A. LANTERI AND A. L. TIRONI KODAI MATH. J. 27 (2004), 299–320 REDUCIBLE HYPERPLANE SECTIONS OF THREEFOLDS: TWO COMPONENTS OF SECTIONAL GENUS ZERO* Antonio Lanteri and Andrea Luigi Tironi Abstract By using adjunction theory, we describe the smooth complex projective threefolds admitting a simple normal crossing divisor of the form A þ B among their hyperplane sections, both components A and B being smooth surfaces with sectional genus 0, and one of them being nef or at worst an exceptional divisor of the first reduction mapping. Introduction Projective manifolds with an irreducible hyperplane section being a special variety have been studied since longtime [BS]. However the corresponding study for a reducible hyperplane section consisting of a simple normal crossing divisor whose components are special varieties was started only recently by Chandler, Howard and Sommese [CHS]. Results from [CHS] seem to indicate that classification in this setting can be harder for varieties of low dimension than in higher dimensions. In particular, let X H P N be a smooth projective variety, let L be its hyperplane bundle and suppose that jLj contains an element A þ B where A; B are smooth irreducible divisors meeting transversally (simple smooth decomposition). If both ðA; LA Þ and ðB; LB Þ have sectional genus zero, then the structure of ðX ; LÞ is known, as well as the description of A and B, if dim X b 4 ([CHS], [LT]). In this paper we address the classification problem in the same set-up, when dim X ¼ 3. We would like to mention that in the course of our study we received the preprint of [BCS], where a similar set up is considered. In particular [BCS, (5.3)] provides the list of all possible numerical invariants concerning A; B, and A V B inside them, when both pairs ðA; LA Þ; ðB; LB Þ are rational scrolls meeting along a smooth curve of positive genus. The case when this curve is rational is not considered there, since, as the authors say, it fits into a general result formulated for any dimension in [CHS]. However, in our opinion, the three dimensional setting deserves to be further studied also in this case. * 2000 Mathematics Subject Classification. Primary 13J30, 14J26; Secondary 14C20, 14N30. Keywords and phrases. 3-folds, line bundles (ample and spanned), adjunction theory, simple normal crossing divisors, sectional genus, special varieties. Received January 16, 2004. 299 300 antonio lanteri and andrea luigi tironi Here, assuming that at least one of A and B is a nef divisor (seminef decomposition) we obtain a classification result, working in a slightly more general setting than that described above. In fact, the adjunction theoretic classification techniques we use allow us to assume that the line bundle L is simply ample and spanned and not necessarily very ample. The result is as follows. Theorem 0.1. Let L be an ample and spanned line bundle on a connected projective manifold X of dimension three. Assume that there is a seminef simple smooth decomposition A þ B A jLj. If gðA; LA Þ ¼ gðB; LB Þ ¼ 0 then ðX ; LÞ and A; B, after renaming, are as in the following cases: 1. a scroll over P 1 , with A being a P 1 -bundle over P 1 and B a fiber of the scroll projection; 2. a quadric fibration q : X ! P 1 , with both A and B being P 1 -bundles over P 1 via q; 3. X ¼ PðVÞ, where V is an ample and spanned vector bundle of rank 2 on F1 with Chern classes c1 ðVÞ ¼ ½3l þ bf
, c2 ðVÞ ¼ 2b  4 for an integer b b 5, where l and f denote the ð1Þ-section and a fiber of F1 , and L is the tautological line bundle on X. Moreover, A G F1 is a section of the scroll projection p, B ¼ PðVg Þ where g is a smooth element of the linear system j2l þ 2fj on F1 , and there exists a non-splitting exact sequence 0 ! ½B
A ! pjA V ! LA ! 0; furthermore h ¼ A V B is isomorphic to g on A and is a section on B; 4. X ¼ PðVÞ, where V is an ample and spanned vector bundle of rank 2 on P 2 with Chern classes c1 ðVÞ ¼ 4, c2 ðVÞ ¼ 5, and L is the tautological line bundle on X. Let p : X ! P 2 be the scroll projection; then A G F1 , pjA : A ! P 2 being the contraction of the ð1Þ-section of A, B ¼ PðVg Þ G P 1  P 1 , g being a smooth conic; furthermore h ¼ A V B is a section of B with h 2 ¼ 0 and on A it is a smooth element of the linear system j2E þ 2F j, where E and F denote the ð1Þ-section and a fiber, respectively; 5. ðP 1  P 2 ; OP 1 P 2 ð1; 2ÞÞ, with A A jOP 1 P 2 ð1; 0Þj and B a general element of jOP 1 P 2 ð0; 2Þj; 6. ðP 1  P 1  P 1 ; OP 1 P 1 P 1 ð1; 1; 1ÞÞ, with A A jOP 1 P 1 P 1 ð1; 0; 0Þj and B A jOP 1 P 1 P 1 ð0; 1; 1Þj, up to reordering the factors; 7. ðPðTP 2 Þ; xTP 2 Þ, where xTP 2 stands for the tautological line bundle, or equivalently, X A jOP 2 P 2 ð1; 1ÞX j and L ¼ OP 2 P 2 ð1; 1ÞX with A A jOP 2 P 2 ð1; 0ÞX j and B A jOP 2 P 2 ð0; 1ÞX j; 8. KX þ 2L is nef and big and ðX ; LÞ admits ðP 3 ; OP 3 ð2ÞÞ as first reduction, the reduction morphism r : X ! P 3 being the blowing-up at s a 1 points: if s ¼ 0 then A; B A jOP 3 ð1Þj, while if s ¼ 1, i.e. r is the blowing-up at a point p, then, up to renaming, either A is the proper transform of a quadric cone having vertex at p and B is the exceptional divisor, or A is the proper transform of a plane through p and B that of a plane not containing p. Actually we prove a little bit more, since the seminefness of the decomposition A þ B is only used to manage the case when both polarized manifolds reducible hyperplane sections of threefolds 301 ðA; LA Þ; ðB; LB Þ are scrolls. The remaining cases are settled by Theorems 3.3, 4.1 and 4.2. On the other hand, contrary to what is known in dimension 2, we do not know concrete examples of polarized threefolds ðX ; LÞ admitting a simple smooth decomposition A þ B A jLj not satisfying the assumption in Theorem 0.1. In fact, all pairs ðA; LA Þ; ðB; LB Þ appearing in [BCS, (5.3)] do not satisfy this assumption; however the corresponding structure of ðX ; LÞ, if any, is completely unknown. Here is a sketch of the proof. The assumption that A þ B is a seminef decomposition implies that the hinge curve h ¼ A V B is rational. Then we can rely on the adjunction theoretic structure of ðX ; LÞ, provided by [CHS] to obtain a precise description of the varieties at hand via a case-by-case analysis. In particular, in Section 2 we prove that if ðX ; LÞ is a scroll over a smooth surface, then there are only three possibilities, namely cases 3, 4 and 5 in Theorem 0.1. Moreover, for each variety appearing in Theorem 0.1 we also list all sectional genus zero decompositions of the reducible elements of jLj. This work was done in the framework of the National Research Project ‘‘Geometry on Algebraic Varieties’’, supported by the MIUR of the Italian Government (Cofin 2002). 1 Background material We work over the field of complex numbers C and we use the standard notation from algebraic geometry. The tensor products of line bundles are denoted additively. The pull-back i  F of a vector bundle F on a projective variety X by an embedding i : Y ! X is denoted by FY . We denote by KX the canonical bundle of a smooth projective variety X . A polarized manifold is a pair ðX ; LÞ consisting of a smooth projective variety X and an ample line bundle L on X . For polarized manifolds we will use the adjunction theoretic terminology of [BS]. In particular we say that ðX ; LÞ is a scroll over a smooth projective variety to mean that it is a scroll in the adjunction theoretic sense. As a consequence, e.g., the smooth quadric surface ðQ 2 ; OQ 2 ð1ÞÞ ¼ ðP 1  P 1 ; OP 1 P 1 ð1; 1ÞÞ is not considered as a scroll over P 1 . Let ðX ; LÞ be a smooth complex projective threefold polarized by an ample and spanned line bundle. Suppose that jLj contains a divisor A þ B where A; B are irreducible smooth surfaces meeting transversally along a smooth curve. We say that A þ B is a smooth decomposition for L. Of course one can also consider smooth decompositions consisting of more than two components. Therefore, sometimes we use the expression simple smooth decomposition (ssd for short) to emphasize that there are just two components. We are interested in ssd A þ B A jLj such that gðA; LA Þ ¼ gðB; LB Þ ¼ 0: Under this assumption both pairs ðA; LA Þ; ðB; LB Þ are polarized surfaces of sec- 302 antonio lanteri and andrea luigi tironi tional genus zero, hence they belong to one of the following two classes [F1, (12.1) and (5.10)]: A :¼ fðP 2 ; OP 2 ðuÞÞ; u ¼ 1; 2; ðQ 2 ; OQ 2 ð1ÞÞg and B :¼ fðFe ; ½C0 þ bf
Þ; e b 0g: ð1:0Þ For the class B of rational scrolls we used standard symbols; of course the integer b has to satisfy the ampleness condition b > e [Ha, p. 380]. Let h :¼ A V B. We call h the hinge curve; by assumption it is a smooth curve. Its genus is given by 2gðhÞ  2 ¼ ðKX þ A þ BÞAB ¼ ðKX þ LÞAB: We recall the following lemma. Lemma 1.1 ([T, Lemma 1]). Let ðX ; LÞ be a threefold polarized by an ample and spanned line bundle L. Let A þ B A jLj be a ssd. Then gðX ; LÞ ¼ gðA; LA Þ þ gðB; LB Þ þ ABL  1. We need some structure results to describe smooth decompositions occurring for some standard varieties of adjunction theory. Lemma 1.2. Let ðX ; LÞ be a threefold polarized by an ample and spanned line bundle L. Assume that ðX ; LÞ is a scroll over a smooth curve C and let A þ B A jLj be a ssd. Then, after renaming, ðA; LA Þ G ðP 2 ; OP 2 ð1ÞÞ, ðB; LB Þ is a scroll over C or ðQ 2 ; OQ 2 ð1ÞÞ, and gðX ; LÞ ¼ gðB; LB Þ ¼ gðCÞ. Proof. Let p : X ! C be the scroll projection and let F G P 2 be a general fiber of p. We have ½A
F ¼ OP 2 ðaÞ and ½B
F ¼ OP 2 ðbÞ, for some integers a; b b 0. Since OP 2 ð1Þ ¼ LF ¼ ½A
F þ ½B
F ¼ OP 2 ða þ bÞ; it follows, after renaming, that a ¼ 0, b ¼ 1. This implies that ðA; LA Þ G ðF ; LF Þ G ðP 2 ; OP 2 ð1ÞÞ and ðB; LB Þ is a scroll over C or, possibly, ðQ 2 ; OQ 2 ð1ÞÞ, if gðCÞ ¼ 0. Moreover ABL ¼ ½B
F ½L
F ¼ LF2 ¼ 1 and then by Lemma 1.1 we deduce that gðX ; LÞ ¼ gðB; LB Þ ¼ gðCÞ. r Lemma 1.3. Let ðX ; LÞ be a Del Pezzo threefold polarized by an ample and spanned line bundle L. Assume that A þ B A jLj is a ssd. Then gðA; LA Þ ¼ gðB; LB Þ ¼ 0 and ðX ; LÞ is one of the following pairs: 1. ðP 1  P 1  P 1 ; OP 1 P 1 P 1 ð1; 1; 1ÞÞ; 2. ðPðTP 2 Þ; xTP 2 Þ, where TP 2 is the tangent bundle to P 2 and xTP 2 stands for the tautological line bundle; 3. ðX ; LÞ has ðP 3 ; OP 3 ð2ÞÞ as first reduction, with X being P 3 blown-up at one point at most. reducible hyperplane sections of threefolds Proof. 303 By the genus formula we have that 2gðhÞ  2 ¼ ðKX þ LÞAB ¼ ABL < 0; hence ABL ¼ 2. By Lemma 1.1 we thus get 1 ¼ gðX ; LÞ ¼ gðA; LA Þ þ gðB; LB Þ þ 1; which gives gðA; LA Þ ¼ gðB; LB Þ ¼ 0. (3.4)]. r Then the assertion follows from [CHS, Remarks. In case 1 of Lemma 1.3 we get a ssd A þ B A jLj, by choosing, e.g., A A jOP 1 P 1 P 1 ð1; 0; 0Þj and B A jOP 1 P 1 P 1 ð0; 1; 1Þj. For this decomposition, ðA; LA Þ G ðQ 2 ; OQ 2 ð1ÞÞ and ðB; LB Þ A B. In case 2, X can be described alternatively as a smooth element of jOP 2 P 2 ð1; 1Þj and L ¼ OP 2 P 2 ð1; 1ÞX . This shows that the only possible decomposition for L is A þ B, where A A jOP 2 P 2 ð1; 0ÞX j and B A jOP 2 P 2 ð0; 1ÞX j, giving a scroll structure on both components. As to 3, note that the case with r : X ! P 3 the blowing-up at a point p does really occur. Recall that L :¼ r  ðOP 3 ð2ÞÞ  E, where E is the exceptional divisor. Here we list all possible decompositions for L. Let B 0 H P 3 be an element of jOP 3 ð2Þj containing p. a) Suppose that B 0 is irreducible and let B ¼ r1 ðB 0 Þ be its proper transform. If B 0 is smooth at p, then B ¼ r  B 0  E A jLj is an irreducible element. If B 0 is a quadric cone with vertex at p, then B ¼ r  B 0  2E A jL  Ej. Moreover B G F2 with ðr  ðOP 3 ð1ÞÞÞB ¼ ½C0 þ 2f
, since r : B ! B 0 is the minimal desingularization. Here C0 is a section of minimal self-intersection and f is a fiber of F2 . So, letting A ¼ E, we get the ssd A þ B A jLj. Note that the curve h ¼ A V B is C0 on B and a conic on A. This gives and LA ¼ ½A
A þ ½B
A ¼ OP 2 ð1 þ 2Þ ¼ OP 2 ð1Þ LB ¼ ð2r  ðOP 3 ð1ÞÞ  EÞB ¼ ½C0 þ 4f
: Therefore ðA; LA Þ G ðP 2 ; OP 2 ð1ÞÞ, ðB; LB Þ G ðF2 ; ½C0 þ 4f
Þ. b) Suppose that B 0 ¼ B10 þ B20 , where Bi0 A jOP 3 ð1Þj, and let Bi ¼ r1 ðBi0 Þ, i ¼ 1; 2. b1) Let p A B10 nB20 . Then B1 ¼ r  ðB10 Þ  E A jr  ðOP 3 ð1ÞÞ  Ej; moreover B1 G F1 with ðr  ðOP 3 ð1ÞÞÞB1 ¼ ½C0 þ f
. Here C0 is a section of minimal selfintersection and f is a fiber of F1 . Letting A ¼ B2 we get the ssd A þ B1 A jLj. Note that h ¼ A V B1 is a line inside A and an element of jC0 þ f j as a curve on B1 . This gives LA ¼ ½A
A þ ½B1
A ¼ OP 2 ð2Þ and LB1 ¼ ð2r  ðOP 3 ð1ÞÞ  EÞB1 ¼ ½C0 þ 2f
; since E V B1 is C0 . Therefore ðA; LA Þ G ðP 2 ; OP 2 ð1ÞÞ, ðB1 ; LB1 Þ G ðF1 ; ½C0 þ 2f
Þ. 304 antonio lanteri and andrea luigi tironi b2) Let p A B10 V B20 , but B10 0 B20 . Then, arguing as before and letting A ¼ E, we get the non simple smooth decomposition A þ B1 þ B2 A jLj, where ðA; LA Þ G ðP 2 ; OP 2 ð1ÞÞ and ðBi ; LBi Þ G ðF1 ; ½C0 þ 2f
Þ for i ¼ 1; 2. b3) Finally let B10 ¼ B20 . Then, letting A ¼ E we get the non-reduced decomposition A þ 2B1 A jLj, with ðA; LA Þ; ðB1 ; LB1 Þ as in b2). Note that the ssd found in a) and b1) correspond to the two cases mentioned in 8 of Theorem 0.1. Lemma 1.4. Let ðX ; LÞ be a threefold polarized by an ample and spanned line bundle L. Assume that ðX ; LÞ is a quadric fibration over a smooth curve C and that A þ B A jLj is a ssd. Then one of the following cases occurs: 1. A is a fiber, up to renaming, and gðX ; LÞ ¼ gðB; LB Þ þ 1; 2. ðA; LA Þ and ðB; LB Þ are scrolls over C or ðQ 2 ; OQ 2 ð1ÞÞ, and h is a section of both. Moreover, under the assumption that gðA; LA Þ ¼ gðB; LB Þ ¼ 0, only case 2 can occur. Proof. Let q : X ! C be the fibration and let F G P 1  P 1 be a smooth fiber. Suppose that A ¼ F ; then LF ¼ ½B
F ¼ OP 1 P 1 ð1; 1Þ: 2 In particular, ABL ¼ ðLF Þ ¼ 2. Moreover KX þ 2L ¼ q  H 1 tA for some integer t b 1, H being an ample line bundle on C. Hence, by adjunction we get the following expressions: 2gðA; LA Þ  2 ¼ ðKX þ L þ AÞAL ¼ tA 2 L  ABL ¼ ABL ¼ 2; 2gðB; LB Þ  2 ¼ ðKX þ L þ BÞBL ¼ tABL  ABL ¼ ðt  1ÞABL ¼ 2ðt  1Þ: Thus gðA; LA Þ ¼ 0, gðB; LB Þ ¼ t, and then gðX ; LÞ ¼ gðB; LB Þ þ 1, by Lemma 1.1. Now suppose that neither A nor B are fibers. Then both ½A
and ½B
restrict non trivially to the general fiber F and since LF G OP 1 P 1 ð1; 1Þ, up to renaming, we see that ½A
F G OP 1 P 1 ð1; 0Þ, ½B
F G OP 1 P 1 ð0; 1Þ. Let D H C denote the image of the singular fibers of q and set U ¼ Anqj1 A ðDÞ. Since the general fiber of qjA is a P 1 , A is a ruled surface over C. Set e ¼ A V F and f ¼ B V F . Note that 1 ¼ ef ¼ ABF . Hence, from the equality 0 ¼ e 2 ¼ ðAF Þ 2 ¼ ½F
A ½A
A ¼ ½F
A ðLA  ½B
A Þ ¼ ½F
A LA  FAB ¼ eLA  1; we see that e, the general fiber of qjA is a line. Since LA is ample, this implies that every fiber of qjA is a line. Thus ðA; LA Þ is a scroll over C or ðQ 2 ; OQ 2 ð1ÞÞ. Moreover, since h V F ¼ e V f, we have that qjA : h ! C is an isomorphism, i.e., h is a section of ðA; LA Þ. Of course the same argument works for ðB; LB Þ. This proves the first part of the statement. As to the last assertion, note that under our assumption on the sectional genera, in case 1 we would get gðX ; LÞ ¼ 1. According to the classification of polarized manifolds this would imply that ðX ; LÞ is not a quadric fibration [F1, (12.3)], a contradiction. r 305 reducible hyperplane sections of threefolds Example. Let X ¼ P 1  Fe (any e b 0) and L ¼ p1 OP 1 ð1Þ þ p2 ½l þ bf
ðb > eÞ, where p1 ; p2 are the two projections, and l; f are the fundamental section and a fiber of Fe respectively. Note that L is very ample. The composition of p2 with the scroll projection of ðFe ; ½l þ bf
Þ gives a morphism q : X ! P 1 making ðX ; LÞ a fibration with all fibers being smooth quadrics. Actually for any fiber F of q we have F ¼ f  f , where f ¼ p2 ðF Þ and f is a fiber of p2 , hence F G F 0 ¼ P 1  P 1 . For any curve G H Fe we set sG ¼ s V p1 2 ðGÞ, where s is a fiber of p1 , i.e., s A jp1 OP 1 ð1Þj. Thus LF ¼ p1 jF OP 1 ð1Þ þ p2 jF ½l þ bf
¼ ½sf þ f
is the sum of the two rulings, i.e., ðF ; LF Þ ¼ ðQ 2 ; OQ 2 ð1ÞÞ. Let A A jp1 OP 1 ð1Þj and let B A j p2 ½l þ bf
j be a smooth element (there is such a B because ½l þ bf
with b > e is very ample). Then A þ B A jLj is a ssd. Moreover, we have that ðA; LA Þ G ðFe ; ½l þ bf
Þ; ðB; LB Þ G ðP 1  g ¼ F 0 ; ½sg þ ðg 2 Þ f
Þ; where g is a smooth curve in jl þ bfj. Note that sg G g via p2 . Moreover, g 2 ¼ 2b  e b 2 and so ðB; LB Þ is in class B, i.e., it is not ðQ 2 ; OQ 2 ð1ÞÞ. Furthermore h G sg , hence gðhÞ ¼ 0. Since ABL ¼ ABðA þ BÞ ¼ AB 2 ¼ g 2 ¼ 2b  e; recalling Lemma 1.1 we get gðX ; LÞ ¼ 2b  e  1. We would like to note also that in this case we can decompose an element of jLj in at most b þ 2 smooth irreducible components, all of sectional genus zero, by taking A as above and B0 A jp2 lj, Bj A jp2 fj for every j ¼ 1; . . . ; b. Thus we obtain ðA; LA Þ G ðFe ; ½l þ bf
Þ; ðB0 ; LB0 Þ G ðF 0 ; ½sl þ ðb  eÞ f
Þ and ðBj ; LBj Þ G ðF 0 ; ½sf þ f
Þ for j ¼ 1; . . . ; b. Finally, note that for e ¼ 0 we get X ¼ P 1  P 1  P 1 with L ¼ OP 1 P 1 P 1 ð1; 1; bÞ, b b 1. For b ¼ 1 our ðX ; LÞ is a Del Pezzo threefold, while, for b b 2 we get a quadric fibration and gðX ; LÞ ¼ 2b  1. 2 Scrolls over surfaces In this Section we classify (2.0) threefolds X endowed with an ample and spanned line bundle L such that ðX ; LÞ is a scroll over a smooth surface S, and there is a ssd A þ B A jLj with gðA; LA Þ ¼ gðB; LB Þ ¼ 0. Recall that by scroll we mean an adjunction-theoretic scroll. However, by [S, Theorem 3.3 (note that the proof there works also for k ¼ 1)], our ðX ; LÞ is also a classical scroll over S. Let p : X ! S be the scroll projection; since L restricts to every fiber as OP 1 ð1Þ, up to renaming, we have that pjA : A ! S is a birational morphism, while B ¼ p  g, where g H S is a smooth curve. Moreover, since gðA; LA Þ ¼ gðB; LB Þ ¼ 0 we have that ðA; LA Þ as well as ðB; LB Þ are among the pairs listed in ð1:0Þ. In particular this implies that g G P 1 and S is either 306 antonio lanteri and andrea luigi tironi P 2 or Fe for some e b 0, being dominated birationally by A. Moreover there are two possibilities: either i) pjA is an isomorphism, or ii) pjA contracts some ð1Þ-line. Since ðX ; LÞ is a scroll over S, we can write X ¼ PðVÞ where V :¼ p L is an ample and spanned vector bundle of rank 2 on S. Moreover, since L is the tautological line bundle of V, we know that KX þ 2L ¼ p  H, where H ¼ KS þ det V is an ample line bundle on S. By the genus formula we have both 2 ¼ 2gðA; LA Þ  2 ¼ ðKX þ A þ LÞLA ¼ ðKX þ 2L  BÞAL ¼ p  HAL  ABL and 2 ¼ 2gðB; LB Þ  2 ¼ ðKX þ B þ LÞLB ¼ ðKX þ 2L  AÞBL ¼ p  HBL  ABL: The two genera being equal, this gives p  HAL ¼ p  HBL ¼ p  Hp  gL ¼ Hg: ð2:1Þ H det V ¼ p  HL 2 ¼ p  HLðA þ BÞ ¼ 2p  HBL ¼ 2Hg: ð2:2Þ Moreover Theorem 2.1. Let X ; L; A; B be as in ð2:0Þ and let S 0 P 2 . Then X ¼ PðVÞ, where V is an ample and spanned vector bundle of rank 2 on F1 with Chern classes c1 ðVÞ ¼ ½3l þ bf
, c2 ðVÞ ¼ 2b  4 for an integer b b 5, where l and f denote the ð1Þ-section and a fiber of F1 , and L is the tautological line bundle on X. Moreover A G F1 is a section of X, B ¼ PðVg Þ, where g is a smooth element of the linear system j2l þ 2fj, and the scroll projection p induces a non-splitting exact sequence 0 ! ½B
A ! pjA V ! LA ! 0; furthermore h ¼ A V B is isomorphic to g on A and is a section on B. Proof. By what we said before, S ¼ Fe for some e b 0. Moreover, in view of (1.0), we are in case i). Letting X ¼ PðVÞ, where V ¼ p L as before, now we can write det V ¼ ½al þ bf
for some suitable integers a and b. Moreover KX þ 2L ¼ p  H, where H ¼ KFe þ det V ¼ ½ða  2Þl þ ðb  2  eÞf
; is an ample line bundle, hence a > 2 and b > ða  1Þe þ 2; 2 ð2:3Þ in view of [Ha, p. 380]. Since ðA; LA Þ is either ðQ ; OQ 2 ð1ÞÞ or a scroll by (1.0), we can identify A with Fe via the isomorphism pjA and write (up to exchanging the factors in case e ¼ 0) LA ¼ ½l þ tf
for some integer t > e. Thus, in view of the identification h ¼ A V B ¼ A V p g ¼ g reducible hyperplane sections of threefolds 307 induced by pjA , we get ABL ¼ hLA ¼ gðl þ tfÞ: Moreover, since p  HAL ¼ ðpjA HÞLA , ð2:4Þ (2.1) gives Hðl þ tfÞ ¼ Hg: ð2:5Þ Now, as already observed, B ¼ p  g, where g H Fe is a smooth rational curve. Hence, by [Ha, p. 380] an easy check leads to the following possibilities: (1) g ¼ l, (2) g ¼ f, (3) g @ l þ mf, for some integer m b e, (4) e ¼ 0 and g @ ml þ f, for some integer m b 2, (5) e ¼ 1 and g @ 2l þ 2f. We proceed with a case-by-case analysis. Case (1). We have Hg ¼ Hl, but this contradicts (2.5). Case (2). We have Hg ¼ Hf. But (2.5) says that Hg ¼ Hf þ Hðl þ ðt  1ÞfÞ > Hf, a contradiction. Case (3). We have Hg ¼ ða  2Þðm  eÞ þ b  e  2. Moreover (2.5) immediately gives m ¼ t and then ABL ¼ 2m  e, by (2.4). By the condition gðB; LB Þ ¼ 0, recalling also (2.1) we thus get b ¼ ð4  aÞm þ ða  2Þe. Combining this with (2.3) we have ð4  aÞm > e þ 2 > 0 and so, recalling (2.3) we conclude that a ¼ 3. Hence b ¼ m þ e. Replacing these values into the expressions of H det V and Hg, (2.2) gives a numerical contradiction. Case (4). In this case we have H det V ¼ 2ðab  a  bÞ and Hg ¼ a  2 þ mðb  2Þ. So, noting that b > 2 by (2.3), (2.2) gives m ¼ a  1. But (2.5) in turn shows that t ¼ b  3. Thus Hg ¼ mt þ 2m  1. On the other hand (2.4) shows that ABL ¼ mt þ 1. So, recalling also (2.1), the condition gðB; LB Þ ¼ 0 leads to the equality m ¼ 0, which is a contradiction. Thus we are in Case (5). In particular, S ¼ F1 and H ¼ ½ða  2Þl þ ðb  3Þf
. We have H det V ¼ ða  2Þðb  aÞ þ aðb  3Þ and 2Hg ¼ 4ðb  3Þ. So (2.2) gives ða  2Þðb  aÞ ¼ ð4  aÞðb  3Þ. Recalling (2.3) we deduce that 2 a b  a a ð4  aÞðb  3Þ. Thus a ¼ 3 and b b 5. Moreover, we get LA ¼ ½l þ ðb  2Þf
, by (2.5). Now let p : F1 ! P 1 be the ruling projection. Note that ðV n ½2l
Þf ¼ Of l Of ð1Þ for any fiber f, since V is ample and det V ¼ ½3l þ bf]. So L ¼ p ðV n ½2l
Þ is an invertible sheaf on P 1 , and we can put L ¼ OP 1 ðdÞ for an integer d. From the injection p  L ! V n ½2l
we obtain an exact sequence 0 ! ½2l þ df
! V ! Q ! 0; ð2:6Þ where the quotient is the line bundle Q ¼ det V  ½2l þ df
¼ ½l þ ðb  dÞf
. Since V is ample, it follows that Q is ample, hence b  d > 1. Moreover c2 ðVÞ ¼ ð2l þ dfÞðl þ ðb  dÞfÞ ¼ 2b  d  2. Thus the Chern–Wu relation for the tautological line bundle L on X gives L 3 ¼ c1 ðVÞ 2  c2 ðVÞ ¼ ð3l þ bfÞ 2  ð2b  d  2Þ ¼ 4b þ d  7: ð2:7Þ 308 antonio lanteri and andrea luigi tironi On the other hand, since LA2 ¼ ðl þ ðb  2ÞfÞ 2 ¼ 2b  5 and LB2 ¼ L 2 p  g ¼ deg Vg ¼ g det V ¼ 2b, we get L 3 ¼ LA2 þ LB2 ¼ 4b  5. Comparing this with (2.7) we obtain d ¼ 2, hence c2 ðVÞ ¼ 2b  4 and the exact sequence induced by (2.6) on A via the isomorphism pjA becomes 0 ! ½B
A ! pjA V ! LA ! 0. Note that the sequence does not split; otherwise, by restricting (2.6) to l we would get Vl ¼ Ol l Ol ðb  3Þ, contradicting the ampleness of V. r The above result is e¤ective as the following example shows. Example. Let X be the Fano bundle obtained by blowing-up Y ¼ PðTP 2 Þ along a fiber [D, Theorem 3] (see also [SW, case 14 in the Theorem]). Let p : X ! F1 be the projection, and let b : F1 ! P 2 be the induced blowing-up of the basis of Y . With the same notation as above recall that l is the exceptional divisor of b and that b  OP 2 ð1Þ ¼ ½l þ f
. Let V :¼ b  TP 2 n ½f
. Since the exceptional divisor of the blowing-up X ! Y is p  l, we can write X ¼ Pðb  TP 2 n ½l
Þ ¼ PðVÞ: Let L be the tautological line bundle of V on X . formula KX ¼ 2L þ p  ðKF1 þ det VÞ gives Then the canonical bundle 2L ¼ KX þ p  ðl þ 2fÞ: Since KX is ample and p  ðl þ 2fÞ is spanned, this shows that L is ample. Moreover L is spanned, V being so. To see this note that b  TP 2 is very ample on F1 nl and spanned on l. Now, consider a smooth curve g A j2l þ 2fj, and let B ¼ p  g. Then B is a smooth surface isomorphic to Fe for some e b 0. Note that L  B is the tautological line bundle of V n ½g
¼ b  TP 2 n ½2l  f
¼ b  TP 2 ð1Þ n ½l
: Let x ¼ bðlÞ and let Jx H OP 2 be the corresponding ideal sheaf. From the Euler sequence we know that TP 2 ð1Þ is spanned and h 0 ðTP 2 ð1ÞÞ ¼ 3. Then h 0 ðTP 2 ð1Þ n Jx Þ ¼ 1. Moreover, since c2 ðTP 2 ð1ÞÞ ¼ 1, the unique surface S HY corresponding to the non-trivial elements in H 0 ðTP 2 ð1Þ n Jx Þ is a meromorphic section of Y containing exactly one fiber: namely that over x. It thus follows that h 0 ðL  BÞ ¼ h 0 ðb  TP 2 ð1Þ n ½l
Þ ¼ 1. Note that the unique element A A jL  Bj is just the proper transform of S in the blowing-up X ! Y . Therefore A is a smooth surface, which is a meromorphic section of p. Moreover, the number of fibers of p it contains is given by c2 ðV n ½g
Þ. But it is immediate to check that c2 ðV n ½g
Þ ¼ c2 ðb  TP 2 ð1Þ n ½l
Þ ¼ 0: Therefore A is a holomorphic section of p, i.e., pjA : A ! F1 is an isomorphism. On the other hand, when S ¼ P 2 , we get the following very precise description of ðX ; LÞ. reducible hyperplane sections of threefolds 309 Theorem 2.2. Let X ; L; A; B be as in (2.0) and assume that S ¼ P 2 . If i) holds, then ðX ; LÞ ¼ ðP 1  P 2 ; OP 1 P 2 ð1; 2ÞÞ, with, up to renaming, A A jOP 1 P 2 ð1; 0Þj and B a general element of jOP 1 P 2 ð0; 2Þj. If ii) holds, then X ¼ PðVÞ, where V is an ample and spanned vector bundle of rank 2 on P 2 with Chern classes c1 ðVÞ ¼ 4, c2 ðVÞ ¼ 5. Moreover, up to renaming, A G F1 , pjA : A ! P 2 being the contraction of the ð1Þ-section of A, and B ¼ PðVg Þ G P 1  P 1 , g being a smooth conic; furthermore h ¼ A V B is a section of B with h 2 ¼ 0 and on A it is a smooth element of the linear system j2E þ 2F j, where E and F denote the ð1Þ-section and a fiber, respectively. Proof. Set X ¼ PðVÞ with V :¼ p L, as at the beginning of this Section. Since S ¼ P 2 the situation specializes as follows: det V ¼ OP 2 ðaÞ with a b 4 since H ¼ KP 2 þ det V ¼ OP 2 ða  3Þ is an ample line bundle. Moreover g A jOP 2 ðuÞj with u ¼ 1 or 2, since g is a smooth rational curve. By (2.2) we know that Hðdet V  2½g
Þ ¼ 0. So, since H is ample, we get a ¼ 2u and recalling our conditions on a and u we conclude that 4 ¼ a ¼ 2u, i.e., c1 ðVÞ ¼ 4 and g is a smooth conic. In addition, H ¼ OP 2 ð1Þ, p  HLA ¼ p  HLB ¼ 2, by (2.1) and (2.2), and so for the smooth curve h ¼ A V B we get Lh ¼ LAB ¼ 4: ð2:8Þ Since B ¼ p  g we have B ¼ PðVg Þ. Thus, in view of the above, we can write Vg ¼ OP 1 ðe1 Þ l OP 1 ðe2 Þ where 1 a e1 a e2 ¼ 8  e1 . Hence ðB; LB Þ is a scroll over P 1 of invariant e ¼ 8  2e1 , whose degree is given by LB2 ¼ L 2 p  g ¼ deg Vg ¼ 8: ð2:9Þ Using this information together with (2.8) and the conditions in [Ha, p. 380], a straightforward verification shows that e ¼ 0, i.e., B G P 1 P 1 , h is an element of the ruling transverse to the projection pjB , and LB ¼ ½h þ 4f
, where f denotes a fiber of pjB . On the other hand, from the Chern–Wu relation for the tautological line bundle L on X we get L 3 ¼ c1 ðVÞ 2  c2 ðVÞ ¼ 16  c2 ðVÞ: ð2:10Þ 2 Note that ðP ; det VÞ is the first reduction of ðS; LS Þ, where S is a general element of jLj. Thus gðX ; LÞ ¼ gðS; LS Þ ¼ 3. This implies that c2 ðVÞ b 3, due to results of Lanteri–Sommese [LS] and Noma [N]. Recalling (2.9) and (2.10) we thus see that ðA; LA Þ has degree LA2 ¼ L 3  LB2 ¼ 8  c2 ðVÞ a 5: 2 ð2:11Þ First suppose we are in case i). Then ðA; LA Þ ¼ ðP ; OP 2 ðvÞÞ, where v ¼ 1 or 2 in view of (2.11). Moreover, since h corresponds to g in the isomorphism pjA : A ! P 2 , we have that h A jOP 2 ð2Þj; hence 4 ¼ LAB ¼ LA h ¼ 2v: 310 antonio lanteri and andrea luigi tironi Thus v ¼ 2, LA2 ¼ 4 and so (2.11) gives c2 ðVÞ ¼ 4. On the other hand, since A is a section of p there is a surjection from V to a line bundle on P 2 . This gives rise to an exact sequence 0 ! OP 2 ðxÞ ! V ! OP 2 ð yÞ ! 0; with x; y integers and y > 0 since V is ample. conclusions we get ð2:12Þ Combining this with the previous x þ y ¼ c1 ðVÞ ¼ 4 ¼ c2 ðVÞ ¼ xy; hence x ¼ y ¼ 2. On the other hand (2.12) splits, since the first cohomology group of any line bundle on P 2 is trivial; thus V ¼ ðOP 2 ð2ÞÞl2 and this concludes the proof in case i). Now suppose we are in case ii). Then, checking the list in (1.0) we see that, necessarily, A ¼ F1 and pjA is the contraction of the ð1Þ section of A. Let E and F be the ð1Þ-section and a fiber of A, respectively. Since E is a fiber of p, we have LA E ¼ LE ¼ 1. Writing LA ¼ ½xE þ yF
for some integers x; y, we thus get y ¼ x þ 1. Then LA2 ¼ x 2 þ 2x, with x b 1. Comparing this with (2.11) we see that x ¼ 1, hence LA ¼ ½E þ 2F
. Note that Eh ¼ 0; otherwise g ¼ pðhÞ would contain the point p ¼ pðEÞ, and then E would be in A V B, contradicting the irreducibility of h. On the other hand, LA h ¼ 4, by (2.8). So, writing h @ sE þ tF for some integers s; t, these conditions immediately imply that t ¼ s ¼ 2, i.e. h A j2E þ 2F j. r To show that the result of Theorem 2.2 is e¤ective, we produce an example as in case ii). Example. Let V H P 11 be a 4-dimensional classical scroll over P 2 of degree 11; let p : V ! P 2 be the projection, let L be the hyperplane line bundle, and let F ¼ p L be the corresponding very ample vector bundle of rank 3. Such a scroll exists, e.g., take F ¼ OP 2 ð2Þ l OP 2 ð1Þl2 . Note that ‘‘a priori’’ ðV ; LÞ could not be a scroll (in the adjunction theoretic sense). Actually, KV þ 3L ¼ p  H, where H ¼ KP 2 þ det F ¼ OP 2 ðaÞ with a b 0, because c1 ðFÞ b rk F ¼ 3, due to the (very) ampleness of F. However, by [F2, Main theorem], the only exception to the ampleness of KP 2 þ det F is F ¼ OP 2 ð1Þl3 . But, recalling the Chern–Wu relation, this would imply that 11 ¼ L 4 ¼ L 2 p  ðc1 ðFÞ 2  c2 ðFÞÞ ¼ 9  3 ¼ 6; a contradiction. Therefore the pair ðV ; LÞ is a scroll. Now take a general hyperplane section X A jLj of V and let L ¼ LX . Then the pair ðX ; LÞ is clearly a scroll. Remarks. a) Note that B G P 1  P 1 in case ii) of Theorem 2.2. However ¼ 8. So ðB; LB Þ 0 ðQ 2 ; OQ 2 ð1ÞÞ. The same conclusion holds for Theorem 2.1, since we showed there that LB2 ¼ 2b b 10. LB2 reducible hyperplane sections of threefolds 311 b) Note also that in case ii) of Theorem 2.2 we have ½A
A ¼ LA  ½B
A ¼ ½E
, hence A is not nef. The same is true for A in Theorem 2.1, since there ½A
A ¼ LA  ½B
A ¼ ½l þ ðb  4Þf
. 3 The case when both ðA; LA Þ; ðB; LB Þ A A Here we start the proof of Theorem (0.1). For the moment we do not need the assumption that A þ B A jLj is a seminef ssd. First we prove some useful lemmas. Lemma 3.1. Let L be an ample and spanned line bundle on a smooth projective threefold X. Assume that there exists a ssd A þ B A jLj, and let h ¼ A V B be the corresponding smooth hinge curve. Suppose furthermore that ðA; LA Þ G ðP 2 ; OP 2 ðuÞÞ, with u ¼ 1; 2 and B G P 2 or Fe for some e b 0. Then gðhÞ ¼ 0. Proof. Set ½A
A ¼ OP 2 ðdÞ and ½B
A ¼ OP 2 ðdÞ for some integers d; d. Since OP 2 ðuÞ ¼ LA ¼ OP 2 ðd þ dÞ, we get d þ d ¼ u a 2. Note that h A jBA j. So, assuming, by contradiction, that gðhÞ > 0, we get d b 3, which implies d < 0. So, if gðhÞ > 0 we have that dd < 0. On the other hand, looking at the smooth curve h inside B, we have 0 > dd ¼ ½A
A ½B
A ¼ A 2 B ¼ ½A
2B ¼ h 2 : Thus B 0 P 2 . On the other hand, the only irreducible curve on B G Fe with negative self-intersection is the section C0 . Therefore gðhÞ ¼ gðC0 Þ ¼ 0, a contradiction. r Lemma 3.2. Let L be an ample and spanned line bundle on a smooth projective threefold X. Assume that there exists a ssd A þ B A jLj and let h ¼ A V B be the corresponding smooth hinge curve. Suppose furthermore that ðA; LA Þ G ðQ 2 ; OQ 2 ð1ÞÞ and B G Fe for some e b 0. Then gðhÞ ¼ 0. Proof. Set ½B
A ¼ OP 1 P 1 ðx; yÞ for some integers x; y. Then ½A
A ¼ LA  ½B
A ¼ OP 1 P 1 ð1  x; 1  yÞ: Since h A jBA j is an irreducible curve, we see that ðx; yÞ is either ð1; 0Þ; ð0; 1Þ, or x; y are both positive integers. Moreover, if we assume, by contradiction, that gðhÞ > 0, then we get x b 2 and y b 2; otherwise h would be a section of one of the two rulings of P 1  P 1 . We have ½A
A ½B
A ¼ OP 1 P 1 ðx; yÞOP 1 P 1 ð1  x; 1  yÞ ¼ xð1  yÞ þ yð1  xÞ ¼ x þ y  2xy: Note that the function jðx; yÞ ¼ x þ y  2xy is always negative for x b 2, y b 2 (actually jðx; yÞ ¼ xð1  yÞ þ yð1  xÞ a xð1Þ þ yð1Þ a 4Þ. On the other hand, looking at the smooth curve h inside B G Fe , we thus see that 312 antonio lanteri and andrea luigi tironi 0 > jðx; yÞ ¼ ½A
A ½B
A ¼ A 2 B ¼ ½A
2B ¼ h 2 : But this leads to the same contradiction as in Lemma 3.1. r Now we can proceed with a case-by-case analysis of our ssd A þ B A jLj as in Theorem 0.1. (I) Assume that A G P 2 and B G P 2 . Put ½A
B ¼ OP 2 ðd 0 Þ and ½B
A ¼ OP 2 ðdÞ with d; d 0 b 1. By Lemma 3.1 we know that gðhÞ ¼ 0, hence from the equalities dðd  3Þ ¼ ðKA þ ½B
A Þ½B
A ¼ 2gðhÞ  2 ¼ ðKB þ ½A
B Þ½A
B ¼ d 0 ðd 0  3Þ we get d 2  3d þ 2 ¼ d 0 2  3d 0 þ 2 ¼ 0. LA ¼ OP 2 ðuÞ, with u ¼ 1; 2. Then Therefore d; d 0 A f1; 2g. Recall that d 02 ¼ ½A
2B ¼ ½A
A ½B
A ¼ ðLA  ½B
A Þ½B
A ¼ ðu  dÞd: This immediately shows that d 0 ¼ 2 is impossible, while d 0 ¼ 1 implies d ¼ 1, u ¼ 2. Then, letting LB ¼ OP 2 ðvÞ, from the symmetric relation 1 ¼ ½B
2A ¼ ½A
B ½B
B ¼ ½A
B ðLB  ½A
B Þ ¼ v  1; we get v ¼ 2. So ðA; LA Þ G ðB; LB Þ G ðP 2 ; OP 2 ð2ÞÞ. Moreover ABL ¼ ½B
A LA ¼ du ¼ 2 and by Lemma 1.1 we obtain that gðX ; LÞ ¼ 1. So ðX ; LÞ is a Del Pezzo threefold and since L 3 ¼ LA2 þ LB2 ¼ 8 we see from [F1, (8.5)] that ðX ; LÞ G ðP 3 ; OP 3 ð2ÞÞ. (II) Assume, up to renaming, that A G P 2 and B G Q 2 . Put ½B
A ¼ OP 2 ðdÞ with d b 1 and ½A
B ¼ OP 1 P 1 ðx; yÞ for some nonnegative integers x and y. Note that d 2 ¼ ½B
2A ¼ ½B
B ½A
B ¼ ðLB  ½A
B Þ½A
B ¼ x þ y  2xy and dðd  3Þ ¼ 2gðhÞ  2 ¼ ðKB þ ½A
B Þ½A
B ¼ 2xy  2x  2y: Hence 2xy  2x  2y ¼ dðd  3Þ ¼ x þ y  2xy  3d and this gives 3x þ 3y  4xy ¼ 3d b 3: ð3:2Þ Moreover we have that ABL ¼ du ¼ x þ y for u ¼ 1; 2. If u ¼ 1 then d ¼ x þ y and from (3.2) we get xy ¼ 0. Thus we can suppose without loss of generality that x ¼ 0. Moreover, since gðhÞ ¼ 0 by Lemma 3.1, we get y ¼ 1 and so d ¼ 1 by (3.2). Then ABL ¼ x þ y ¼ 1 and by Lemma 1.1 we get gðX ; LÞ ¼ 0. In view of our assumptions it thus follows that ðX ; LÞ is a scroll over P 1 [F1, (12.1) and (5.10)]. If u ¼ 2 then 2d ¼ x þ y and (3.2) gives 3d ¼ 4xy. Moreover, 3x þ 3y  4xy ¼ 2d þ d ¼ x þ y þ d reducible hyperplane sections of threefolds 313 1 a d ¼ 2x þ 2y  4xy: ð3:3Þ and then Since 3d ¼ 4xy 0 0 we get x b 1 and y b 1. If x ¼ 1 then 2  2y ¼ d b 1 and so y a 1=2, but this is a contradiction. In the same way we see that y 0 1, so we can suppose that x; y b 2. But then (3.3) would give 1 a d < 0, which is absurd. (III) Finally assume that both A and B are Q 2 . Put ½A
B ¼ OP 1 P 1 ðx; yÞ and ½B
A ¼ OP 1 P 1 ðz; wÞ for some nonnegative integers x; y; w and z. Since 2xy ¼ ½A
2B ¼ ðLA  ½B
A Þ½B
A ¼ z þ w  2zw and 2zw ¼ ½B
2A ¼ ðLB  ½A
B Þ½A
B ¼ x þ y  2xy; we deduce that x þ y ¼ 2ðxy þ zwÞ ¼ z þ w: ð3:4Þ By Lemma 3.2, the genus formula for h A jAB j gives 2 ¼ 2gðhÞ  2 ¼ ðKB þ ½A
B Þ½A
B ¼ 2xy  2x  2y; hence ðx  1Þð y  1Þ ¼ 0. Similarly, arguing on h A jBA j we get ðz  1Þðw  1Þ ¼ 0. Up to exchanging the factors of A and B we can thus assume that x ¼ z ¼ 1. But then (3.4) gives y ¼ w and 1 þ y ¼ 2ðy þ wÞ ¼ 4y, which is impossible, y being an integer. We sum up the above results in the following statement Theorem 3.3. Let L be an ample and spanned line bundle on a smooth projective threefold X. Assume that there is a ssd A þ B A jLj. If both ðA; LA Þ and ðB; LB Þ are in class A then ðX ; LÞ is either a scroll over P 1 or ðP 3 ; OP 3 ð2ÞÞ. Remark. Note that, at least in cases (I) and (II), we could deduce the result above also from [CHS, (3.10)], in view of Lemma 3.1, after a case-by-case analysis of the polarized threefolds appearing there (e.g., see the argument at the end of Section 5). However we preferred to provide a more direct proof. 4 The case when ðA; LA Þ A A and ðB; LB Þ A B We proceed with a case-by-case analysis also in this context, continuing the enumeration started in Section 3. (IV) Assume that ðA; LA Þ G ðP 2 ; OP 2 ðuÞÞ with u ¼ 1; 2. Put ½B
A ¼ OP 2 ðdÞ with d b 1. Let B G Fe for some e b 0 and let C0 and f be a section of minimal self-intersection and a fiber respectively. Since ðB; LB Þ is a scroll we can write LB ¼ ½C0 þ bf
for some integer b > e. Note that h A jAB j is a smooth curve of genus zero on Fe by Lemma 3.1. Hence by [Ha, p. 380] there are three possibilities: 314 antonio lanteri and andrea luigi tironi (1) ½A
B ¼ ½ f
; (2) ½A
B ¼ ½C0
; (3) ½A
B ¼ ½xC0 þ yf
with x > 0 and y > xe, or e > 0, x > 0 and y ¼ xe. Case (1). We observe that ½A
B LB ¼ ABL ¼ 1 and then gðX ; LÞ ¼ 0 by Lemma 1.1. After ruling out ðP 3 ; OP 3 ð1ÞÞ and ðQ 3 ; OQ 3 ð1ÞÞ, we deduce that ðX ; LÞ is a scroll over P 1 [F1, (12.1) and (5.10)], with ðA; LA Þ G ðP 2 ; OP 2 ð1ÞÞ. Case (2). Since gðhÞ ¼ 0, by the genus formula we obtain 2 ¼ 2gðhÞ  2 ¼ ðKA þ ½B
A Þ½B
A ¼ ðd  3Þd; 2 i.e., d  3d þ 2 ¼ 0. Thus d ¼ 1 or 2. Let d ¼ 1. Thus ½A
A ¼ LA  ½B
A ¼ OP 2 ðu  1Þ with u ¼ 1; 2. then we get ½A
A ¼ OP 2 ð1Þ and so If u ¼ 2 1 ¼ ½A
A ½B
A ¼ ½A
2B ¼ C02 ¼ e a 0; but this is a contradiction. If u ¼ 1 then ½A
A ¼ OP 2 and ½B
A LA ¼ ABL ¼ 1. So we get gðX ; LÞ ¼ 0 by Lemma 1.1, and again ðX ; LÞ is a scroll over P 1 . Now let d ¼ 2. Thus ½A
A ¼ OP 2 ðu  2Þ with u ¼ 1; 2. If u ¼ 2 then 0 ¼ ½A
A ½B
A ¼ ½A
2B ¼ e. Moreover 4 ¼ ½B
A LA ¼ ABL ¼ ½A
B LB ¼ C0 ðC0 þ bf Þ, which gives LB ¼ ½C0 þ 4f
. Therefore ½KX þ 2L
B f ¼ ðKB þ ½A
B þ LB Þ f ¼ 2f 2 ¼ 0: This shows that KX þ 2L cannot be ample. Note that ðX ; LÞ can be neither ðP 3 ; OP 3 ð1ÞÞ nor ðQ 3 ; OQ 3 ð1ÞÞ. Hence, from adjunction theory we know that ðX ; LÞ belongs to a very short list of pairs. If KX þ 2L is not nef, then by [CHS, (3.1)] ðX ; LÞ is (A) a scroll over a smooth curve, but this is absurd, because LA ¼ OP 2 ð2Þ, and this polarization on A is not compatible with Lemma 1.2. Now suppose that KX þ 2L is nef but not big. Then, by [CHS, (3.2)], ðX ; LÞ is one of the following pairs: (B) a Del Pezzo threefold; (C) a quadric fibration over a smooth curve; (D) a scroll over a smooth surface. Since ½B
A LA ¼ ABL ¼ 4; ð4:1Þ Lemma 1.1 shows that gðX ; LÞ ¼ 3. Clearly this rules out case (B). Case (C) cannot occur as well. Actually, let q : X ! C be the quadric fibration over a smooth curve C. Since A G P 2 , A cannot be a fiber of q and so qjA : A ! C is a surjection. But this is impossible since P 2 cannot fibre over a curve. In case (D) we conclude that ðX ; LÞ is a scroll over P 2 as in Theorem 2.2, case i), because A G P 2 . Assume now that KX þ 2L is nef and big. Then by [CHS, (3.5)] reducible hyperplane sections of threefolds 315 (E) there exists the first reduction morphism r : ðX ; LÞ ! ðX 0 ; L 0 Þ and the adjoint bundle KX 0 þ 2L 0 is ample. Since KX þ 2L is not ample, clearly r is not an isomorphism. So, by [CHS, (3.5), (3.6)] we get the following possibilities: (E1 ) one of the two components of A þ B A jLj is an exceptional divisor of r, or (E2 ) r contracts at least one exceptional divisor E and neither A nor B is one of them. But case (E) cannot occur. Indeed, in case (E1 ) we have that A ¼ E is an exceptional divisor of r; then ABL ¼ ½B
E LE ¼ OP 2 ð2ÞOP 2 ð1Þ ¼ 2, which contradicts (4.1). On the other hand, in case (E2 ) clearly A V E ¼ j while B V E is a rational curve on B, which is contracted by r. But this is impossible, since, as we have seen B G F 0 . Finally suppose that u ¼ 1. Then ½A
A ¼ OP 2 ð1Þ, ½B
A ¼ OP 2 ð2Þ and so we get 2 ¼ ½A
A ½B
A ¼ ½A
B2 ¼ e. Since 2 ¼ ABL ¼ ½A
B LB ¼ C0 ½C0 þ bf
¼ e þ b ¼ 2 þ b; we conclude that L 3 ¼ LA2 þ LB2 ¼ 1 þ ½C0 þ 4f
2 ¼ 7 and by Lemma 1.1 we obtain that gðX ; LÞ ¼ 1, i.e., ðX ; LÞ is a Del Pezzo threefold of degree 7. Therefore X is P 3 blown-up at a point p and L ¼ r  OP 3 ð2Þ  E, where r : X ! P 3 is the blowing-up and E ¼ r1 ð pÞ is the exceptional divisor [F1, (8.6)]. Case (3). Since ðu  dÞd ¼ ½A
A ½B
A ¼ ½A
2B ¼ ½xC0 þ yf
2 ¼ xð2y  exÞ > 0 and 1 a d, u a 2, we deduce that d ¼ 1, u ¼ 2 and x ¼ 1. This gives ABL ¼ ½A
2B þ ½B
2A ¼ 2 and by Lemma 1.1 we have that gðX ; LÞ ¼ 1, i.e., ðX ; LÞ is a Del Pezzo threefold. Furthermore, from x ¼ 1 we get ½B
2B ¼ ðLB  ½A
B Þ 2 ¼ ðb  yÞ 2 f 2 ¼ 0. So it follows that LB2 ¼ ðA 2 þ 2AB þ B 2 ÞB ¼ ½A
A ½B
A þ 2½B
2A ¼ 3 and then L 3 ¼ LA2 þ LB2 ¼ 4 þ 3 ¼ 7. This leads to the same conclusion as at the end of case (2). The discussion above proves the following Theorem 4.1. Let L be an ample and spanned line bundle on a smooth projective threefold X. Assume that there is a ssd A þ B A jLj. If ðA; LA Þ G ðP 2 ; OP 2 ðuÞÞ with u ¼ 1; 2 and ðB; LB Þ A B then ðX ; LÞ is one of the following pairs: 1. a scroll over P 1 ; 2. ðP 1  P 2 ; OP 1 P 2 ð1; 2ÞÞ, with A A jOP 1 P 2 ð1; 0Þj and B A jOP 1 P 2 ð0; 2Þj a general element; 3. ðX ; LÞ has ðP 3 ; OP 3 ð2ÞÞ as first reduction with X being P 3 blown-up at one point. 316 antonio lanteri and andrea luigi tironi (V) Suppose now that ðA; LA Þ G ðQ 2 ; OQ 2 ð1ÞÞ. By the genus formula and Lemma 3.2 we get 2 ¼ 2gðhÞ  2 ¼ ðKA þ ½B
A Þ½B
A ¼ ðLA þ ½A
A Þ½B
A ¼ ABL  ½A
2B : We claim that ½A
2B a 0. If not, then the equation above gives ABL < ABL þ ½A
2B ¼ 2, and then Lemma 1.1 would imply gðX ; LÞ ¼ 0. But, clearly, ðX ; LÞ can be neither ðP 3 ; OP 3 ð1ÞÞ, nor ðQ 3 ; OQ 3 ð1ÞÞ, because ðB; LB Þ A B. Moreover ðX ; LÞ cannot be a scroll over a smooth curve, since our ssd A þ B is not compatible with Lemma 1.2. This proves the claim. Note that it cannot be ½A
B ¼ ½ f
. Otherwise we would get 2 ¼ ABL ¼ ½B
2A ¼ ½B
B ½A
B ¼ ½B
B f ¼ ðLB  ½A
B Þ f ¼ LB f ¼ 1; a contradiction. If ½A
B 0 ½C0
, recalling that h A jAB j is a smooth curve, we can write ½A
B ¼ ½aC0 þ bf
, where a > 0, b > ae or e > 0, a > 0 and b ¼ ae [Ha, p. 380]. But in these cases we have ½A
2B ¼ að2b  aeÞ > 0, which contradicts the claim. All this shows that ½A
B ¼ ½C0
. On the other hand, since A G Q 2 ¼ P 1  P 1 , we can write ½B
A ¼ OP 1 P 1 ðx; yÞ for some integers x; y b 0. Recall that h A jBA j has genus zero by Lemma 3.2. So, up to exchanging the factors of A, we can suppose that x ¼ 1. Then we get e ¼ ½A
2B ¼ ½A
A ½B
A ¼ LA ½B
A  ½B
2A ¼ 1  y: Therefore ½B
A ¼ OP 1 P 1 ð1; e þ 1Þ and ½A
A ¼ OP 1 P 1 ð0; eÞ. thus get By adjunction we ½KX þ 2L
A ¼ KA þ LA þ ½B
A ¼ OP 1 P 1 ð0; eÞ and this shows that KX þ 2L is not ample. Arguing as in (IV) and noting that neither A nor B can be an exceptional divisor of the first reduction morphism, we conclude that ðX ; LÞ is one of the following pairs: (A) a scroll over a smooth curve; (B) a Del Pezzo threefold; (C) a quadric fibration over a smooth curve; (D) a scroll over a smooth surface. (E2 ) there exists the first reduction morphism r : ðX ; LÞ ! ðX 0 ; L 0 Þ and the adjoint bundle KX 0 þ 2L 0 is ample; r contracts at least one exceptional divisor E and neither A nor B is one of them. Case (A) cannot occur since the structure of A and B is not compatible with Lemma 1.2. To deal with case (B), let LB ¼ ½C0 þ bf
with b > e. Since gðhÞ ¼ 0, we have that b  e ¼ C0 ðC0 þ bf Þ ¼ ½A
B LB ¼ 2  ½A
2B ¼ e þ 2; i.e. b ¼ 2e þ 2. By Lemma 1.1 we know that ABL ¼ gðX ; LÞ þ 1 ¼ 2. Then e ¼ 0, b ¼ 2, i.e. ðB; LB Þ ¼ ðF 0 ; ½C0 þ 2f
Þ. Therefore L 3 ¼ LA2 þ LB2 ¼ 2 þ LB2 ¼ 6 and by [F1, (8.7)] we get ðX ; LÞ ¼ ðP 1  P 1  P 1 ; OP 1 P 1 P 1 ð1; 1; 1ÞÞ. reducible hyperplane sections of threefolds 317 In case (C), let q : X ! C be the quadric fibration morphism. Then by Lemma 1.4 we deduce that C G P 1 . Case (D) does not occur since the present polarization of A is not compatible with that of the component P 1  P 1 arising in the proofs of Theorems 2.1 and 2.2 (see Remark a) at the end of Section 2). Finally, case (E2 ) is not possible. Indeed, since LE ¼ ½A
E þ ½B
E ¼ OP 2 ð1Þ and A G P 1  P 1 , we deduce that ½A
E ¼ OP 2 and ½B
E ¼ OP 2 ð1Þ. Thus EB H B is a ð1Þ-curve and so this gives EB ¼ C0 , B G F1 and by arguing as in case (B), we see that LB ¼ ½C0 þ 4f
. Thus we get 3 ¼ C0 ðC0 þ 4f Þ ¼ EB LB ¼ EB 2 þ EBA ¼ BE2 ¼ 1; but this is absurd. We can sum up the above results in the following Theorem 4.2. Let L be an ample and spanned line bundle on an irreducible projective manifold X of dimension three. Assume that there is a ssd A þ B A jLj. If ðA; LA Þ G ðQ 2 ; OQ 2 ð1ÞÞ and ðB; LB Þ A B then ðX ; LÞ is one of the following pairs: 1. a quadric fibration over P 1 ; or 2. ðP 1  P 1  P 1 ; OP 1 P 1 P 1 ð1; 1; 1ÞÞ. 5 The case when both ðA; LA Þ; ðB; LB Þ A B When both components of the ssd A þ B A jLj are rational scrolls we need a further condition to control the genus of the smooth curve h ¼ A V B. This is provided by the notion of seminef decomposition which we recall for the convenience of the reader (see [T, §1]). Definition. Let ðX ; LÞ be a polarized manifold of dimension n b 3 and let A þ B A jLj be a ssd. We say that A þ B is a seminef divisor or shortly a seminef ssd, if at least one of A and B is a nef divisor or at worst an exceptional divisor of the first reduction mapping of ðX ; LÞ. Remark. Note that all ssd of sectional genera zero allowed for the pairs ðX ; LÞ listed in Theorems 2.2, 3.3, 4.1 and 4.2 are seminef. So these results remain unchanged under the extra assumption that A þ B A jLj is a seminef ssd. In particular this gives cases 1, 2, 5, 6 and 8 in Theorem 0.1. The next lemma shows the role of the seminefness assumption. Lemma 5.1. Let L be an ample and spanned line bundle on a smooth projective threefold X and let A þ B A jLj be a seminef ssd. If both ðA; LA Þ; ðB; LB Þ A B, then the corresponding hinge curve h ¼ A V B has genus gðhÞ ¼ 0. Note that in the present setting, neither A nor B can be an exceptional divisor of the first reduction morphism. Hence, due to the assumption, one of 318 antonio lanteri and andrea luigi tironi them, say A, is nef. Then the assertion above is an obvious corollary of the following more general result. Lemma 5.2. Let L be simply an ample line bundle on a smooth projective threefold X, and suppose that A þ B A jLj is a ssd. If A is a nef divisor and ðA; LA Þ is a scroll over a smooth curve of genus p, then the hinge curve h has genus gðhÞ ¼ 0 or p. Proof. Let C0 and f be a section of minimal self-intersection and a fiber of A respectively. Since h A jBA j, we can write ½B
A 1 ½xC0 þ yf
(numerical equivalence) for some integers x; y, where x ¼ hf b 0. On the other hand, since ðA; LA Þ is a scroll, we have that LA 1 ½C0 þ tf
for a suitable integer t > 0. Thus ½A
A ¼ LA  ½B
A 1 ½ð1  xÞC0 þ ðt  yÞ f
; where 1  x ¼ Af b 0, due to the nefness of A. Now, if gðhÞ a p, it can only be gðhÞ ¼ 0 or p, by the Riemann–Hurwitz theorem. So, let gðhÞ > p. Then x b 2, otherwise h would be either a fiber (if x ¼ 0) or a section of A (if x ¼ 1), which implies gðhÞ ¼ 0 or p, respectively. But this clearly contradicts the inequality x a 1 obtained before. r The fact that gðhÞ ¼ 0 under the assumption that A þ B A jLj is a seminef ssd with gðA; LA Þ ¼ gðB; LB Þ ¼ 0 also follows from [T, Lemma 6]. However, we preferred to present here a direct proof of this fact in line with that of Lemmas 3.1 and 3.2. To conclude the proof of Theorem 0.1, consider our seminef ssd A þ B A jLj. If KX þ 2L is not nef and big, we know that ðX ; LÞ belongs to a very precise list of pairs [S]. On the other hand, if KX þ 2L is nef and big, let r : X ! X 0 be the reduction morphism. Since ðA; LA Þ and ðB; LB Þ are in B, r can contract neither A nor B. So, one of them, say A, is a nef divisor and then we deduce from Lemma 5.1 that gðhÞ ¼ 0. Then [CHS, (3.10)] applies. Note that case 5 in [CHS, (3.10)] cannot occur. Thus we obtain that ðX ; LÞ is one of the following threefolds: (A) a scroll over a smooth curve; (B) a Del Pezzo threefold; (C) a quadric fibration over a smooth curve; (D) a scroll over a smooth surface; (E) KX þ 2L is nef and big and there exists the first reduction morphism r : ðX ; LÞ ! ðX 0 ; L 0 Þ; moreover, the adjoint bundle KX 0 þ 2L 0 is ample and neither A nor B is a fiber of r. Let A 0 ¼ rðAÞ and B 0 ¼ rðBÞ. The following cases can occur: (E1) ðX 0 ; L 0 Þ G ðQ 3 ; OQ 3 ð2ÞÞ. (E2) ðX 0 ; L 0 Þ G ðP 3 ; OP 3 ð3ÞÞ; (E3) X 0 is a P 2 -bundle over a smooth curve Y , u : X 0 ! Y with 2KX 0 þ 3L 0 ¼ u  H for an ample line bundle H on Y . Thus ðF ; LF0 Þ ¼ reducible hyperplane sections of threefolds 319 ðP 2 ; OP 2 ð2ÞÞ for a general fiber F . In the present setting there are two possibilities: (E3-i) after renaming, if necessary, A 0 is a fiber of u, Y ¼ P 1 , and B 0 meets a general fiber in a smooth conic; or (E3-ii) both A 0 and B 0 are P 1 -bundles over Y ¼ P 1 via u, each meeting a fiber of u in a line. Since ðA; LA Þ and ðB; LB Þ are in B, case (A) is not possible. Case (B) occurs: in fact, by combining [CHS, (3.4)] with Lemma 1.3 and the subsequent remark we see that ðX ; LÞ ¼ ðPðTP 2 Þ; xTP 2 Þ. Case (C) can also occur, and the base curve is P 1 by Lemma 1.4, since both A and B are P 1 -bundles over P 1 . Case (D) leads to the pairs in Theorem 2.1 and in case ii) of Theorem 2.2. We observe that cases (E1) and (E2) cannot occur since gðA; LA Þ ¼ gðB; LB Þ ¼ 0. Case (E3-i) is not possible. Since LA0 0 ¼ LF0 ¼ OP 2 ð2Þ we have A 0 B 0 L 0 ¼ ½B 0
F LF0 ¼ ðLF0 Þ 2 ¼ 4 and so by Lemma 1.1 we obtain that gðX 0 ; L 0 Þ ¼ 3. Moreover 2KX 0 þ 3L 0 ¼ u  H 1 tF for some integer t b 1 and then by the genus formula we get 8 ¼ 2ð2gðX 0 ; L 0 Þ  2Þ ¼ 2ðKX 0 þ 2L 0 ÞðL 0 Þ 2 ¼ ð2KX 0 þ 3L 0 þ L 0 ÞðL 0 Þ 2 ¼ tðLF0 Þ 2 þ ðL 0 Þ 3 ¼ 4t þ ðL 0 Þ 3 b 4t þ 1: Thus t ¼ 1 and ðL 0 Þ 3 ¼ 4. But this gives 4 ¼ ðL 0 Þ 3 ¼ ðLF0 Þ 2 þ ðLB0 0 Þ 2 ¼ 4 þ ðLB0 0 Þ 2 which is absurd since L 0 is ample. Finally, also case (E3-ii) is not possible. Since p ¼ ujA 0 : A 0 ! P 1 is a P 1 bundle and ðLA0 0 Þf ¼ OP 1 ð1Þ for every fiber f of p, we know that KA 0 þ 2LA0 0 ¼ p  H 0 1 tf for some integer t. As observed before, we can assume without loss of generality that A is nef, hence A 0 is nef. Therefore ðKX 0 þ 2L 0 ÞA 0 f ¼ ðKA 0 þ LA0 0 þ ½B 0
A 0 Þ f a ðKA 0 þ 2LA0 0 Þ f ¼ t½ f
2 ¼ 0; but this is a contradiction since we know that KX 0 þ 2L 0 is ample. Summing up the discussion above we get cases 2 again, 3, 4 and 7 of Theorem 0.1. Combining this with the remark before Lemma 5.1 concludes the proof of Theorem 0.1. References [BS] M. C. Beltrametti, A. J. Sommese, The Adjunction Theory of Complex Projective Varieties, Expositions in Mathematics, vol. 16, W. de Gruyter, Berlin–New York, 1995. [BCS] M. C. Beltrametti, K. A. Chandler, and A. J. Sommese, Reducible hyperplane sections, II, Kodai Math. J. 25 (2002), no. 2, 139–150. [CHS] K. A. Chandler, A. Howard and A. J. Sommese, Reducible hyperplane sections I, J. Math. Soc. Japan 51 (1999), 887–910. 320 antonio lanteri and andrea luigi tironi [D] I. V. Dëmin, Three dimensional Fano manifolds representable in the form of line bundles (Russian), Izv. Acad. Nauk SSSR, 44 (1980); English translation in Math. USSR Ivz. 17 (1981), 219–226. Addendum to this paper in Izv. Acad. Nauk SSSR, 46 (1982); English translation in Math. USSR Ivz. 20 (1983), 625–626. [F1] T. Fujita, Classification Theories of Polarized Varieties, London Math. Soc. Lecture Notes Series, vol. 155, Cambridge Univ. Press, Cambridge, 1990. [F2] T. Fujita, On adjoint bundles of ample vector bundles, in Complex Algebraic Varieties, Bayreuth 1990, Lecture Notes in Math. 1507 (1992), 105–112, Springer–Verlag, New York. [Ha] R. Hartshorne, Algebraic Geometry, Springer–Verlag, New York, 1978. [LS] A. Lanteri, A. J. Sommese, A vector bundle characterization of P n , Abh. Math. Sem. Univ. Hamburg 58 (1988), 89–96. [LT] A. Lanteri, A. L. Tironi, On reducible hyperplane sections of 4-folds, J. Math. Soc. Japan 53 (2001), 559–563. [N] A. Noma, Classification of rank-2 ample and spanned vector bundles on surfaces whose zero loci consist of general points, Trans. Amer. Math. Soc. 342 (1994), 867–894. [S] A. J. Sommese, On the adjunction theoretic structure of projective varieties, in Proceedings of the Complex Analysis and Algebraic Geometry Conference, Göttingen 1985, Lecture Notes in Math. 1194 (1986), 175–213, Springer–Verlag, New York. [SW] M. Szurek, J. A. Wiśniewski, Fano bundles of rank 2 on surfaces, Compositio Math. 76 (1990), 295–305. [T] A. L. Tironi, Reducible hyperplane sections of 4-folds: seminef decompositions with low sectional genera, Istituto Lombardo (Rend. Sc.), A 134 (2000), 47–58. Authors’ address: Dipartimento di Matematica ‘‘F. Enriques’’—Università Via C. Saldini, 50 I-20133 Milano, Italy lanteri@mat.unimi.it atironi@mat.unimi.it