Table of Contents
ISRN Geometry
VolumeΒ 2011, Article IDΒ 214853, 5 pages
http://dx.doi.org/10.5402/2011/214853
Research Article

On Modular Ball-Quotient Surfaces of Kodaira Dimension One

Departement Mathematik, ETH Zürich, HG J65, RÀmistraße 101, 8092 Zürich, Switzerland

Received 12 April 2011; Accepted 30 April 2011

Academic Editors: D.Β Franco and E. H.Β Saidi

Copyright Β© 2011 Aleksander Momot. 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.

Abstract

Let Ξ“βŠ‚ππ”(2,1) be a lattice which is not co-ompact, of finite covolume with respect to the Bergman metric and acting freely on the open unit ball πβŠ‚β„‚2. Then the toroidal compactification 𝑋=Ξ“\𝐁 is a projective smooth surface with elliptic compactification divisor 𝐷=𝑋\(Ξ“\𝐁). In this short note we discover a new class of unramifed ball quotients 𝑋. We consider ball quotients 𝑋 with kod(𝑋)=1 and β„Ž1(𝑋,π’ͺ𝑋)=1. We prove that each minimal surface with finite Mordell-Weil group in the class described admits an Γ©tale covering which is a pull-back of 𝑋6(6). Here 𝑋6(6) denotes the elliptic modular surface parametrizing elliptic curves 𝐸 with 6-torsion points π‘₯,𝑦 which generate 𝐸[6].

1. Introduction

Let the symbol 𝒯 denote the class of complex projective smooth surfaces 𝑋 which contain pairwise disjoint elliptic curves 𝐷1,…,π·β„Žπ‘‹ such that β‹ƒπ·π‘ˆ=𝑋⧡𝑖 admits the open unit ball πβŠ‚β„‚2 as universal holomorphic covering. As explained in [1], 𝒯 forms the β€œgeneric” class of compactified ball-quotient surfaces. There are several motivations to study surfaces in 𝒯 without assuming that the fundamental group πœ‹1(π‘ˆ,βˆ—) with its PoincarΓ© action on 𝐁 is an arithmetic lattice of 𝐏𝐔(2,1); we refer to [2] or to the introduction of [1]. Since the discovery of blown-up abelian surfaces in 𝒯 by Hirzebruch and Holzapfel some years ago (cf. [3]) there have been no further examples of surfaces of special type in 𝒯. In this short note we present new examples of modular surfaces π‘‹βˆˆπ’― of Kodaira dimension kod(𝑋)=1.

In what follows we only consider complex projective smooth surfaces. An elliptic surface is an elliptic fibration πœ‹βˆΆπ‘‹β†’πΆ of a surface 𝑋 over a smooth curve 𝐢. If πœ‹ is an elliptic fibration, then a smooth fiber 𝐹 of πœ‹ is an elliptic curve and hence isomorphic to β„‚/(β„€+πœβ„€) for a 𝜏 in the upper half plane. The 𝑗-invariant of an elliptic surface πœ‹βˆΆπ‘‹β†’πΆ is the unique morphism from 𝑋 to the projective line which maps a smooth fiber 𝐹 of πœ‹ to 𝑗(𝐹)=𝑗(𝜏). An elliptic surface with finite Mordell-Weil group MW(𝑋) of sections is called extremal if the rank 𝜌(𝑋) of the NΓ©ron-Severi group of 𝑋 equals β„Ž1,1(𝑋). Particular examples of elliptic surfaces arise in the following way. To each pair of positive integers (π‘š,𝑛)βˆ‰{(1,1),(1,2),(2,2),(1,3),(1,4),(2,4)}(1.1) there exists a modular surface πœ‹π‘›(π‘š)βˆΆπ‘‹π‘›(π‘š)→𝐢𝑛(π‘š) in the sense of Shioda [4] which is extremal, has no multiple fibers and nonconstant 𝑗-invariant. The fibration πœ‹π‘›(π‘š) has the following properties. (i)The Mordell-Weil group MW(𝑋𝑛(π‘š)) is isomorphic to β„€/π‘šβ„€Γ—β„€/𝑛℀. (ii)𝐢𝑛(π‘š) is the (compactified) curve Γ𝑛(π‘š)⧡ℍ where Γ𝑛(π‘š)βŠ‚π’π₯2(β„€) is the group ξƒͺ;ξƒͺ≑ξƒͺξƒ°ξƒ―ξƒ©π‘Žπ‘π‘π‘‘π‘Žπ‘π‘π‘‘1βˆ—01modπ‘š,𝑏≑0mod𝑛.(1.2)(iii)The curve 𝐢𝑛(π‘š) parametrizes triples (𝐸,π‘₯,𝑦) of elliptic curves 𝐸 and points π‘₯∈𝐸[π‘š],π‘¦βˆˆπΈ[𝑛] such that |β„€π‘₯+℀𝑦|=π‘šπ‘›.(iv)There are sections 𝜎1, 𝜎2 of order π‘š, respectively, 𝑛 generating the Mordell-Weil group, such that a point π‘βˆˆΞ“π‘›(π‘š)⧡ℍ corresponds to the triple 𝐸=fiberover𝑐,π‘₯=𝐸∩𝜎1,𝑦=𝐸∩𝜎2.(1.3)(v)All singular fibers of πœ‹π‘›(π‘š) are of type πΌπ‘˜ in Kodaira's notation and they lie over the cusps of π‘βˆˆπΆπ‘›(π‘š). A representant of 𝑐 in β„šβˆͺ{∞} is stabilized by a matrix π›ΎβˆˆΞ“ which is an 𝐒π₯2(β„€)-conjugate of ξƒͺ1π‘˜01.(1.4)

By [5] each extremal elliptic surface πœ‹βˆΆπ‘‹β†’πΆ with nonconstant 𝑗-invariant, no multiple fibers and Mordell-Weil group MW(𝑋) isomorphic to β„€/π‘šβ„€Γ—β„€/𝑛℀, where (π‘š,𝑛) is as above, allows a cartesian diagram of finite maps214853.eq.001(1.5)

With this perspective we formulate our main result. A complex projective smooth surface 𝑋 is irregular if β„Ž1(𝑋,π’ͺ𝑋)>0. An irregular surface of Kodaira dimension kod(𝑋)=1 admits an up to isomorphism unique elliptic fibration to a curve of genus β„Ž1(𝑋,π’ͺ𝑋). If β„Ž1(𝑋,π’ͺ𝑋)=1, then this elliptic fibration coincides with the Albanese morphism. Finally, we remark that β„Ž1(𝑋6(6),π’ͺ𝑋6(6))=1 and that 𝐢6(6) is an elliptic curve.

Theorem 1.1. Let 𝑋 be an irregular minimal surface with kod(𝑋)=1, elliptic fibration πœ‹βˆΆπ‘‹β†’πΆ and empty or finite Mordell-Weil group. (1)The surface 𝑋 is in 𝒯 if and only if the following holds. The curve 𝐢 is elliptic and there exists an isogeny ξ‚πœˆβˆΆπΆβ†’πΆ of elliptic curves such that 𝑋=π‘‹Γ—πœˆξ‚πΆ is isomorphic to a pull-back 𝑋6(6)Γ—πœ‡ξ‚πΆ arising from an isogeny ξ‚πœ‡βˆΆπΆβ†’πΆ6(6) with the property that deg𝜈=6degπœ‡/πœ’(𝑋)≀36. (2)Assume that 𝑋 is a surface in 𝒯 isomorphic to a pull-back 𝑋6(6)Γ—πœ‡ξ‚πΆ. Then the compactification divisor 𝐷 of 𝑋 consists of the 36 sections of πœ‹; each section has self-intersection number βˆ’πœ’(𝑋)=βˆ’6(degπœ‡); the fibration πœ‹ admits 12(degπœ‡) singular fibers of type 𝐼6, and each component of an 𝐼6 intersects 𝐷 in precisely 6 points; one has 𝜌(𝑋)=60(degπœ‡)+2 and 𝑋 is extremal.

2. Some Basic Properties of Surfaces in 𝒯

We cite two results on ball-quotient surfaces which will be needed for the proof of the theorem. The first result is essentially [6, Theorem  3.1] restricted to dim𝑋=2 with attention to sign conventions, except the assertion on semistability. The latter assertion follows from [7]. A reduced effective divisor on a surface 𝑋 is called semistable if it has normal crossings and if every rational smooth prime component intersects the remaining components in more than one point. If 𝑇,𝐷 are divisors on 𝑋, we say that 𝑇 is ample modulo 𝐷 if 𝑇2>0 and if the intersection number 𝐢𝑇 is positive for each curve 𝐢 on 𝑋 not supported in 𝐷.

Theorem 2.1 (see [6, 7]). Let 𝑋 be a smooth projective surface and π·βŠ‚π‘‹ a divisor with normal crossings. Suppose that 𝐾𝑋+𝐷 is big and ample modulo 𝐷. Then 𝑐21ξ€·Ξ©1𝑋(log𝐷)≀3𝑐2ξ€·Ξ©1𝑋(log𝐷),(2.1) with equality if and only if 𝑋⧡𝐷 is an unramified ball quotient Γ⧡𝐁 and 𝐷 is semistable.

There is a canonical exact sequence 0⟢Ω1π‘‹βŸΆΞ©1𝑋(log𝐷)resβˆ’βˆ’β†’π’ͺ𝐷⟢0,(2.2) where res is the PoincarΓ© residue map. With this one proves that 𝑐1(Ξ©1𝑋(log𝐷))=[𝐷]βˆ’π‘1(𝑋)∈𝐻2(𝑋,β„‚) and 𝑐2(Ξ©1𝑋(log𝐷))=𝑐2(𝑋)βˆ’(𝑐1(𝑋),[𝐷])+([𝐷],[𝐷])∈𝐻4(𝑋,β„‚). Therefore, 𝑐21ξ€·Ξ©1𝑋=𝐾(log𝐷)𝑋+𝐷2.(2.3) If Ξ“ξ…žβŠ‚Ξ“ is a neat normal subgroup with finite index in Ξ“, then Ξ“ξ…žβ§΅π is compactified by a smooth elliptic divisor, and Γ⧡𝐁 is compactified by a divisor 𝐷. As 𝐷 is the quotient π·ξ…ž/𝐺, where 𝐺=Ξ“/Ξ“ξ…ž, it is a normal curve. Hence, 𝐷 is smooth and consists of elliptic curves, for rational curves cannot appear because of semistability. Thus, if equality holds in the theorem, then 𝐷 is smooth. Moreover, one can show that 𝐾𝑋+𝐷 is big and ample modulo 𝐷.

Lemma 2.2. Let 𝑋 be in 𝒯 with compactification divisor 𝐷 and consider an irreducible curve πΏβŠ‚π‘‹. If 𝐿 is smooth rational, then |𝐿∩𝐷|β‰₯3. And if 𝐿 is a smooth elliptic curve, then |𝐿∩𝐷|β‰₯1.

Proof. If the statement is wrong for some rational curve 𝐿, then the induced holomorphic map of universal coverings ξ„Ÿξ„ŸπΏβ§΅π·β†’π=𝑋⧡𝐷 yields a contradiction to Liouville's theorem.

3. Proof of the Theorem

We begin by proving (1) and suppose that 𝑋 is an irregular minimal surface of Kodaira dimension kod(𝑋)=1 in 𝒯. Then 𝑋 is an elliptic surface in a unique way. We denote by πœ‹βˆΆπ‘‹β†’πΆ the elliptic fibration and assume that its Mordell-Weil group is empty or finite. As above, 𝐷 is the compactification divisor of 𝑋. Moreover, we let 𝐹 be the numerical class of a fiber of πœ‹. Since 𝐾𝑋+𝐷 is ample modulo 𝐷, it follows that 𝐹 has positive intersection with 𝐷. Thus, a component of 𝐷 dominates 𝐢. The theorem of Hurwitz implies that 𝐢 is an elliptic curve and β„Ž1(𝑋,π’ͺ𝑋)=1. Moreover, after transition to an etale cover ξ‚πœˆβˆΆπΆβ†’πΆ and performing a base change 𝑋=π‘‹Γ—πœˆξ‚πΆ, we achieve that every 𝐷𝑖 is a section as soon as it dominates 𝐢 [1, Lemma  3.3]. We will assume for the time being that 𝑋𝑋=; once we have shown that (1) is true if 𝑋𝑋=, it will be easy to obtain (1) in the general case. If 𝑋𝑋=, then the curves 𝐷𝑖 are all sections, because they are disjoint. In this case we have the following.

Lemma 3.1. The identities 36πœ’(𝑋)=π·πΉβ‹…πœ’(𝑋)=βˆ’π·2 and 𝐷𝐹=36 hold.

Proof. The canonical bundle formula implies that 𝐾𝑋=πœ‹βˆ—(𝔠) with a Weil divisor π” βˆˆDiv(𝐢). Moreover, β„Ž0(𝑋,π‘šπΎπ‘‹)=β„Ž0(𝐢,π‘šπ” ). The theorem of Riemann-Roch yields β„Ž0(𝑋,𝐾𝑋)=deg𝔠>0. It results from the adjunction formula that 𝐷2𝑖=βˆ’deg𝔠=βˆ’β„Ž0𝑋,𝐾𝑋=βˆ’πœ’(𝑋).(3.1) Hence, βˆ’π·2βˆ‘π·=βˆ’2𝑖=π·πΉπœ’(𝑋). Furthermore, 12πœ’(𝑋)=𝑐2(𝑋) by Noether's formula. So, Theorem 2.1 yields the remaining identities.

We consider the Mordell-Weil group MW(𝑋)=MW(𝑋). By assumption, MW(𝑋) is finite and equals MWtor(𝑋). It follows from the previous lemma that |MWtor(𝑋)|β‰₯𝐷𝐹=36. We next prove the following lemma of general interest.

Lemma 3.2. Let πœ‹βˆΆπ‘‹β†’πΆ be a minimal elliptic surface over an elliptic curve 𝐢. Assume that kod(𝑋)β‰₯1 and that each rational curve πΏβŠ‚π‘‹ intersects at least one section of πœ‹. Suppose moreover that |MWtor(𝑋)|β‰₯33. Then the following assertions hold. (1)MW(𝑋) is a torsion group isomorphic to β„€/6β„€Γ—β„€/6β„€. (2)All singular fibers of πœ‹ are semistable of type 𝐼6 and each rational curve πΏβŠ‚πΌπ‘› intersects 6 distinct sections of πœ‹.(3)𝑋 has 2πœ’(𝑋) singular fibers. (4)𝜌(𝑋)=β„Ž1,1(𝑋)=10πœ’(𝑋)+2.

Proof. Statement (1) follows directly from [8, equation  (4.8)]. If MW(𝑋) is a torsion group, then its sections do not intersect each other. So, their sum is a smooth divisor 𝐷. Moreover, [8, Lemma  1.1] implies that all singular fibers are of type 𝐼𝑛 for some 𝑛>0. If π»π‘›βŠ‚π‘€(𝑋) is the nontrivial isotropy group of a node π‘₯βˆˆπΌπ‘› then 𝐻𝑛 and MWtor(𝑋)/𝐻𝑛 are cyclic by [8, Lemma  2.2]. Because of (1) we thus have |𝐻𝑛|=6 for all isotropy groups 𝐻𝑛. Let π‘†βˆˆMW(𝑋) be the neutral section. By the proof of [8, Lemma  2.2], 𝐻𝑛 consists of those sections which intersect the prime component πΏβŠ‚πΌπ‘› containing π‘†βˆ©πΌπ‘›. However, since by assumption we may take any section to be the neutral element of MW(𝑋), we have 𝐿𝐷=6 for each component πΏβŠ‚πΌπ‘›. As 𝐷𝐼𝑛=36, we get 𝑛=6. This yields (2). Recalling that βˆ‘πΌπ‘›π‘›=𝑐2(𝑋), we find for the number 𝑑 of singular fibers of πœ‹π‘‘=2πœ’(𝑋)=2𝑔(𝐢)βˆ’2+rankMW(𝑋)+2πœ’(𝑋).(3.2)
We receive (3). Finally, according to [8, Proposition  1.6] the last equality holds if and only if 𝜌(𝑋)=β„Ž1,1(𝑋), so that 𝑋 is extremal. An easy calculation shows now (4).

As explained in Section 1, 𝑋 is isomorphic to a pull-back 𝑋6(6)Γ—πœ‡πΆ. This shows (1) in the theorem if 𝑋𝑋=. Next we withdraw the additional assumption that 𝑋𝑋= from the beginning of the proof and let ξ‚πœˆβˆΆπΆβ†’πΆ be an isogeny of minimal degree such that 𝑋 is a pull-back 𝑋6(6)Γ—πœ‡πΆ. We are left to show that degπœˆβ‰€36 and πœ’(𝑋)β‹…deg𝜈=6degπœ‡. The former estimate is clear, because over a curve π·π‘–βŠ‚π‘‹ there lie ≀36 curves ξ‚π·π‘–βŠ‚ξ‚π‘‹. The latter equality holds, because πœ’(𝑋)=6degπœ‡/deg𝜈 by the lemma below. This yields (1) in the general case. Statement (2) in the theorem results from Lemma 3.2, the fact that πœ‡ is Γ©tale and the following lemma.

Lemma 3.3. The modular surface 𝑋6(6) has invariant πœ’(𝑋6(6))=6.

Proof. This is a consequence of the formulae in [9, page 77f].

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