∈lmf∗A−C0 which is very ample and irreducible. We put H≔f∗H0+lC0. Then, H∈lmA.
The locusf−1H coincides with Suppf∗H=SuppH0+lC0. Thus, f−1H⊃f−1C.



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