Smooth Complete Toric Threefolds with No Nontrivial Nef Line Bundles

174 Proc. Japan Acad., 81, Ser. A (2005) [Vol. 81(A),

Smooth complete toric threefolds with no nontrivial nef line bundles

By Osamu Fujino∗) and Sam Payne∗∗) (Communicated by Shigefumi Mori, m. j. a., Dec. 12, 2005)

Abstract: We describe all of smooth complete toric threefolds of Picard number 5 with no nontrivial nef line bundles, and show that no such examples exist with Picard number less than 5. Key words: Projectivity; toric geometry.

1. Introduction. Let X be a complete al- no nontrivial nef line bundles. See [1], [9, 5.3.14], [11, gebraic variety. A line bundle L on X is said to 3.3], and [14]. Recently, the first author has given ex- be nef if (L · C), the degree of L restricted to C, is amples of complete, singular toric varieties with arbi- nonnegative for every curve C ⊂ X. In particular, trary Picard number that have no nontrivial nef line any ample line bundle on X is nef, as is the trivial bundles [5]. Motivated by the observation that all of line bundle OX . Hence complete varieties with no the standard examples of smooth, complete, nonpro- nontrivial nef line bundles are nonprojective. Many jective toric varieties are proper over projective va- constructions of smooth or mildly singular nonpro- rieties, he conjectured that every smooth, complete jective varieties use toric geometry; in combinatorial toric variety has a nontrivial nef line bundle. The language, they correspond to nonregular triangula- second author found counterexamples to this conjec- tions of totally cyclic vector configurations, which ture, but the examples were relatively complicated, are also referred to as nonprojective or noncoher- with Picard number 11 and higher [15]. In general, ent triangulations. The standard examples of com- it is known that every smooth complete toric variety plete nonprojective toric varieties [1], [4], [7, II.2E], with Picard number less than 4 is projective [8]. [13, pp. 84–85] arise from nonregular triangulations This note describes the simplest possible exam- of boundary complexes of polytopes; such varieties ples of smooth complete toric threefolds with no non- admit proper birational morphisms to projective trivial nef line bundles, which have Picard number 5. toric varieties, and hence have nontrivial nef line These examples consist of two infinite collections of bundles. If ∆ is the fan over the faces of a trian- combinatorially equivalent varieties, plus one excep- gulation of the boundary complex of some polytope tional example. The constructions yield similar ex- P , then X(∆) is proper and birational over the pro- amples with any Picard number greater than 5, and jective toric variety XP ◦ corresponding to the polar we show that no such examples exist with Picard polytope P ◦, and the pullbacks of ample line bundles number less then 5. This resolves [5, Problem 5.2], from XP ◦ are nontrivial nef line bundles on X(∆). which asked for which Picard numbers there exist If a variety has no nontrivial nef line bundles, smooth complete algebraic varieties with no nontriv- then it admits no nonconstant morphisms to projec- ial nef line bundles, for the case of toric threefolds. tive varieties. One example of such a variety, due to 2. Examples. We begin by giving examples Fulton, is the singular toric threefold corresponding of smooth toric threefolds with no nontrivial nef line to the fan over a cube with one ray displaced, which bundles and Picard number 5. has no nontrivial line bundles at all [6, pp. 25–26, Example 1. Let Σ be the fan in R3 whose 72]. A similar example is due to Eikelberg [3, Exam- rays are generated by ple 3.5]. The literature also contains a number of examples of Moishezon spaces of Picard number 1 with v1 = (1, 0, 0), v2 = (0, 1, 0), v3 = (0, 0, 1) v4 = (0, −1, −1), v5 = (−1, 0, −1), v6 = (−2, −1, 0), 2000 Mathematics Subject Classification. Primary 14M25; Secondary 14C20. and whose maximal cones are ∗) Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, Aichi 464-8602. hv1, v2, v3i, hv1, v2, v4i, hv2, v4, v5i, hv2, v3, v5i, ∗∗) University of Michigan, Department of Mathematics, 2074 East Hall, 530 Church St., Ann Arbor, MI 48109, USA. hv3, v5, v6i, hv1, v3, v6i, hv1, v4, v6i, hv4, v5, v6i. No. 10] Smooth complete toric threefolds with no nontrivial nef line bundles 175

The nonzero cones of Σ are the cones over the faces d1 + d2 + d3 + d4 + d5 + d6 ≥ of the following nonconvex polyhedron. 2d1 + d2 + 2d3 + d4 + 2d5 + d6.

Since all of the di are nonnegative, it follows that d1 = d3 = d5 = 0. Substituting d1 = d5 = 0 in the ﬁrst inequality gives d2 = d4 = 0. Since {v1, . . . , v5} 3 positively spans R , we have PD = {0}, where PD is the polytope deﬁned by

3 PD = {u ∈ R : hu, vii ≥ −di for all i}. Since D is nef,

−di = min{hu, vii: u ∈ PD}, for all i. Therefore, D = 0. The above argument shows that any com- 3 plete fan in R containing the cones hv1, v2, v4i, hv2, v3, v5i, and hv1, v3, v6i corresponds to a complete toric variety with no nontrivial nef line bundles. In particular, star subdivisions of the remain- Let ∆ be the fan obtained from Σ by successive ing cones in ∆ lead to examples of smooth complete star subdivisions along the rays spanned by v7 = toric threefolds with no nontrivial nef line bundles (−1, −1, −1) and v8 = (−2, −1, −1). and any Picard number greater than or equal to 5. We claim that X = X(∆), the toric threefold Example 1 is the toric variety labeled as (8-12) corresponding to the fan ∆ with respect to the lattice in [12, p. 79], as can be checked by setting n = Z3 ⊂ R3, is smooth and complete and that X has no (−1, −1, −1), n0 = (1, 0, 0), and n00 = (0, 0, 1). nontrivial nef line bundles. It is easy to check that Example 2. Let a be an integer, with a 6= |∆| = R3, so X is complete, and that each of the 0, −1. Let ∆ be the fan whose rays are spanned by 12 maximal cones of ∆ is spanned by a basis for Z3, v = (1, 0, 0), v = (0, 1, 0), v = (0, 0, 1), so X is smooth. It remains to show that X has no 1 2 3 v = (0, −1, −a), v = (0, 0, −1), v = (−1, 1, −1), nontrivial nef line bundles. 4 5 6 P v7 = (−1, 0, −1), v8 = (−1, −1, 0), Suppose D = diDi is a nef Cartier divisor, where Di is the prime T -invariant divisor cor- and whose maximal cones are responding to the ray spanned by v . Since nef i hv , v , v i, hv , v , v i, hv , v , v i, hv , v , v i, line bundles on toric varieties are globally generated, 1 2 3 1 3 4 1 4 5 1 5 6 O(D) has nonzero global sections. Therefore we may hv1, v2, v6i, hv2, v3, v8i, hv3, v4, v8i, hv4, v5, v8i, assume that D is eﬀective, i.e. each di ≥ 0. We will hv5, v6, v7i, hv5, v7, v8i, hv6, v7, v8i, hv2, v6, v8i. show that D = 0. Note that hv1, v2, v4i is a cone in ∆, and It is straightforward to check that X(∆) is smooth and complete. We claim that X(∆) has no nontrivial P v5 = v2 + v4 − v1. nef line bundles. Suppose D = diDi is an eﬀective nef divisor on X(∆). The convexity of Ψ gives the It then follows from the convexity of the piecewise D following inequalities: linear function ΨD associated to D that d1 + d8 ≥ ad3 + d4, d1 + d5 ≥ d2 + d4. 2d4 + d6 ≥ (2a + 1)d5 + d8, Similarly, d2 + d5 ≥ d1 + d6,

d2 + d6 ≥ 2d3 + 2d5, d3 + d6 ≥ 2d2 + d8, d3 + d4 ≥ 2d1 + d6, d2 + d4 ≥ −ad3,

Adding the three above inequalities, we have d1 + d8 ≥ d4 − ad5. 176 O. Fujino and S. Payne [Vol. 81(A),

We claim that d3 = d5 = 0. When a > 0, the claim so d3 = d4 = 0. Since {v2, . . . , v7} positively spans 3 follows from the first four inequalities, since R , it follows that PD = {0}, and hence D = 0. The cases where a or b is negative, or both, are 2(d1 + d8) + 2d4 + d6 + 2(d2 + d5) + d3 + d6 ≥ proven similary, using the inequalities abve, together 2(ad + d ) + (2a + 1)d + d + 3 4 5 8 with the following inequalities, which also come from 2(d1 + d6) + 2d2 + d8. the convexity of ΨD: Similarly, the claim follows from the last four in- d3 + d6 ≥ −ad7 + d8, equalities when a < −1. Substituting 0 for d3 and d4 + d8 ≥ (1 − a)d5, d5 in the middle two inequalities gives d1 = d2 = 3 d + d ≥ d − bd , d8 = 0. Since {v1, v2, v3, v5, v8} positively spans R , 4 7 1 6 it follows that PD = {0}, and hence D = 0. So X(∆) d1 + d3 ≥ (1 − b)d2. has no nontrivial nef line bundles. The varieties in Example 2 are labeled (8-50) The varieties in Example 3 are labeled (8-130) in [12, p. 78], as can be checked by setting n = in [12, p. 79], as can be checked by setting n = (1, 0, 0), n0 = (0, 1, 0), and n00 = (0, 0, 1). (1, −1, 0), n0 = (0, 1, 0), and n00 = (0, 0, 1). Example 3. Let a and b be nonzero integers, 3. Classification. Using the classification of with (a, b) 6= (±1, ±1). Let ∆ be the fan whose rays minimal smooth toric threefolds with Picard number are spanned by at most 5, due to Miyake and Oda [12, §9], and Na-

v1 = (−1, b, 0), v2 = (0, −1, 0), v3 = (1, −1, 0), gaya [10], we show that Examples 1, 2, and 3 give all v4 = (−1, 0, −1), u5 = (0, 0, −1), v6 = (0, 1, 0), such varieties that have no nontrivial nef line bun- v7 = (0, 0, 1), v8 = (1, 0, a), dles. Recall that a smooth toric variety is minimal if it is not the blow up of a smooth toric variety along and whose maximal cones are a smooth torus invariant center. hv1, v2, v4i, hv2, v3, v4i, hv3, v4, v5i, hv4, v5, v6i, Theorem. Every smooth complete toric three- hv1, v4, v6i, hv1, v6, v7i, hv1, v2, v7i, hv2, v3, v7i, fold with no nontrivial nef line bundle and Picard number at most 5 is isomorphic to one described in hv3, v5, v8i, hv5, v6, v8i, hv3, v7, v8i, hv6, v7, v8i. Examples 1, 2, and 3, above. It is straightforward to check that X(∆) is smooth It will suﬃce to show that there are no minimal and complete. We claim that X(∆) has no nontrivial smooth complete toric threefolds with no nontriv- nef line bundles. Suppose D = P d D is an eﬀective i i ial nef line bundles and Picard number less than 5, nef divisor on X(∆). We now consider the case where and that every minimal smooth complete nonprojec- a and b are both positive, and show that D = 0. The tive toric threefold with Picard number 5 that is not convexity of Ψ gives the following inequalities: D listed in Examples 1, 2, and 3, admits a nonconstant

d3 + d6 ≥ ad5 + d8, morphism to a projective variety.

d2 + d8 ≥ d3 + ad7, Lemma. There are no minimal smooth complete toric threefolds with no nontrivial nef line bun- d1 + d5 ≥ bd6 + d4, dles and Picard number less than 5. d4 + d7 ≥ d1 + bd2. Proof . By the classiﬁcation theorem of Miyake, Since (a, b) 6= (1, 1), either a > 1 or b > 1. We claim Oda, and Nagaya, every minimal smooth complete that d2 = d5 = d6 = d7 = 0. If a > 1, then adding toric threefold with Picard number less than 5 is iso- the above inequalities gives d5 = d7 = 0, and sub- morphic to either P3, a P2-bundle over P1, a P1- stituting 0 for d5 and d7 in the last two inequalities bundle over a projective toric surface, or one of the gives d2 = d6 = 0. If b > 1, then adding the above following inequalities gives d2 = d6 = 0, and substituting 0 (a) The projective toric variety X(∆), where ∆ is for d2 and d6 in the ﬁrst two inequalities gives d5 = the fan in R3 whose rays are generated by d7 = 0. This proves the claim. Now, convexity of ΨD gives v1 = (1, 0, 0), v2 = (0, 1, 0), v3 = (0, 0, 1), v4 = (0, −1, a), d2 + d5 ≥ d3 + d4, v5 = (0, 0, −1), v6 = (0, 1, −1), v7 = (−1, 2, −1), No. 10] Smooth complete toric threefolds with no nontrivial nef line bundles 177

for some integer a, and whose maximal cones v1 = (0, 1, 0), v2 = (0, −1, −1), are v3 = (1, 0, 0), v4 = (0, 0, 1), v5 = (−1, 0, −1), v6 = (−1, −2, −2), hv1, v2, v3i, hv1, v3, v4i, hv1, v4, v5i, hv1, v5, v6i, v7 = (−1, −1, −1), v8 = (−1, −1, 0), hv , v , v i, hv , v , v i, hv , v , v i, hv , v , v i, 1 6 7 1 2 7 2 3 7 3 4 7 and whose maximal cones are hv4, v5, v7i, hv5, v6, v7i. hv1, v2, v3i, hv1, v3, v4i, hv1, v4, v5i, hv1, v5, v6i,

(b) The toric variety X associated to the fan in R3 hv1, v2, v6i, hv2, v3, v8i, hv3, v4, v8i, hv4, v5, v8i,

whose rays are generated by hv5, v6, v7i, hv5, v7, v8i, hv6, v7, v8i, hv2, v6, v8i.

v1 = (−1, 0, 0), v2 = (0, −1, 0), v3 = (0, 0, −1), In this case, ∆ is a refinement of the complete v4 = (1, 0, 1), v5 = (0, 1, 1), fan whose set of rays is {v1, v3, v4, v7}. Therefore v6 = (1, 1, 1), v7 = (1, 1, 0), X(∆) admits a proper birational morphism to P3. and whose maximal cones are (c) (8-8) The toric variety X(∆) associated to the fan in R3 whose rays are generated by hv1, v2, v3i, hv2, v3, v4i, hv2, v4, v5i, hv1, v2, v5i, v1 = (0, 0, 1), v2 = (1, 0, 0), hv4, v5, v6i, hv1, v3, v7i, hv3, v4, v7i, hv4, v6, v7i, v3 = (0, −1, −1), v4 = (−1, −2, −1), hv5, v6, v7i, hv1, v5, v7i. v5 = (0, 1, 0), v6 = (0, 0, −1), In this case, X is not projective [12, Proposi- v7 = (−1, −2, −2), v8 = (−1, −1, −2), tion 9.4], but the anticanonical bundle −KX is and whose maximal cones are nef. hv1, v2, v3i, hv1, v3, v4i, hv1, v4, v5i, hv1, v2, v5i, Therefore, up to isomorphism, there is only one non- hv , v , v i, hv , v , v i, hv , v , v i, hv , v , v i, projective smooth complete toric threefold with Pi- 2 5 6 3 4 7 2 3 8 3 7 8 card number less than 5, and it has a nontrivial nef hv4, v7, v8i, hv4, v5, v8i, hv5, v6, v8i, hv2, v6, v8i. line bundle. In this case, ∆ is a subdivision of the com- Proof of Theorem. It remains to show that plete fan whose rays are v1, v2, v5, v7. Therefore every minimal smooth complete nonprojective toric X(∆) admits a proper birational morphism to threefold of Picard number 5 that is not isomorphic P(1, 1, 2, 2). to one in Examples 1, 2, or 3 has a nontrivial nef line (d) (8-11) The toric variety X(∆) associated to bundle. By the classification theorem of Miyake and the fan in R3 whose rays are generated by Oda, and Nagaya, and by the second Remark in [12, v = (1, 0, 0), v = (0, −1, 0), p. 80], any such variety is isomorphic to one of the 1 2 v = (0, 0, 1), v = (1, 1, a), varieties listed below. In each case, it will suffice to 3 4 v = (0, 0, −1), v = (0, −1, −1), either exhibit a specific nontrivial nef line bundle, or 5 6 v = (−1, −2, −1), v = (0, 1, b), to give a nonconstant morphism to a projective vari- 7 8 ety. Note that the varieties X(∆) as in Examples 2 for some integers a and b, and whose maximal and 3, but with a = 0 or b = 0, are not minimal. cones are The labels such as (8-50) are the labels given to the hv1, v2, v3i, hv1, v3, v4i, hv1, v4, v5i, hv1, v5, v6i, varieties in [12, pp. 78–79]. Different labels with the same numbers, such as (8-50) and (8-500), indicate hv1, v6, v7i, hv1, v2, v7i, hv2, v3, v7i, hv5, v6, v7i, that the corresponding varieties have combinatori- hv3, v4, v8i, hv4, v5, v8i, hv5, v7, v8i, hv3, v7, v8i. ally equivalent fans. In this case, the projection (x, y, z) 7→ (x, y) 0 (a) (8-5 ) The toric variety X = X(∆) as Exam- gives a map from ∆ to the complete fan in R2 ple 2, with a = −1. In this case, −KX −D4 −D8 whose rays are generated by (1, 0), (0, 1), (1, 1), is nef. and (−1, −2). Therefore X(∆) admits a proper (b) (8-500) The toric variety X(∆) associated to surjective morphism to the blow up of P(1, 1, 2) the fan in R3 whose rays are generated by at a point. 178 O. Fujino and S. Payne [Vol. 81(A),

0 (e) (8-13 ) The toric variety X = X(∆) as in Ex- hv3, v5, v8i, hv5, v7, v8i, hv6, v7, v8i, hv2, v6, v8i. ample 3, with (a, b) = (±1, ±1). In this case, −KX is nef. In this case, the projection (x, y, z) 7→ x gives a (f) (8-1300) The toric vatiety X(∆) associated to map from ∆ to the complete fan in R. There- the fan in R3 whose rays are generated by fore X(∆) admits a proper surjective morphism to P1. v1 = (1, 1, b), v2 = (1, 0, 0), v3 = (0, −1, 0), v4 = (0, 0, −1), v5 = (−1, a, d), v6 = (0, 1, 0), Acknowledgments. The ﬁrst author is par- v7 = (0, 0, 1), v8 = (−1, c, d + 1), tially supported by the Sumitomo Foundation and by for some integers a, b, c, and d, and whose max- the Grant-in-Aid for Scientiﬁc Research # 17684001 imal cones are from JSPS.

hv1, v2, v4i, hv2, v3, v4i, hv3, v4, v5i, hv4, v5, v6i, The second author is supported by a Graduate Research Fellowship from the NSF. hv1, v4, v6i, hv1, v6, v7i, hv1, v2, v7i, hv2, v3, v7i,

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