Abstract
In this study, we give an alternative and elementary proof to Tsuji’s criterion for a Cartier divisor to be numerically trivial.
1. Introduction
In this article, every algebraic variety is proper over the field of complex numbers .
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]). Let a surjective morphism between complete varieties. Let be a nef Cartier divisor on and some subvarieties of , such that and a subset of which is a union of countably many proper Zariski-closed subsets. Assume that(1)for some , for every curve on (2) for every irreducible curve on , such that Then, 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 (Theorem 1). 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]). Let be a surjective morphism from a complete surface to a complete curve . Let be a nef Cartier divisor on . Assume that(1)For some , for every curve on (2) for some irreducible curve on , such that Then, is numerically trivial.
Remark 1. In the statement of Lemma 1, condition (1) immediately implies that is numerically trivial on every general fiber of the morphism 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 Theorem 1
Proof. We prove the assertion by induction on .
First, we take a commutative diagram as shown in Figure 1 with the following properties:(1) and are nonsingular projective varieties(2) is a birational morphism(3) is a generically finite morphism(4) is a morphism with only connected fibersThere exists some irreducible component of , such that . We set .
The locus is included in a union of at most countably many proper Zariski-closed subsets of (Proposition 1). Thus, we obtain a union of countably many proper Zariski-closed subsets of with the following two properties:(1) is numerically trivial on every fiber of over (2) for every irreducible curve on , such that It suffices to prove that for every irreducible curve on . We fix an irreducible curve on .
Case 1. . This case divides into Subcases 1 and 2.
Subcase 1. and is a point.
We have from (1).
Subcase 2. and is a curve. This subcase divides into Subcases 3 and 7.
Subcase 3. , is a curve and . This subcase divides into Subcases 4, 5, and 6.
We note that for some irreducible curve on .
Subcase 4. , is a curve, , and .
. Thus, .
Subcase 5. , is a curve, , and .
Because from (2), Lemma 1 implies that is numerically trivial, and thus, .
Subcase 6. , is a curve, , and .
Because the codimension , we have an irreducible hyperplane section of that includes and (Proposition 2). Then, is numerically trivial from the induction hypothesis. Consequently, .
Subcase 7 (cf. [[4], 2.1.2]). , is a curve, and .
Let be the set of irreducible components of . We note that for some irreducible curve on . Thus, is numerically trivial for some , such that from the property (2) and from the induction hypothesis.
If , then from the connectedness of fibers of , and therefore, for some .
Thus, or .
If and , then from the connectedness of fibers of , and therefore, for some . From this argument, we obtain the following properties:(1)(2) for all with (3)The fact that is numerically trivial and that implies that is numerically trivial from the induction hypothesis. The fact that is numerically trivial and that implies that is numerically trivial from the induction hypothesis. This argument implies that is numerically trivial. Because , we have that . Consequently, .
Case 2. . This case divides into Subcases 8 and 11.
Subcase 8. and . This subcase divides into Subcases 9 and 10.
Subcase 9. , , and .
Lemma 1 implies that is numerically trivial, and thus, .
Subcase 10. , , and .
Because , we have that for an irreducible hyperplane section of from the property (2) of the divisor . The Hodge index theorem implies that is numerically trivial. Thus, .
Subcase 11. and .
Because the codimension , there exists an irreducible hyperplane section of that includes (Proposition 2). We may assume that . Note that, from Case 1, for every irreducible curve on , such that . Thus, is numerically trivial from the induction hypothesis. Consequently, .□
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. Let be a surjective morphism between projective varieties and a nef Cartier divisor on .
We assume that for some , the intersection number for every irreducible curve on .
Then, the locus is included in a union of at most countably many proper Zariski-closed subsets of .
Proof. There exists some ample divisor on . Assume that is an irreducible curve on , such that (i.e., is a point) and that . There exists some irreducible component of the universal scheme for the Hilbert scheme of , such that includes , where is the point , which represents the subscheme of , and Figure 2 shows the projections and and the property that . We set .
First, we consider the normalization , , and of the morphism . Next, consider the Stein factorization of the morphism .
Last, consider the flattening , , and of the morphism , where the morphism is birational and the variety is nonsingular. We note that the morphism is flat and with only connected fibers.
We put .
Thus, we have the commutative diagram, as shown in Figure 3.
From the flatness of the morphism , the intersection number for every fiber of the morphism because . Thus, for every fiber of the morphism , the morphism contracts to one point from the connectedness of . In other words, is included in some fiber of .
There exists some ample divisor on . Of course, for every curve on . Because the morphism is birational, we have that is not a point (i.e., ) for some fiber of . From the flatness, for every fiber of . Thus, every fiber of cannot be contracted to a point by the morphism .
There exists some fiber of , such that . Then, because the morphism maps to a point . Consequently, because does not contract to a point by the morphism . Thus, . From the flatness of the morphism , the intersection number is for every fiber of the morphism .
We note that every fiber of is mapped in some fiber of . In other words, every fiber of is swept out by fibers of .
So, for every fiber of , the locus is swept out by connected curves , such that is one point and that the intersection number . We note that we consider as and that from the connectedness of . Thus, . Consequently, . In other words, is disjoint with .
The countability of the irreducible components of the Hilbert scheme of implies the assertion.
Proposition 2. Let be a nonsingular projective variety and a Zariski-closed subset with codimension . Then, there exists some irreducible hyperplane section , such that .
Proof. We take some ample divisor on . We have a birational morphism , such that is a nonsingular projective variety, that is divisorial with only simple normal crossings and that there exists an effective divisor with the property that and is -ample. Then, is ample for a sufficiently large integer . For a sufficiently large and divisible integer , the divisor is very ample, and there exists a member which is very ample and irreducible. We put . Then, .
The locus coincides with . Thus, .
Data Availability
No data were used to support this study.
Disclosure
The updated version of the manuscript is presented in arXiv:2109.02 034v1 ([8]).
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
The author was supported by the research grant of Gifu Shotoku Gakuen University in the years 2019 and 2020.