Strong Approximation over Function Fields
By studying -curves on varieties, we propose a geometric approach to strong approximation problem over function fields of complex curves. We prove that strong approximation holds for smooth, low degree affine complete intersections with the boundary smooth at infinity.
Key words and phrases:strong approximation, affine varieties, -curve, stable log map, complete intersection
2010 Mathematics Subject Classification:Primary 14G05, 14G25, Secondary 14M10
cm \newdir ¿*!/-5pt/@¿
Given a variety over a number field, the existence of rational points (integral points) and their distributions (weak approximation, strong approximation) are extensively studied by number theorists. In general, these problems are very difficult and lacking of complete solutions.
The analogue between number fields and function fields of Riemann surfaces suggests that one may start with geometric function field case first. A great progress has been made along this direction on the arithmetics of projective varieties with lots of rational curves during the past fifteen years. Let be a function field of a Riemann surface, and consider a projective variety defined over . If the geometric fiber is rationally connected, then admits rational points [GHS03, dJS03]. When the geometric fiber is rationally simply connected, weak approximation holds for , see [dJS06, Has10] for the definitions and results. Furthermore, it is expected that weak approximation holds for rationally connected varieties [KMM92, HT06]. As the first example, all the geometric conditions above hold for projective spaces.
While the results above focus on the projective case, number theorists study arithmetics of open varieties such as linear algebraic groups and affine hypersurfaces as well. This paper is an attempt to build a parallel theory of integral points on open varieties over from the logarithmic geometry point of view.
Since affine spaces satisfies strong approximation [Ros02, Theorem 6.13], it is natural to search geometric substitute for affine spaces that strong approximation holds. As the non-proper generalization of rationally connected varieties, -connected varieties have been studied in our previous works [CZ14b, CZ14a, CZ15, Zhu15]. On one hand side, this program is influenced by Iitaka’s philosphy and the works of Keel-McKernan [KM99] by studying the birational geometry of pairs. On the other hand side, we replace the non-proper smooth varieties by their logarithmically smooth (equivalently toroidal) compactifications, and view -curves as a morphism of logarithmic schemes. The theory of stable log maps developed in [Che14, AC14, GS13] provides a frame work for the study of -curves.
In this paper, we introduce the notion -simple-connectedness using the stable log map compactification. This is a key to our approach for the strong approximation conjecture over . Our proposal is parallel to the approach of Hassett [Has10] and de Jong-Starr [dJS06] for weak approximation of rationally simply connected varieties.
Let be a function field of a smooth, irreducible complex algebraic curve. Let be a log smooth projective variety over . Assume the following hold:
satisfies weak approximation over .
There exists a curve class and a geometrically irreducible component of the moduli space of two pointed -curves defined over such that
a general point of parametrizes a smoothly embeded -curve.
is dominant with rationally connected geometric generic fiber.
Then strong approximation holds for the interior over .
We refer to Section 1.1 for the notations and terminologies of the above theorem. The formulation of strong approximation is defined in Section 2. If we call Condition (2) above -simple-connectedness (with respect to the curve class ), the theorem above states that strong approximation holds for -simply-connected -varieties if weak approximation holds. Furthermore, -simple-connectedness is a geometric condition, and only depends on the interior.
Affine spaces are the first class of examples of -simply-connected varieties because any pair of points can be joined by a unique affine line.
Affine spaces are -simply-connected. Thus strong approximation holds for affine spaces over .∎
By studying the geometry of -conics on complete intersection, we give a bound for low degree smooth complete intersection pairs to be -simply connected.
Let be a smooth complete intersection pair of type in . Assume that is not the affine space. Then the general fiber of the evaluation morphism defined in (1.1.1)
is a smooth complete intersection in of type
In particular, the general fiber is rationally connected if .
We refer to Section 1.1 for the notations and terminologies of the above proposition. Combining this result with the works of [Has10, dJS06] on weak approximation of low degree complete intersection in projective spaces, we conclude that
Strong approximation holds for the interior of any smooth complete intersection pair of type in with
Strong approximation holds for the universal cover of the interior of any smooth complete intersection pair of type in with
It is known that there exists rational curves on a smooth projective rationally connected variety through any finite number of points. But the analogy for -connected varieties are wildly open. Our results on strong approximation provides an interesting class of examples from this point of view.
Let be a smooth complete intersection pair of type in with Then there exists an -curve passing through any -tuple of points on .
In Section 2, we will state the geometric version of strong approximation, and proof Theorem 1.1. In Section 3 and 4, we analyze the moduli space of -lines and -conics, and conclude the proof of Theorem 1.4, and Corollary 1.5 and 1.6 in Section 5.
1.1. Notations and terminologies
Capital letters such as , , , and , ect. are reserved for log schemes with the corresponding underlying schemes denoted by , , , and . For any log scheme , denote by the open locus with the trivial log structure.
An -map is a genus zero stable log map with precisely one marked point with a non-trivial contact order. An -curve is an -map with an irreducible source curve, whose image has non-trivial intersection with the open locus of the target with the trivial log structure. We call an -curve an -line or an -conic if the curve class of the -curve is the line or the conic curve class respectively.
For any log scheme , any curve class on , and an positive integer , denote by the moduli stack of -maps to with curve class , and markings with the trivial contact order. Then is a log stack with the canonical log structure. Denote by the underlying stack obtained by removing the log structure of . We have the evaluation morphism induced by the -markings with the trivial contact order
where the right hand side is -copies of .
Let be the moduli space of -pointed, genus zero stable maps to with curve class .
Let be a smooth complete intersection in () of type with . Let be a smooth hypersurface section of degree . We call the pair a smooth complete intersection pair of type . We denote by the log scheme associated to the pair . Denote by Denote by the line class on .
The authors would like to thank Xuanyu Pan for explaining his thesis work [Pan13].
2. Strong Approximation
2.1. The arithmetic formulation
We first recall the adelic formulation of strong (weak) approximation over function fields of curves, see [Has10].
Let a smooth irreducible projective curve over with function field . For each place , denote by the completion of at . Let be a nonempty finite set of places of , the ring of -integers. Denote by the ring of adeles over all places outside , where the product is the restricted product, i.e. all but finite number of factors are in . The ring has two natural topologies: the first one is the product topology, and the second one is the adelic topology, with a basis of open sets given by where for all but finitely many .
Let be a geometrically integral algebraic variety over . Denote by be the set of -rational points, and be the restricted product of . Thus, the adelic points admits the product topology and adelic topology, i.e., locally inheriting from adelic affine spaces.
We say that strong approximation (respectively, weak approximation) holds for away from if the inclusion
is dense in the adelic topology (respectively, product topology). To be more precise, this is equivalent to for any finite set of places containing , any integral model over of , and any open set under the adelic topology for each place , the image of via the diagonal map in
is not empty. We say that strong approximation holds for if strong approximation holds for away from any nonempty .
The above definition does not depend on the choice of model. Strong approximation implies weak approximation. The converse also holds when is proper over .
2.2. The geometric formulation
The geometric setting of weak approximation has been formulated and studied in [HT06]. We next translate Definition 2.1 into the geometric setting. To apply logarithmic geometry, we would like to replace the open variety by a proper log smooth variety with the log trivial part .
Let be a smooth, proper, and log smooth variety over . Denote by its log trivial open subset. A proper model of is a family of log schemes:
is a smooth projective curve with the trivial log structure;
is proper flat over ;
the generic fiber of is .
We say such model is regular if is a smooth variety. This can always be achieved via resolution of singularities.
Let be the log trivial open subset of a proper, smooth, log smooth variety defined over . Then strong approximation holds for away from is equivalent to the following statement:
Given any proper regular model of as in Definition 2.2, any finite set of places such that is smooth and log smooth over , any smooth points in for can be realized by a section of which is integral (i.e., away from the boundary) over .
2.3. Proof of Theorem 1.1
Step 1 To prove the theorem, it suffices to verify the statement in Proposition 2.3. Let denote the open subset over which has geometrically irreducible, and rationally connected fibers whose general points parametrizing -curves.
is integral over ;
the associated rational point, still denoted by , lies in .
Step 3 Since weak approximation holds over , we may choose a general section
the associated rational point, still denoted by , lies in .
Step 4 The fiber is a geometrically irreducible rationally connected variety defined over whose general points parametrize -curves. By [GHS03, KMM92], there exists a rational point on parametrizing a smooth embedded -curve. This rational point gives a generic -ruled surface in , denoted by . By construction, the surface contains:
the section integral over , and
the section . In particular, is smooth at for all .
Since strong approximation holds for away from [Ros02, Theorem 6.13], we can find a section with for all and integral over .∎
3. -lines through a general point
3.1. A deformation result
Let be a projective log smooth variety. For any curve class and a subscheme with either a closed point or the empty set, there are finitely many sub-varieties of such that if is an -curve with curve class through , and , then is free. In particular, an -curve through and a general point of with curve class is free.
where is the evaluation morphism induced by the marking with the trivial contact order.
Let be the irreducible component of with the universal morphism . Let
where is an open and dense subset such that is smooth over , and all closures are taken in .
Consider an -curve of curve class with for any . Let be the component containing . By construction, the universal morphism is dominant, and intersects . Same argument as in [Kol96, Chapter II 3.10] implies that is free. ∎
Notations and assumptions as in Proposition 3.1, any -curve passing through and a very general point of is free.
This follows from Proposition 3.1 by taking into account all choices of curve classes. ∎
3.2. -lines on smooth complete intersection pairs
Consider the smooth complete intersection pair as in Notation 1.7. In this subsection, we study the evaluation morphism
A general fiber of is smooth and projective.
Every nonempty connected component of a general fiber is of expected dimension .
The first statement follows from Proposition 3.1. Since every -map with line class in a general fiber is free, the dimension is calculated by the Euler characteristic of the pullback of the log tangent bundle.
Next we would like to describe the general fiber of explicitly in equations. Fix a general point . Let be the fiber over of the evaluation morphism:
We consider the restriction of the boundary evaluation morphism on :
If , the morphism is a closed immersion, and the image of is an irreducible, smooth complete intersection in of type
Let be any affine scheme. For any scheme , we denote by .
The scheme is determined by its -points:
Assume for simplicity . Consider a -point . A -line joining and can be expressed as
where is the parameter of the line and for each .
Let be the defining equation of for with . Restricting them on the line equation of , we have
where is a homogeneous polynomial of degree . The condition implies that . The condition is equivalent to the vanishing of for each , which gives a complete intersection of type
Similarly, let be the defining equation of . Restricting them on the line equation of , we have:
where is a homogeneous polynomial of degree . The point lying outside implies that . Note that is indeed . The condition being an -line is equivalent to the vanishing of the polynomials , i.e., a complete intersection of type
Now we define a complete interesection in defined by the equations:
Since , is automatically a closed subscheme of .
To summarize, we proved the following.
The image of the morphism
lies in .
Proof of Proposition 3.4.
It suffices to prove that is isomorphic to under , i.e., every -point of is the image of a unique -point of under . This follows from the fact that any -point of gives a -family of lines via the projection:
where the target is the Hilbert scheme of lines through . Furthermore, such family of lines meet the boundary exactly once, hence is a family of -lines. ∎
A general fiber of is a nonempty, irreducible, and smooth complete intersection if is log Fano, or equivalently, .
4. Moduli of -conics through general two points
For the rest of this section, we work with the following assumption.
Let be a smooth complete intersection pair in of type with for each .
The goal of this section is to study general fibers of the two-pointed evaluation morphism
given by the two marked points with the trivial contact order. The proof of Theorem 1.3 will be concluded at the end of this section.
For later use, denote by the fiber of (4.0.1), and the corresponding log scheme with the minimal log structure pulled back from . When there is no confusion of the pair of points , we will simply write and , and omit the subscripts.
4.1. Smoothness of the moduli
We first observe the following:
The line through a general pair of points in is not contained in .∎
For a general pair of points , the boundary divisor parameterizes maps with the following properties:
The underlying curve consists of three irreducible genus zero components for , with precisely two nodes joining and for .
Each component contains a marking with the trivial contact order for .
has three special points given by the contact marking and two nodes.
contracts the component to a point .
The restriction is an embedding of two free -lines for .
The characteristic sheaf is a locally constant sheaf with fiber .
By Lemma 4.2, the boundary parameterizes stable log maps with property (1) – (4). Statement (5) of follows from Proposition 3.1 and the general choice of . Property (6) and (7) follow from the definition of minimality, see [Che14, Construction 3.3.3], [AC14, Section 4], and [GS13, Construction 1.16]. ∎
For a general pair of points , the fiber is a log smooth scheme with the smooth boundary divisor .
Let be the open sub-stack parameterizing free -maps with the curve class . Lemma 4.3(7) implies that . Furthermore, the morphism is smooth along the boundary divisor parametrizing reducible conics by Lemma 4.3(6). By generic smoothness, we conclude that is smooth with a smooth boundary divisor . In particular, the pair is log smooth. ∎
4.2. A lifting property in the transversal case
We pause here to study the lifting of a special type of usual stable maps to stable log maps. Here is a slightly general result that fits our need.
Let be a log smooth variety over with a smooth boundary divisor . Consider a family of genus zero usual stable maps with two markings and over an arbitrary base scheme such that
The family is obtained by gluing two families of smooth rational curves and along the markings and .
Each has two markings and for .
The restriction is a family of -curves over intersecting transversally along for .
Then there exists up to a unique isomorphism, a unique family of genus zero minimal stable log maps such that
The underlying scheme of is .
The family of stable log maps has one contact marking , and two other markings with the trivial contact order.
The family of usual stable maps obtained by removing log structures on , forgetting the contact marking , and then stabilizing, is .
We divide the proof into the following two lemmas. We first prove the local existence.
Our construction here is similar to the case of [CZ15, Proposition 2.2] but for a family of maps. We take a family of smooth rational curves with three markings , , and . Such a family is necessarily trivial, and we thus have . We have a family of nodal rational curves
obtained by gluing with via the identification of the markings
Now the underlying stable map over lifts uniquely to the underlying stable map
over by contracting the component .
Consider the projectivized normal bundle with two boundary divisors and corresponding to the normal bundles and respectively. Here is the normal bundle of in . Consider the expansion obtained by gluing and via the identification . We next want to lift to a stable map such that
the composition is compatible with .
is a family of relative stable map tagent to only along with multiplicity , and intersecting transversally only along and .
Replace by an étale cover, we may assume that the pull-back along is a trivial family of rational curves over . Note also that the morphism needed factors through with the corresponding tangency along and . Since is also a trivial family, to construct , it suffices to select a meromorphic section on with two simple zeros along and , and a pole of order along . Such a meromorphic section clearly exists. This yields the usual stable map with the desired properties.
Finally, by [Kim09], see also the construction of [CZ15, Proposition 2.2], since the usual stable map intersects the boundary transversally along , it lifts to a unique log stable map over a log scheme in the sense of [Kim09]. Here is the log scheme with the underlying structure , and the canonical log structure as in [Ols03], and has underlying structure . Since there is a natural projection of log schemes , composition this projection with we obtain the stable log map as needed. ∎
The uniqueness in Proposition 4.5 holds.
It suffices to show the uniqueness locally. Shrinking , we may again assume base scheme is affine. It sufficies to verify that the lift constructed in Lemma 4.6 is unique.
Assume that we have two different liftings and . We first notice that except the freeness in (5), all other statements in Lemma 4.3 applies to both and . In particular, the two liftings and have the same underlying stable map (4.2.1) over constructed in the proof of Lemma 4.6.
We first compare the two stable log maps over the contracted component . Shrinking , we may assume that is generated by a global section . By Lemma 4.3(6) and further shrinking , we may assume that is generated by a global section for . By choosing the generators appropriately, we may assume over we have
where is a meromorphic function on with only first order poles along and , and second order zero along .
We now focus on the node of for . Let and be the canonical log structure on and associated to the family . Shrinking again, we assume is generated by global sections and corresponding to smoothing nodes and respectively. By Lemma 4.3(7), the log curves and has log structure near node defined by pulling the standard log structure along the correspondences respectively:
where . Here we identify with its image in .
Since is an embedding of a family of -curves, the two sections and are identified with the image of , where is the image of . This in particular means that we have a canonical identification
We thus conclude that the morphism of log structures induced by the correspondence induces an isomorphism of the two log curves and . In view of (4.2.2), this further induces an isomorphism of the two stable log maps . Such isomorphism is canonical from the discussion above. ∎
4.3. Forgetful morphism to moduli of usual stable maps
Now consider the moduli space of usual stable maps with two markings . Consider the -evaluation morphism
induced by the two markings. Given a pair of points , denote by the fiber of (4.3.1) over . When there is no danger of confusion, we will write instead of . Denote by the locus parameterizing maps with reducible domain curves. Recall that
[dJS06, Lemma 5.1] For a general choice of , the scheme is smooth with a smooth divisor .
We then consider the forgetful morphism
obtained by sending a stable log map to its underlying stable map, forgetting the marking with non-trivial contact order, then stabilizing. This induces a forgetful morphism of the fibers
Fixing a general choice of , the forgetful morphism is an embedding of a closed sub-scheme. Furthermore, it induces an embedding of closed sub-scheme .
Note that if a usual stable map intersects the boundary of at a single smooth point of the source curve, then it lifts to an -map in a unique way. Thus, the morphism is an embedding. It remains to consider around the locus of stable log maps with reducible domain curves.
By Assumption (4.1) and Proposition 4.5, the forgetful morphism is injective on the level of closed points. Since by our assumptions both and are smooth, it remains to verify the injectivity of tangent vectors.
We fixed a minimal stable log map over a geometric point . It suffices to consider the case that is reducible, and denote by the image of in .
Consider the morphism of fibers . Recall that for any smooth variety , the tangent bundle can be identified with . Now that injectivity of tangent vectors follows from applying the uniqueness of Proposition 4.5 to families over . ∎
4.4. Pull-back of the boundary divisor
Fix a general choice of . The log smooth scheme has its log structure given by the boundary divisor .
Since the locus of with reducible domain curves form the smooth divisor , to show that the log structure of is same as the log structure given by the smooth divisor , it suffices to verify is log smooth. Thus, it suffices to verify is log smooth, where is the Artin stack of genus zero pre-stable curves with two markings equipped with the canonical log structure of curves. Since the morphism is strict, the log smoothness is equivalent to the smoothness of the underlying maps . This follows from that parameterizes free usual stable maps. ∎
Fix a general choice of . There is a canonical morphism of log schemes
compatible with in (4.3.2). Furthermore, .
Now consider a family of minimal stable log maps corresponding to an -point of . Let be the underlying stable map over , and be the image of in . Denote by and the family of log curves over and with the canonical log structure. We first notice that there is a canonical commutative diagram of log schemes
To see this, we may shrink , and put auxiliary markings on the non-contracted component of such that