Abstract
Let 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 . Then the toroidal compactification is a projective smooth surface with elliptic compactification divisor . In this short note we discover a new class of unramifed ball quotients . We consider ball quotients with kod and . We prove that each minimal surface with finite Mordell-Weil group in the class described admits an étale covering which is a pull-back of . Here denotes the elliptic modular surface parametrizing elliptic curves with 6-torsion points which generate [6].
1. Introduction
Let the symbol denote the class of complex projective smooth surfaces which contain pairwise disjoint elliptic curves such that admits the open unit ball as universal holomorphic covering. As explained in [1], forms the “generic” class of compactified ball-quotient surfaces. There are several motivations to study surfaces in without assuming that the fundamental group with its Poincaré action on is an arithmetic lattice of ; we refer to [2] or to the introduction of [1]. Since the discovery of blown-up abelian surfaces in by Hirzebruch and Holzapfel some years ago (cf. [3]) there have been no further examples of surfaces of special type in . In this short note we present new examples of modular surfaces of Kodaira dimension .
In what follows we only consider complex projective smooth surfaces. An elliptic surface is an elliptic fibration of a surface over a smooth curve . If is an elliptic fibration, then a smooth fiber of is an elliptic curve and hence isomorphic to for a in the upper half plane. The -invariant of an elliptic surface is the unique morphism from to the projective line which maps a smooth fiber of to . An elliptic surface with finite Mordell-Weil group of sections is called extremal if the rank of the Néron-Severi group of equals . Particular examples of elliptic surfaces arise in the following way. To each pair of positive integers there exists a modular surface in the sense of Shioda [4] which is extremal, has no multiple fibers and nonconstant -invariant. The fibration has the following properties. (i)The Mordell-Weil group is isomorphic to . (ii) is the (compactified) curve where is the group (iii)The curve parametrizes triples of elliptic curves and points such that .(iv)There are sections , of order , respectively, generating the Mordell-Weil group, such that a point corresponds to the triple (v)All singular fibers of are of type in Kodaira's notation and they lie over the cusps of . A representant of in is stabilized by a matrix which is an -conjugate of
By [5] each extremal elliptic surface with nonconstant -invariant, no multiple fibers and Mordell-Weil group isomorphic to , where is as above, allows a cartesian diagram of finite maps
(1.5)
With this perspective we formulate our main result. A complex projective smooth surface is irregular if . An irregular surface of Kodaira dimension admits an up to isomorphism unique elliptic fibration to a curve of genus . If , then this elliptic fibration coincides with the Albanese morphism. Finally, we remark that and that is an elliptic curve.
Theorem 1.1. Let be an irregular minimal surface with , elliptic fibration and empty or finite Mordell-Weil group. (1)The surface is in if and only if the following holds. The curve is elliptic and there exists an isogeny of elliptic curves such that is isomorphic to a pull-back arising from an isogeny with the property that . (2)Assume that is a surface in isomorphic to a pull-back . Then the compactification divisor of consists of the 36 sections of ; each section has self-intersection number ; the fibration admits singular fibers of type , and each component of an intersects in precisely 6 points; one has and is extremal.
2. Some Basic Properties of Surfaces in
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 with attention to sign conventions, except the assertion on semistability. The latter assertion follows from [7]. A reduced effective divisor on a surface 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 are divisors on , we say that is ample modulo if and if the intersection number is positive for each curve on not supported in .
Theorem 2.1 (see [6, 7]). Let be a smooth projective surface and a divisor with normal crossings. Suppose that is big and ample modulo . Then with equality if and only if is an unramified ball quotient and is semistable.
There is a canonical exact sequence where is the Poincaré residue map. With this one proves that and . Therefore, If is a neat normal subgroup with finite index in , then is compactified by a smooth elliptic divisor, and is compactified by a divisor . As is the quotient , where , it is a normal curve. Hence, is smooth and consists of elliptic curves, for rational curves cannot appear because of semistability. Thus, if equality holds in the theorem, then is smooth. Moreover, one can show that is big and ample modulo .
Lemma 2.2. Let be in with compactification divisor and consider an irreducible curve . If is smooth rational, then . And if is a smooth elliptic curve, then .
Proof. If the statement is wrong for some rational curve , then the induced holomorphic map of universal coverings yields a contradiction to Liouville's theorem.
3. Proof of the Theorem
We begin by proving (1) and suppose that is an irregular minimal surface of Kodaira dimension in . Then is an elliptic surface in a unique way. We denote by the elliptic fibration and assume that its Mordell-Weil group is empty or finite. As above, is the compactification divisor of . Moreover, we let be the numerical class of a fiber of . Since is ample modulo , it follows that has positive intersection with . Thus, a component of dominates . The theorem of Hurwitz implies that is an elliptic curve and . Moreover, after transition to an etale cover and performing a base change , we achieve that every is a section as soon as it dominates [1, Lemma 3.3]. We will assume for the time being that ; once we have shown that (1) is true if , it will be easy to obtain (1) in the general case. If , then the curves are all sections, because they are disjoint. In this case we have the following.
Lemma 3.1. The identities and hold.
Proof. The canonical bundle formula implies that with a Weil divisor . Moreover, . The theorem of Riemann-Roch yields . It results from the adjunction formula that Hence, . Furthermore, by Noether's formula. So, Theorem 2.1 yields the remaining identities.
We consider the Mordell-Weil group . By assumption, is finite and equals . It follows from the previous lemma that . We next prove the following lemma of general interest.
Lemma 3.2. Let be a minimal elliptic surface over an elliptic curve . Assume that and that each rational curve intersects at least one section of . Suppose moreover that . Then the following assertions hold. (1) is a torsion group isomorphic to . (2)All singular fibers of are semistable of type and each rational curve intersects 6 distinct sections of .(3) has singular fibers. (4).
Proof. Statement (1) follows directly from [8, equation (4.8)]. If is a torsion group, then its sections do not intersect each other. So, their sum is a smooth divisor . Moreover, [8, Lemma 1.1] implies that all singular fibers are of type for some . If is the nontrivial isotropy group of a node then and are cyclic by [8, Lemma 2.2]. Because of (1) we thus have for all isotropy groups . Let be the neutral section. By the proof of [8, Lemma 2.2], consists of those sections which intersect the prime component containing . However, since by assumption we may take any section to be the neutral element of , we have for each component . As , we get . This yields (2). Recalling that , we find for the number of singular fibers of
We receive (3). Finally, according to [8, Proposition 1.6] the last equality holds if and only if , so that is extremal. An easy calculation shows now (4).
As explained in Section 1, is isomorphic to a pull-back . This shows (1) in the theorem if . Next we withdraw the additional assumption that from the beginning of the proof and let be an isogeny of minimal degree such that is a pull-back . We are left to show that and . The former estimate is clear, because over a curve there lie ≤36 curves . The latter equality holds, because by the lemma below. This yields (1) in the general case. Statement (2) in the theorem results from Lemma 3.2, the fact that is étale and the following lemma.
Lemma 3.3. The modular surface has invariant .
Proof. This is a consequence of the formulae in [9, page 77f].