On the geometry of thin exceptional sets in Manin’s conjecture
Abstract.
Manin’s Conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set. We study whether the exceptional set in Manin’s Conjecture is a thin set using the minimal model program and boundedness of log Fano varieties.
1. Introduction
Let be a smooth projective variety defined over a number field and let be a big and nef adelically metrized line bundle on with associated height function . For any subset , define the counting function
Manin’s Conjecture predicts that the asymptotic behavior of the counting function as increases is controlled by certain geometric invariants of . Let denote the cone of pseudoeffective divisors. Define
and
the codimension of the minimal  
Recall that a thin subset of is a finite union , where each is a morphism that is generically finite onto its image and admits no rational section. The following version of Manin’s Conjecture was first suggested by Peyre in [Peyre03, Section 8] and explicitly stated in [BL16, Conjecture 1.4] and [LeRudulier, Conjecture 1.1]:
Manin’s Conjecture.
Let be a smooth projective variety over a number field with ample anticanonical class . Let be a big and nef adelically metrized line bundle on . There exists a thin set such that one has
(1.1) 
where is Peyre’s constant as in [Peyre] and [BT].
See [Peyre16] for a different version of Manin’s conjecture using the notion of freeness. Previous statements of the conjecture ([BM], [Peyre], [BT]) gave a “closed set” version: instead of removing a thin set , they removed only the points lying on a closed subset . It turns out that the closed set version of Manin’s Conjecture is false. All known counterexamples to the closed set version of Manin’s Conjecture for arise from two kinds of geometric incompatibilities:

a Zariski dense set of subvarieties such that
in the lexicographic order (as in [BTcubic]), or

a generically finite dominant morphism such that
in the lexicographic order (as in [LeRudulier]).
In either case, the expected growth rate of points of bounded height on is larger than that on and the pushforward of these points can form a dense set on .
The following conjecture predicts that such geometric incompatibilities can not obstruct the thin set version of Manin’s Conjecture.
Conjecture 1.1.
Let be a smooth uniruled variety over a number field and let be a big and nef divisor on . Consider morphisms where is a smooth projective variety and is generically finite onto its image. As we vary over all such morphisms such that either is not big or
in the lexicographic order, the points
are contained in a thin subset of .
Remark 1.2.
We also address the case when
However, the image of points on such varieties need not lie in a thin set. In fact, to obtain the correct Peyre’s constant one may need to allow contributions of such points. We give an example of this behavior in Example LABEL:exam:_Sano.
We make partial progress towards Conjecture 1.1, building on [BT], [HTT15], [LTT14], [HJ16]. Our approach can be broken down into two steps.

First, we prove a boundedness statement over using the minimal model program.

Second, we prove a thinness statement over using Hilbert’s Irreducibility Theorem.
We emphasize that our results are not just theoretical, but can be used to understand the exceptional set in specific examples. We discuss the example of [BTcubic] (Example 5.2) and the example of [LeRudulier] (Examples LABEL:lerudulierexample2 and LABEL:lerudulierexample), as well as many others in Sections 6, LABEL:fanothreefoldsec, and LABEL:examplesec.
Example 1.3.
Let be a smooth Fano threefold over a number field which has geometric Picard rank and index , degree , and is general in moduli. Then:

The set of lines (i.e. curves with ) sweeps out a divisor on .

The set of conics (i.e. curves with ) is parametrized by a surface , forming a family , and the morphism is generically finite dominant of degree .
Corollary LABEL:coro:_fano3fold shows that the conclusion of Conjecture 1.1 holds on with thin set where is the ramification divisor of .
For the rest of the introduction, we describe more precisely the geometry underlying thinness of point contributions in certain situations.
1.1. Families of subvarieties
The behavior of the constant for subvarieties was worked out by [LTT14] and [HJ16]. Using the recent results of [DiCerbo16], [HJ16] proves that over an algebraically closed field of characteristic there is a closed proper subset such that all subvarieties with are contained in .
In this paper we extend the analysis to handle the constant. We first show that to understand all subvarieties of with larger constants, only finitely many families of subvarieties need to be considered. The following theorem shows that for any subvariety lying outside of a fixed closed subset of , either the values for are smaller than those for or is covered by subvarieties lying in a fixed bounded family. This is the optimal statement, since such need not themselves form a bounded family.
Theorem 1.4.
Let be a smooth uniruled projective variety of dimension over an algebraically closed field of characteristic and let be a big and semiample divisor on . There is a proper closed set and a finite collection of families of subvarieties with the evaluation map satisfying the following: for any subvariety of , either

,

in the lexicographic order, or

is the image of the closure of the locus by for some , and the general fiber over satisfies in the lexicographic order.
Furthermore, if denotes a resolution of the general fiber of and is the corresponding map, we have that .
In particular, the pairs achieve a maximum in the lexicographic order as varies over all subvarieties of not contained in .
The families of Theorem 1.4 have interesting properties. We prove that for each such family the map is generically finite. Furthermore, the family satisfies a basic dichotomy: either the map has degree , or over an open subset of the parameter space there is a monodromy action on the local system describing relative NéronSeveri groups as in [KM92].
When we work over a number field, the dichotomy above yields thinness of point contributions: either the map is generically finite of degree , or we can leverage the monodromy action using the Hilbert Irreducibility Theorem. Combining everything, we obtain:
Theorem 1.5.
Let be a geometrically uniruled smooth projective variety of dimension defined over a number field and let be a big and semiample divisor on . Suppose that and is rigid. As we vary over all geometrically integral subvarieties defined over such that either is not big or
in the lexicographic order, the points
are contained in a thin subset of .
Remark 1.6.
The two geometric conditions in Theorem 1.5 are quite natural in our situation. First, the restriction on the Picard group ensures that the value does not change when we pass to to apply the minimal model program. Second, when then by [LTT14] the fibers of the canonical map have values at least as large as . In this situation it makes sense to first study Conjecture 1.1 on the fibers, and then to understand the Iitaka fibration as a last step.
1.2. Generically finite maps
We next turn to the geometric consistency of Manin’s Conjecture with respect to generically finite maps. The key conjecture is:
Conjecture 1.7.
Let be a smooth uniruled projective variety over an algebraically closed field of characteristic and let be a big and nef divisor on . Up to birational equivalence, there are only finitely many generically finite covers such that and .
We note that by the recent results of [birkar16] the degree of the maps in Conjecture 1.7 is bounded, giving a weaker finiteness statement. The condition is a technical condition necessary for finiteness to hold. When this condition fails, (and hence ) is covered by subvarieties with higher values, and one can instead study the geometry of these subvarieties.
Our approach to proving Conjecture 1.7 is to reduce to the finiteness of the étale fundamental group of Fano varieties as in [Xu14], [GKP16]. We conjecture that any generically finite morphism as in Conjecture 1.7 is birationally equivalent to a morphism of Fano varieties that is étale in codimension . We are able to solve this problem in dimension and obtain partial results in dimension 3:
Theorem 1.8.
Let be a smooth projective surface over an algebraically closed field of characteristic and let be a big and nef divisor on . Then any generically finite cover such that and is birationally equivalent to a cover which is étale in codimension , and there are only finitely many such covers up to birational equivalence.
Furthermore if then .
Theorem 1.9.
Let be a smooth Fano threefold over an algebraically closed field of characteristic such that has index or has Picard rank , index and is general in the moduli. Then there is no dominant generically finite map of degree such that .
The precise analogue of Conjecture 1.7 for number fields does not hold due to the presence of twists. However, we can still hope to show thinness of point contributions. Due to the finiteness in Conjecture 1.7 over an algebraic closure, it suffices to consider the behavior of rational points for all twists of a fixed map.
Theorem 1.10.
Let be a smooth projective variety over a number field satisfying and let be a big and nef divisor on . Suppose that is a generically finite cover from a smooth projective variety satisfying . As we let vary over all elements of such that the corresponding twists satisfy
in the lexicographic order, the set
is contained in a thin subset of .
In fact, when the induced extension of function fields is not Galois, then the contributions of all twists lies in a thin set. In the Galois case, we must leverage the comparison of values to apply the Hilbert Irreducibility Theorem.
Remark 1.11.
As remarked above, Example LABEL:exam:_Sano shows that the statement of Theorem 1.10 fails if we allow twists such that
Nevertheless this fact is still compatible with Manin’s conjecture and Peyre’s constant as Example LABEL:exam:_Sano demonstrates.
The paper is organized as follows. In Section 2, we recall the geometric invariants appearing in Manin’s conjecture and the notion of balanced divisors. In Section 3, we collect several results from the literature on the minimal model program and birational geometry. Then we turn to families of subvarieties and prove Theorem 1.4 and Theorem 1.5 in Section 4 and Section 5. In Section 6, we study generically finite covers for surfaces and prove Theorem 1.8. In Section LABEL:fanothreefoldsec, we study generically finite covers for Fano folds and prove Theorem 1.9. In Section LABEL:sec:_twists, we study contributions of twists and prove Theorem 1.10. In Section LABEL:examplesec, we explore several examples to illustrate our study.
Acknowledgments. The authors would like to thank I. Cheltsov and C. Jiang for answering our questions about Fujita invariants and Fano threefolds. They also would like to thank B. Hassett, D. Loughran, and Y. Tschinkel for answering our questions about twists. Finally they would like to thank M. Kawakita for his email communications regarding lengths of divisorial contractions on threefolds. We thank D. Loughran, J. M^{c}Kernan, and M. Pieropan for useful comments. Lehmann is supported by an NSA Young Investigator Grant. Tanimoto is supported by Lars Hesselholt’s Niels Bohr professorship.
2. Balanced divisors
Here we recall some basic definitions of geometric invariants appearing in Manin’s conjecture, and the notion of balanced divisors.
2.1. Geometric invariants
Here we assume that our ground field is a field of characteristic zero, but not necessarily algebraically closed. In this paper, a variety defined over means a geometrically integral separated scheme of finite type over .
Suppose that is a smooth projective variety defined over . We denote the NéronSeveri space on by and its rank by . If is a base extension to an algebraic closure , then in general, we have , and the strict inequality can happen. We define the cone of pseudoeffective divisors as the closure of the cone of effective divisors in and we denote it by .
Definition 2.1.
Let be a smooth projective variety defined over and a big Cartier divisor on . The Fujita invariant is
For a singular projective variety , we define to be the Fujita invariant of the pullback of to a smooth model. This is welldefined because the invariant does not change upon pulling back via a birational morphism [HTT15, Proposition 2.7]. Note that since cohomology of line bundles is invariant under flat base change, we have . Also by [BDPP], if and only if is geometrically uniruled.
For the purposes of Manin’s Conjecture, it is often useful to add an assumption on the adjoint divisor .
Definition 2.2.
Let be a pair of a projective variety and a big and nef Cartier divisor defined over . We say that the adjoint divisor for is rigid, or that is adjoint rigid, if there exists a smooth resolution such that is rigid, i.e., the Iitaka dimension of is zero. Again this definition does not depend on the choice of the resolution .
Definition 2.3.
Let be a smooth projective variety defined over and a big Cartier divisor on . We define the invariant by
the codimension of the minimal  
Again this is a birational invariant upon pulling back [HTT15, Proposition 2.10], so we define this invariant for singular projective varieties via passage to a smooth model. In general the constant is not preserved by base extension: we may have .
2.2. Balanced divisors
Here we assume that our ground field is an algebraically closed field of characteristic zero. The following notion is introduced in [HTT15, Definition 3.1]
Definition 2.4.
Let be a uniruled projective variety and a big Cartier divisor on . Suppose that we have a morphism from a projective variety which is generically finite onto its image. Then is weakly balanced with respect to if

is big, and;

we have the inequality
in the lexicographic order.
When the strict inequality holds, we say that is balanced with respect to .
We say that is (weakly) balanced with respect to general subvarieties if there exists a proper closed subset such that is (weakly) balanced with respect to any subvariety not contained in .
We say that is (weakly) balanced with respect to generically finite covers if is (weakly) balanced with respect to every surjective generically finite morphism .
We say that is (weakly) balanced with respect to general morphisms if there exists a proper closed subset such that is (weakly) balanced with respect to any generically finite morphism whose image is not contained in .
It is also convenient to consider the following notion, which restricts attention to the values:
Definition 2.5.
Let be a uniruled projective variety and a big Cartier divisor on . Suppose that we have a generically finite morphism onto its image. Then is balanced (respectively strongly balanced) with respect to if

is big, and;

we have the inequality :
We say that is (strongly) balanced with respect to general subvarieties if there exists a proper closed subset such that is (strongly) balanced with respect to any subvariety not contained in . Other notions for generically finite morphisms and covers are defined in a similar way.
3. Preliminaries
We collect some useful results for our analysis. In this section we work over an algebraically closed field of characteristic zero.
3.1. The Minimal Model Program
Definition 3.1.
A pair is a normal factorial projective variety with an effective divisor . A pair is called a factorial terminal log Fano pair if has only terminal singularities and is ample.
Theorem 3.2 (relative version of Wilson’s theorem).
Let be a projective morphism between irreducible varieties and a big and nef Cartier divisor on . Then there exists an effective divisor such that for any sufficiently small , is ample.
Proof.
There is an open neighborhood of in consisting of big divisors. In particular, there is some big divisor such that is ample for any sufficiently small . If we replace by for some sufficiently ample divisor on , then we can find an effective divisor that is linearly equivalent to without losing the desired property. ∎
Let be a smooth projective morphism between irreducible varieties. According to [KM92, Proposition 12.2.5], there exists a local system over in the analytic topology (or étale topology) with a finite monodromy. When for any , we have an isomorphism for any by Hodge theory.
Remark 3.3.
The two results above in [KM92] are stated over in the analytic topology, but one can prove the same results over any algebraically closed field of characteristic zero in the étale topology by a comparison theorem.
First suppose that we have a smooth projective morphism defined over an algebraically closed subfield such that any fiber satisfies . Consider the following functor:
It follows from [KM92, Proposition 12.2.3] that this functor is represented by a proper separated unramified algebraic space . Then we can define the sheaf of sections of with open support in the étale topology. We claim that this satisfies the properties we stated above. Indeed, after base change to and considering the sheaf in the analytic topology, forms a local system with finite monodromy action. Then we have a finite étale cover over such that the local system is trivialized by . This means that is a local system in étale topology, and stalks in the analytic topology and étale topology coincide. Finally algebraic fundamental groups and NéronSeveri groups are invariant under base change of algebraically closed fields, so our assertion follows from results over in the analytic topology.
For an arbitrary algebraically closed field of characteristic zero, a standard argument in algebraic geometry shows that our family is defined over a subfield which admits an embedding into , so our assertion again follows from a comparison theorem.
Theorem 3.4 (the relative MMP vs the absolute MMP).
Let be a smooth projective morphism from a smooth irreducible variety to an irreducible variety and a big and nef Cartier divisor on . Assume that for any and the monodromy action on is trivial. Consider the relative MMP over
Then for a general , is the absolute MMP. If is not an isomorphhism, then we have

the map is a divisorial contraction if is;

the map is a flip if is;

the map is a Mori fiber space if is.
.
Proof.
This follows from [dFH09, Theorem 4.1] and its proof. ∎
Theorem 3.5.
Let be a smooth projective variety and a big and nef divisor on . Suppose that is adjoint rigid. Then there is a birational contraction to a factorial terminal weak Fano variety such that
Proof.
We apply MMP and obtain a birational contraction such that . By [LTT14, Lemma 3.5], we have . As explained in [LTT14, Lemma 2.4 and Proposition 2.5], is a factorial terminal log Fano variety. Hence it follows from [BCHM] that is a Mori dream space. Since is nef, the pushforward is in the movable cone. Thus we can find a small factorial modification such that is big and nef. We claim that has only terminal singularities.
Indeed, as explained in [LTT14, Lemma 2.4], we can find an effective divisor such that the pair is a terminal pair. Let be an ample effective divisor on and we denote its strict transform on by . Then for a sufficiently small , we still have that is a terminal pair. We apply MMP to . The result must be since is ample and is factorial. Since is an isomorphism in codimension one, decomposes into a composition of flips. We conclude that is a terminal pair. It is easy to see that
Thus our assertion follows. ∎
3.2. Explicit Fujitatype statements
To understand the balanced property for example, it is crucial to have effective Fujitatype results. One approach is to use the work of [Reider] for surfaces and its extensions to dimension (see for example [Lee99]). However, we are usually forced to work with divisors, so it will be more useful to rely on effective volume bounds for singular Fano varieties.
Proposition 3.6.
Let be a smooth projective variety and let be a big and nef divisor on .

Suppose . If then is big.

Suppose . If and for every curve through a general point of then is big. Furthermore, if is rigid, then we have .

Suppose . If and for every surface through a general point and for every curve through a general point then is big.
Proof.
In each case we need to show that if satisfies the given criteria then . The proof is by induction on the dimension. We focus on (3), since the cases (1), (2) use exactly the same argument but are easier.
Suppose that is not rigid. We run the MMP and obtain the minimal model . Let be the semiample fibration associated to . After applying a resolution of indeterminacy, we may assume that is a birational morphism. Let be a general fiber of . Then we have . It follows from the theorem in lower dimensions that .
Assume that is rigid. Again we apply the MMP with scaling of and obtain the minimal model . Fix a small and continue to run the MMP with scaling of . The result is a factorial terminal threefold admitting a Mori fibration whose general fiber is a terminal Fano variety such that is trivial along the fibers.
If the fibers of have dimension or , then by resolving we may as well suppose that the rational map is a morphism. Then for a general fiber of . By the theorem in lower dimensions, we have .
When is a point, is a terminal Fano of Picard rank . In [Nami], Namikawa showed that every terminal Gorenstein Fano 3fold can be deformed to a smooth Fano 3fold, in particular we have . Moreover Prokhorov proved that the degree of a factorial terminal nonGorenstein Fano 3fold of Picard rank one is bounded by ([Pro]). All together, we have . Since we have , we conclude that . ∎
We can phrase the result to look more like Reider’s theorem:
Theorem 3.7.
Let be a smooth surface and let be a big and nef divisor on . Suppose that for every rational curve passing through a general point. Then is big.
Proof.
Let be a minimal model for . By the curve condition, we only need to consider the case when is rigid. We continue the MMP, and obtain with a Mori fiber structure . If , then our curve condition implies that . Hence we assume that is a smooth del Pezzo surface of Picard rank one, i.e., and .
There is an ample rational curve on satisfying – one can simply take a general line on . Since the classes of and are proportional, by our condition on curves we see that . ∎
Corollary 3.8.
Let be a smooth projective 3fold and let be a big and nef divisor on . If and for every rational curve through a general point then is big.
3.3. Rationally connected varieties
By an argument of [Nak, II.5.15 Lemma], the kernel in the following lemma does not depend on the choice of general fiber .
Lemma 3.9.
Let be a normal factorial variety and suppose that is a morphism whose general fiber is irreducible and rationally connected. Let denote the kernel of the restriction map for a general fiber . Then is spanned by a finite collection of effective irreducible vertical divisors.
Proof.
See the proof of [LTT14, Theorem 4.5]. ∎
4. Families of varieties
In this section we address the geometric behavior of the constants for families of subvarieties. The following proposition is useful for us.
Proposition 4.1.
Let be a morphism of finite type from a noetherian scheme to an irreducible noetherian scheme. Then
is constructible in . In particular, if the generic point of is contained in , then there exists a nonempty open set such that .
Proof.
See [stacks] Lemma 36.22.7., Lemma 36.21.5., and Lemma 27.2.2. ∎
Definition 4.2.
A morphism between irreducible varieties is called a family of varieties if the generic fiber is geometrically integral. A family of projective varieties is a projective morphism which is a family of varieties. A family of subvarieties is a family of projective varieties admitting a morphism which restricts to a closed immersion on every fiber of .
From now on in this section we work over a fixed algebraically closed field of characteristic zero.
4.1. Variation of constants in families
We show that the geometric invariants are constant for general members of a family of projective varieties.
Let be a uniruled projective variety and a big Cartier divisor on . The adjoint divisor for is the divisor for some smooth resolution . The Iitaka dimension of the adjoint divisor does not depend on the choice of .
Theorem 4.3 ([Hmx13]).
Let be a family of projective varieties. Suppose that is a big and nef Cartier divisor on . Then there exists a nonempty open subset such that the invariant is constant for , and the Iitaka dimension of the adjoint divisor for is constant for .
Proof.
By resolving and throwing away a closed subset of the base, we may suppose that every fiber is smooth. By Theorem 3.2, there is a fixed effective divisor such that for any sufficiently small we have where is ample. Fix a positive rational number . After a further blowup resolving and after replacing by a linearly equivalent divisor, we may suppose that has simple normal crossing support and is log canonical. Furthermore, after throwing away a closed subset of the base we may suppose that every component of dominates .
We then apply the invariance of log plurigenera as in [HMX13, Theorem 1.8]. We conclude that for any sufficiently divisible and for every fiber the value of is independent of . The desired conclusion is immediate. ∎
Proposition 4.4.
Let be a family of projective varieties. Suppose that is a big and nef Cartier divisor on . Assume that for a general member , the adjoint divisor is rigid. Then there exists a nonempty subset such that is constant over .
Proof.
By resolving and throwing away a closed subset of the base, we may assume that (i) every fiber is smooth for , (ii) is constant over , (iii) is rigid for every . It follows from [LTT14, Proposition 2.5] that every fiber is rationally connected, so we conclude that for any . [KM92, Proposition 12.2.5] indicates that the sheaf forms a local system in the étale topology (see Remark 3.3) with a finite monodromy action. Replacing by a finite étale cover and taking a base change, we may assume that the monodromy action is trivial, and in particular, we have a natural isomorphism for a general , and dually . Then we apply the relative MMP over and obtain a birational contraction map to a relative minimal model over . It follows from Theorem 3.4 that for a general , is a minimal model of MMP, hence [LTT14, Lemma 3.5] implies that . This is constant for a general by combining Theorem 3.4 with the constancy of . ∎
Remark 4.5.
It is interesting to ask whether the adjoint rigidity of is necessary in Proposition 4.4. If we make no assumption on , then the most we can ask for is constancy of the value for a very general fiber (consider for example a family of K3 surfaces where the Picard rank jumps infinitely often), and it should be possible to prove such a statement using invariance of plurigenera. However, if is uniruled, then it seems likely that the value should again be constant on a general fiber.
4.2. Universal families of subvarieties breaking the balanced property
In this section, we construct the universal families of subvarieties breaking the balanced property. Let us recall some results from [LTT14] and [HJ16].
Lemma 4.6.
Let be a smooth uniruled variety and a big and nef divisor on . Suppose that the adjoint divisor is not rigid. Let be the canonical fibration associated to . Let be a general fiber of . Then we have
In particular, is not balanced with respect to .
Proof.
See [LTT14, Theorem 4.5] and its proof. ∎
Theorem 4.7 ([Hj16] Corollary 2.15).
Let be a smooth projective variety of dimension and let be a semiample big divisor on . Fix a positive real number . As we vary over all subvarieties of , there are only finitely many values of which are at least .
Proof.
We may rescale to assume it is Cartier. By an argument of Siu, is pseudoeffective for any resolution of a subvariety of . Thus is bounded above by as we vary over all subvarieties . By replacing by , we may ensure that . We may then apply [HJ16, Corollary 2.15] to an appropriate rescaling of . ∎
Corollary 4.8.
Let be a projective variety of dimension and let be a semiample big divisor on . Fix a positive real number . The set of all subvarieties that are adjoint rigid, not contained in , and satisfy are parametrized by a bounded subset of .
Proof.
By [HJ16, Corollary 2.15] we may suppose after rescaling that is Cartier, is basepoint free, and for every subvariety of . Our goal is to show that the degree of is bounded. Thus, we may apply a resolution of singularities so that and are smooth.
Applying the MMP, we obtain a birational contraction to a factorial terminal weak Fano variety . (See Theorem 3.5.) Note that . Furthermore, the coefficients of the divisor can only attain a finite number of values. [HX14, Theorem 1.3] shows that the pairs are parametrized by a bounded family. We deduce that there are only finitely many possible values of as we vary using invariance of log plurigenera. Then
and our assertion follows. ∎
Using this corollary and arguing as in [LTT14, Theorem 4.8] one obtains:
Theorem 4.9 ([Hj16] Theorem 1).
Let be a smooth uniruled projective variety and let be a semiample big divisor on . There is a closed subset such that any subvariety with is contained in .
Next we define a family of subvarieties breaking the balanced property.
Definition 4.10.
Let be a projective variety and a big and nef divisor on . A family of subvarieties is called a family of subvarieties breaking the balanced property if

the evaluation map is dominant,

is normal,

for a general member of , we have ,

for a general member of , the adjoint divisor for is rigid, and

for a general member of , .
We say that a family of varieties breaks the balanced property if the same conditions hold, except we only require that restricted to every fiber of is generically finite onto its image.
Theorem 4.11.
Let be a smooth uniruled projective variety of dimension and a big and semiample divisor on . Then there exists a proper closed subset and a finite number of families of subvarieties breaking the balanced property with evaluation maps such that for every subvariety breaking the balanced property, either

or,

there exist and a subvariety such that satisfies and is birational onto its image. In this case the fibers of break the balanced property for .
We call the universal families of subvarieties breaking the balanced property.
Proof.
Let be a proper closed subset as in the statement of Theorem 4.9. By enlarging if necessary, we may assume that contains . We consider the locus parametrizing irreducible and reduced subvarieties where satisfies:

is not contained in ;

the adjoint divisor for is rigid;

we have and .
It follows from Corollary 4.8 that is a bounded family. Let be the Zariski closure of in . Let be irreducible components of and the pullback of the universal family on . We denote the evaluation map for each family by . If is not dominant, then we add the closure of the image of to , so that we may assume that for any , is a family of subvarieties breaking the balanced property (after taking the normalization of if necessary).
We claim that the and we constructed satisfy the desired properties. Let be a subvariety such that is not contained in and is not balanced with respect to . If the adjoint divisor for is rigid, then it is easy to see that is a member of for some . Suppose that the adjoint divisor for is not rigid. Let be a smooth resolution and be the canonical fibration associated to . Let be the graph of the rational map . It follows from the universal property of that there exists a dense open set and a morphism such that is the pullback of the universal family on . Lemma 4.6 implies that this actually factors through for some . We denote the image of by . It follows from the construction that is the closure of and that is birational on . ∎
Remark 4.12.
The universal families of subvarieties breaking the balanced property constructed in the proof of Theorem 4.11 actually satisfy a stronger universal property. Suppose that we have a family of subvarieties breaking the balanced property . Then there exist a dense open subset and a morphism for some such that is isomorphic to . In particular the families are unique up to birational isomorphisms.
Proof of Theorem 1.4:.
Choose and the families as the in the statement of Theorem 4.11. By Definition 4.10.(4), the general fiber of satisfies the desired adjoint rigidity. Suppose that is a subvariety of which breaks the balanced property and is not contained in . According to Theorem 4.11, there is a locus for some such that the closure of the image of is equal to , and the covering family of fibers breaks the balanced condition for . This is exactly the desired statement. ∎
It follows that the constants achieve a maximum as we vary over all subvarieties. This is of course a necessary condition for the geometric consistency of Manin’s Conjecture. More precisely:
Corollary 4.13.
Let be a smooth uniruled projective variety of dimension and let be a big and semiample divisor on . As varies over all subvarieties of not contained in the values achieve a maximum in the lexicographic order.
4.3. Geometric properties of families
We next discuss the geometric results underlying thin set results for families of subvarieties.
Proposition 4.14.
Let be a smooth uniruled projective variety of dimension and let be a big and nef divisor on such that is adjoint rigid. Suppose that is a family of subvarieties of dimension breaking the balanced condition via a dominant map such that the induced rational map is birational onto its image. Then is generically finite.
Proof.
Suppose that is not generically finite so that a general fiber is positive dimensional. After resolving singularities of and shrinking , we may assume that our family is smooth. After taking a finite étale base change of , we may assume that the monodromy action on is trivial. After shrinking if necessary, we assume that is smooth and the induced map is quasi finite. We apply the relative MMP and obtain a relative minimal model and we denote the exceptional divisors of by and their union by . It follows from Theorem 3.4 that for a general member of , the support of the rigid divisor numerically equivalent to is contained in . After shrinking if necessary, we may assume that this holds for any member of .
Choose a general complete intersection variety in of dimension . Then is a subvariety of with dimension at least , and by generality we know that the general point of is not contained in . Furthermore, we claim that
Indeed, if we let be the composition of the first projection with , and be the composition of the second projection with , then it suffices to show that . Note that the fibers of have dimension at least , since there is at least a onedimensional family of subvarieties parametrized by through each point of . Since is dominant, its restriction to any general subvariety of the parameter space is also dominant. Altogether we obtain the claim.
By cutting by hyperplanes, we can choose a general subvariety of dimension such that

the evaluation map