Uniformity for integral points on surfaces, positivity of log cotangent sheaves, and hyperbolicity
We prove that the Lang-Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with positive log cotangent sheaf are uniformly bounded. This generalizes work of Abramovich and Abramovich-Matsuki. In addition, we show that (1) all subvarieties of a surface with positive log cotangent bundle are of log general type, and that (2) smooth quasi-projective surfaces with positive and globally generated log cotangent have finitely many integral points, generalizing a theorem of Moriwaki.
One of the most intriguing consequences of Lang’s Conjecture, proved in [CHM], is that the number of rational points on curves of genus over a number field is not only finite, but is also bounded by a constant that depends only on and (see Theorem 2.2). The original ideas of Caporaso, Harris, and Mazur, have since been extended and generalized leading to proofs that similar uniformity statements, conditional on the Lang Conjecture, hold in higher dimensions (see Section 2.1 for more details).
A natural question addressed in [Aell] is the following: does the Lang-Vojta Conjecture (Conjecture 2.3) imply similar uniformity statements for integral points? Abramovich showed this cannot hold unless one restricts the possible models used to define integral points. This led Abramovich to define the notion of stably integral points (see Definition 4.1) to prove uniformity results, conditional on the Lang-Vojta Conjecture, for integral points on elliptic curves, and together with Matzuki in [AM] for integral points on principally polarized abelian varieties (PPAVs). Roughly speaking, stably integral points are -integral points which remain integral after stable reduction.
The goal of this paper is to generalize uniformity statements for stably integral points to pairs of log general type. The first result we obtain is the generalization of the results of [Aell] to arbitrary stable pointed curves. Unless stated otherwise, will denote a number field, and a finite set of places of containing the Archimedean ones.
Theorem 1.1 (see Theorem 4.6).
Assume the Lang-Vojta Conjecture. If is a pointed stable curve over , then the set of stably -integral points on is uniformly bounded.
To obtain analogous results in higher dimensions one needs to tackle an extra problem: stable models do not exist due to the absence of a semistable reduction theorem over arbitrary Dedekind domains. As a result, there is no canonical choice of model for higher dimensional algebraic varieties.
Instead, using recent results on moduli of stable pairs (see Definition 3.11), the higher dimensional analogue of the moduli of stable pointed curves, we define “good” models which play the role of stable models for curves, and actually generalize them (see Section 5). This allows us to define a notion of moduli-stably-integral points (-integral points, see Definition LABEL:def:ms), integral points which are integral with respect to a fixed model of the moduli space.
The second result we obtain is that -integral points on families of stable pairs lie in a subscheme whose degree is uniformly bounded, which generalizes a theorem of Hassett [Hassett, Theorem 6.2].
Theorem 1.2 (see Corollary LABEL:cor:subscheme).
Assume the Lang-Vojta Conjecture. Suppose that is a stable family over a smooth variety with integral and openly log canonical (see Definition 3.7) general fiber over . For all , there exists a proper closed subscheme containing all -integral points of whose irreducible components have uniformly bounded degree.
There are two obstacles to obtaining uniformity. The first, which appears also in the case of rational points, is the presence of curves with non-positive Euler characteristic, which contain infinitely many integral points. One natural approach to circumvent this problem is by asking for positivity of the log cotangent sheaf. In particular, if a smooth proper variety has ample cotangent, then all subvarieties are of general type [Lazarsfeld, 6.3.28]. The extension of this property to the logarithmic setting is much more intricate. First, the log cotangent sheaf is never ample (see Proposition LABEL:rmk:neverample).
Instead, one can ask that the log cotangent sheaf is almost ample (see Definition LABEL:def:dbarample) – asking that the sheaf is as positive as possible. For surfaces we prove the following.
(see Proposition LABEL:prop:ampleness and Corollary LABEL:cor:slcample for slc extension). Let be a log canonical surface pair with almost ample log cotangent. Then all pairs with , such that is not contained in are of log general type.
From here, the standard way to conclude uniformity from Theorem 1.2, is to run an induction argument once you answer “yes” to the following question, thus overcoming the second obstacle:
Do -integral points satisfy the subvariety property – i.e. are -integral points for a pair lying on a pair with also -integral points for ?
The above question was answered affirmatively for abelian surfaces using Néron models [AM]. Without having an explicit model to work with in higher dimensions, we reduce the question to a sufficient geometric criterion. In general, this is quite subtle. However, assuming positivity of the log cotangent sheaf, we are able to verify the subvariety property, thus proving uniformity.
Theorem 1.4 (see Corollary LABEL:cor:unif_2).
Assume the Lang-Vojta Conjecture. Let be a log canonical stable surface pair with good model such that
is an effective -Cartier divisor with ample and
each fiber of has almost ample log cotangent (see Definition LABEL:def:dbarample),
then there exists a constant where is the volume of , such that the set of -integral points of has cardinality at most , i.e.
Remark 1.5 (see Remark LABEL:rmk:uniformity(2)).
Assuming the Lang-Vojta conjecture, our methods give a proof for uniformity under any assumption that guarantees that all subvarieties are of log general type. We argue that asking for almost ample log cotangent is the most natural from a geometric standpoint.
It is also natural to ask whether positivity on the log cotangent sheaf implies finiteness statements for integral points. Indeed, Moriwaki proved that projective varieties over a number field with ample and globally generated have finitely many points [moriwaki, Theorem E]. We prove the following generalization.
Theorem 1.6 (see Theorem LABEL:thm:moriwakimain).
Let be a smooth quasi-projective surface with log smooth compatification over a number field . If the log cotangent sheaf is globally generated and almost ample, then for any finite set of places the set of -integral points is finite.
Theorem 1.7 (see Theorem LABEL:thm:moriwakisub).
Let be a log smooth surface over . If the log cotangent sheaf is globally generated, then for any finite set of places , every irreducible component of is geometrically irreducible and isomorphic to a semi-abelian variety.
Obstacles to proving many of the above theorems for stem from, e.g. the presence of singular subvarieties, and subtleties in defining their log cotangent sheaves (see Remark LABEL:rmk:higherdimbir and Remark LABEL:rmk:higherdim).
In the process of proving uniformity, we stumbled upon the following example, which discusses hyperbolicity in families. In particular, positivity of the log cotangent sheaf on the normalization of every fiber is not enough to guarantee hyperbolicity is a closed condition.
(See Example LABEL:ex:counterexample) We show the existence of a stable family where is a curve, the generic fiber is a normal surface with almost ample log cotangent (and therefore hyperbolic), while the special fiber , although having almost ample log cotangent on the normalization, contains a curve in the non-normal locus which is not of log general type.
One main ingredient in this paper, following the ideas of [CHM], is a Fibered Power Theorem, proved in [fpt], which gives the analogue for pairs of the main Theorem of [Afpt].
Theorem 1.9 (see [fpt]).
There are three appendices. In Appendix LABEL:sec:stacks, we define the stack of stable pairs over . This is probably known, but we include it for lack of reference. In Appendix LABEL:app:sheaves we show there exists an almost ample log cotangent sheaf on the universal family of the moduli stack. Appendix LABEL:app gives an alternative definition of -integral points that does not depend on the choice of models of stacks.
We thank Brendan Hassett for suggesting this problem and Dan Abramovich for his constant support and suggestions. This paper has benefited from discussions with Jarod Alper, Dori Bejleri, Damian Brotbek, Ya Deng, Gabriele Di Cerbo, János Kollár, Sándor Kovács, Wenfei Liu, Yuchen Liu, Zsolt Patakfalvi, Fabien Pazuki, Sönke Rollenske, David Rydh, Karl Schwede, Jason Starr, and Bianca Viray. We thank Ariyan Javanpeykar for discussions leading to the notion of good models. We thank Max Lieblich for help verifying Example LABEL:ex:counterexample. Research of Turchet supported in part by funds from NSF grant DMS-1553459. Research of Ascher supported in part by funds from NSF grant DMS-1162367/11500528 and an NSF Postdoctoral fellowship.
In this paper, always denotes a number field.
2. Previous results
In this section we discuss previous results on uniformity of rational and integral points.
2.1. Uniformity for rational points
Faltings proved that for projective curves over of genus , the set is finite [Falt]. In higher dimensions there is a conjectural analogue:
Conjecture 2.1 (Bombieri-Lang (surfaces), Lang (), [Lang] and [Lang2]).
Let be an algebraic variety defined over . If is of general type, then the set is not Zariski-dense.
[CHM] showed that Conjecture 2.1 implies that in Faltings’ Theorem is not only finite, but is also uniformly bounded by a constant that does not depend on the curve .
[CHM] Let be a number field and an integer. Assume Lang’s Conjecture. Then there exists a number such that for any smooth curve defined over of genus the following holds:
Pacelli [Pacelli] (see also [Aquadratic]), proved that only depends on and . More recently, cases of Theorem 2.2 have been proven unconditionally ([krz], [stoll] and [paz]) depending on the Mordell-Weil rank of the Jacobian of the curve and for [paz], on an assumption related to the Height Conjecture of Lang-Silverman. It has also been shown that families of curves of high genus with a uniformly bounded number of rational points in each fiber exist [dnp].
Näive translations fail in higher dimensions as subvarieties can contain infinitely many rational points. However, one can expect that after removing such subvarieties the number of rational points is bounded. Hassett proved that for surfaces of general type this follows from Conjecture 2.1, and that the set of rational points on surfaces of general type lie in a subscheme of uniformly bounded degree [Hassett].
The idea behind both proofs is the following: consider a family whose general fibers are general type curves (resp. surfaces) over a base , the proof reduces to showing that the number of rational points in the fibers is uniformly bounded. If the total space of the family is itself a variety of general type, the Lang Conjecture trivially implies the uniformity statement. In general this is not the case, but if the family has maximal variation in moduli, then the dominant irreducible component of a high enough fibered power will be of general type. Conjecture 2.1 and an induction argument will then give uniformity. In general, it is always true that for big enough admits a dominant map to a variety of general type. From this, one can conclude the result in a similar fashion. This can be applied to a “global” family of curves to obtain the result of [CHM].
The algebro-geometric result alluded to above is known as a fibered power theorem and was shown for curves in [CHM], for surfaces [Hassett] and in general by Abramovich [Afpt]. The pairs analogue is Theorem 1.9 ([fpt]). In higher dimensions, similar uniformity statements hold conditionally on Lang’s Conjecture, and follow from the fibered power theorem under some additional hypotheses that take care of the presence of subvarieties that are not of general type ([AV]).
2.2. Uniformity of Integral Points
The analogue of Faltings’ Theorem for quasi-projective curves is Siegel’s Theorem– every affine curve of positive Euler characteristic possesses a finite number of -integral points. There is a conjectural generalization to higher dimensions, that extends Lang’s Conjecture 2.1 to the quasi-projective case:
Conjecture 2.3 (Lang-Vojta).
Let be a quasi-projective variety and let be a model over the -integers. If is openly of log general type, then is not Zariski dense.
A natural question, is whether Conjecture 2.3 implies a uniform bound on the set of -integral points for quasi-projective curves openly of log general type (see Definition 3.2). Abramovich ([Aell, 0.3]) gave a counterexample: he constructed an elliptic curve, where the number of -integral points in the complement of the origin grow arbitrarily when one suitably changes the model. However, imposing minimality conditions on the model leads to statements similar to [CHM]. In particular, if one considers stable models for the quasi-projective curves over a number field, then the cardinality of the set of -integral points of this model, called stably-integral points, is uniformly bounded, conditional on Conjecture 2.3. This was extended to PPAVs of [AM].
Both results rely on the existence of good models for elliptic curves and abelian varieties. While this can be extended to arbitrary stable curves, it is not clear how to define stable models in arbitrary dimensions outside of the abelian case. However, it was observed by Abramovich and Matzuki that stably integral points admit a nice moduli interpretation (see Section 5).
Unconditional results for uniformity of integral points in certain classes of curves, coming from Thue Equations, were proved in [LT], given some bound on the Mordell-Weil rank of the Jacobian.
As Vojta’s Conjecture implies Lang’s Conjecture, and thus a uniform version of the conjecture, one can ask if Vojta’s conjecture implies a uniformity statement for heights. This was shown by Ih for curves [ih], and was generalized to certain families of hyperbolic varieties [ari].
3. Preliminaries and Notations
The ring of -integers, i.e. the set will be denoted by .
Given an algebraic variety defined over , a model of over is a separated scheme together with a flat map of finite type such that the generic fiber is isomorphic to , i.e. .
Given a quasi-projective variety we will use the following definition for openly of log general type.
(see [fpt, Definition 1.3]) A quasi-projective variety is openly of log general type if there exists a desingularization and a projective compactification with a divisor of normal crossings, such that is big.
The above definition is independent of both the choice of desingularization and of the compactification. From both the viewpoint of birational geometry and of integral points on quasi projective varieties, it has become natural to consider pairs of a variety and a divisor.
A pair is the datum of a projective variety and a -divisor which is a linear combination of distinct prime divisors.
We will often say that a pair , with a projective variety and a normal-crossings divisor is openly of log general type if the quasi-projective variety is. In some applications we will require some conditions on the singularities of the pair.
A pair has log canonical singularities (or is lc) if is normal, is -Cartier, and there is a log resolution (see [kom, Notation 0.4(10)]) such that
where all the and the sum goes over all irreducible divisors on . The pair has canonical singularities if all .
A pair is openly canonical if has canonical singularities.
An lc pair is openly log canonical if it is openly canonical.
For a pair that is openly log canonical, being openly of log general type is equivalent to the condition that is big, in particular this can be checked without referring to a log-resolution of singularities as in Definition 3.2. We note that the notion of a model extends naturally to pairs:
Consider a pair with a -Cartier divisor over . A model for over , is a model of together with an (effective) -Cartier divisor whose restriction to the generic fiber is isomorphic to . In other words, a model for is the datum of a model of and a compatible model of .
Models of pairs can be used to define integral points with respect to a divisor.
Consider a pair , with a Cartier divisor, and a model over . An -integral point is a section such that the support of is contained in . An -integral point of a quasi-projective variety is an -integral point for the pair .
As mentioned in Section 2.2, in order to obtain uniformity results for integral points, it is necessary to restrict the possible models under consideration. We recall here some definitions that will be useful later. First we need a crucial definition.
A pair is semi-log canonical (slc) if is reduced and , the divisor is -Cartier and the following hold:
is Gorenstein in codimension one, and
if is the normalization, then the pair is log canonical, where denotes the preimage of and denotes the preimage of the double locus on .
Slc pairs with ample log canonical sheaf generalize stable pointed curves to higher dimension.
A pair is stable if the -Cartier line bundle is ample, and the pair is semi-log canonical. A stable family is a flat family over a normal variety such that
avoids the generic and codimension one singular points of every fiber,
is -Cartier, and
is a stable pair for all .
We end this subsection by introducing notation for fiber powers of families of stable pairs.
Given a stable family , denote by the fibered power of over , where is the (unique) irreducible component of the fiber power which dominates , and where is the -th projection.
The moduli space of stable pairs is constructed and proven to have projective coarse space in [kp]. It requires a choice of invariants , where is the dimension of the pairs, is their volume, and is a coefficient set satisfying the DCC condition.
4. Uniformity for log stable curves
Abramovich observed [Aell] that uniformity statements for integral points on curves cannot hold without any restrictions on the model (see [AM, 0.3] - for an example and discussion), and instead one should consider stable models. Such a choice provides good models (see section 5 for a more general framework that extends to higher dimension) which possess “positivity” properties preventing the appearance of non-hyperbolic components in the model. As a result, such positivity does not allow the number of integral points to grow arbitrarily. Abramovich’s notion of stably integral points for the complement of the origin in an elliptic curve can be easily generalized to any stable pointed curve as follows.
Let be a stable pointed curve defined over . A point is called a stably -integral point if there exists a finite extension and a stable model over such that is a -integral for . Equivalently .
Given Definition 4.1 one can ask whether the same uniformity results proved in [Aell] hold more generally for a pair openly of log general type. To prove this we introduce the following:
Let be a family of pointed stable curves over . Given a subset , denote by the fibered power of over . Then is -correlated if there exists an such that is contained in a proper closed subset of .
The importance of -correlated sets for uniformity questions is apparent in the following:
Let a family of projective irreducible curves and let be an -correlated subset of . There exists a nonempty open set and an such that for every , .
See [CHM, Lemma 1.1] , [Aell, Lemma 1], or [AM, Lemma 1.1.2]. ∎
In view of Lemma 4.3, in order to prove uniformity for stably integral points on curves openly of log general type, we need to prove that the set of stably integral points is -correlated. We start by stating the following lemma, which will be important throughout.
Let be a pair defined over and let be a dominant morphism. Then, given a proper model over , there exists and a model over such that extends to a map .
The extension property follows from spreading out techniques, noting that the extensions to models of , , and are compatible after possibly enlarging . ∎
Recall that a stable pair (Definition 3.11) in dimension one is a projective curve with at worst nodal singularities and a reduced divisor disjoint from the singular locus. Therefore we obtain:
Let be a family of stable pointed curves with smooth general fiber and let be the set of stably integral points of the family. Then the Lang-Vojta Conjecture implies that is -correlated for some large enough.
We apply Theorem 1.9 to obtain a positive dimensional pair openly of log general type and a dominant morphism which restricts to a regular map on the complement . Applying Conjecture 2.3 to implies that there exists a proper closed subset containing all the -integral points. By Lemma 4.4 there exists a proper closed subset containing all the -integral points. Thus . ∎
Using Noetherian induction on the base we can prove the following:
Assume the Lang-Vojta Conjecture. For all stable pointed curves defined over , the number of stably -integral points on is uniformly bounded.
Following [CHM], we apply Proposition 4.5 to a “global” family of stable pointed curves as follows: let be the genus of ; by assumption we may assume that is irreducible. The stability condition implies that is a positive integer. In particular, there exists , which does not depend on , such that each stable curve of genus and reduced divisor with and , can be embedded in using the linear system . The theory of Hilbert Schemes gives the existence of a family with sections defined over such that given any pair as before, there exists a -rational point such that . This family can be constructed by taking the closure of the locus of pluri-log canonical curves and its restriction to the universal family in the corresponding Hilbert scheme. By construction, the general fiber of is a smooth curve openly of log general type (in particular it is stable). Therefore, Proposition 4.5 applies, so the set of stably -integral points of is -correlated for large enough. By Lemma 4.3, this implies the existence of an open subset such that for every -rational , there exists a non-negative integer such that
Finally one applies Noetherian induction on the dimension of to obtain a similar bound for all fibers in the family . More explicitly one defines to be the union of all irreducible components of whose generic point is a smooth curve and considers the corresponding restricted family . Applying Lemma 4.3 to this new family gives the existence of on open set where the stably integral points of the fibers are uniformly bounded, possibly by a different constant . This inductively gives a chain of base schemes such that and therefore stabilizes after a finite number of steps. For each one has a uniform bound given by a constant outside an open subset . Taking to be the maximum of all shows that the -integral points of any fiber are at most . Since the family has been chosen to be global, it follows that such a bound holds for all stable curves, and thus this proves the theorem. ∎
5. Good models
In this section we extend the notion of stably integral points to higher dimensions. First we need to construct models that play the role of stable models, since the latter are not known to exist for . The main idea behind our construction is to fix models for the moduli stack of stable pairs and construct the models of the pairs as base changes of the models of the stacks, noting that models for stacks are completely analogous to those of varieties. The starting point is the following observation of Abramovich-Matsuki that gives a nice moduli interpretation of stably integral points.
Proposition 5.1 ([Am], Proposition 3.1.3).
Let be a PPAV defined over and let . Consider the associated moduli map Then is stably -integral if and only if is an -integral point in .
This implies that one way to characterize stably -integral points is to look at their image in an appropriate moduli space and test their integrality with respect to a model of such a moduli stack.
Contrary to the moduli space of PPAVs, the moduli space of stable pairs has not been defined over . We remedy this by fixing a model over a Dedekind domain and show that our results are independent of the choice of such a model.
5.1. Construction of good models
By Appendix LABEL:sec:stacks the moduli stack of stable pairs can be defined over . We now make a choice of models of such stacks as follows: choose a ring of integers of a number field and models of and over such that in the following diagram