Integral points and effective cones of moduli spaces of stable maps

Consider the Fulton-MacPherson configuration space of $n$ points on $\P^1$, which is isomorphic to a certain moduli space of stable maps to $\P^1$. We compute the cone of effective ${\mathfrak S}_n$-invariant divisors on this space. This yields a geometric interpretation of known asymptotic formulas for the number of integral points of bounded height on compactifications of $\SL_2$ in the space of binary forms of degree $n\ge 3$.


Introduction
In this paper, we compute the S n -invariant cone of effective divisors of the Fulton-MacPherson configuration space of n points on P 1 . This space is isomorphic to the moduli space M 0,n (P 1 , 1) of stable maps of degree one from genus zero curves with n marked points to P 1 . We also compute the effective cone of the generic fiber of the natural map M 0,n (P 1 , 1)/S n → M 0,n /S n .
Our motivation is to provide a geometric explanation of a formula, obtained by Duke, Rudnick and Sarnak, giving the asymptotic behavior of the number of binary forms of degree n with fixed discriminant and bounded integral coefficients. This fits into a larger program to predict and prove asymptotic formulas for the number of rational and integral points of bounded height on algebraic varieties.
We introduce a counting function for integral points on an algebraic variety as follows: given a variety U over a ring of integers o and functions g 1 , ..., g n , regular on U, define where · is a valuation on o. This is finite only when the functions g j give an embedding of U.
It is most natural to interpret the functions g j as sections of a line bundle L on a projective compactification X ⊃ U defined over the fraction field F of o. The fact that the sections embed U implies that L is big, i.e., is contained in the interior of the effective cone Λ eff (X) of X. Therefore, in order to describe all natural counting functions on open subsets of X we need to compute its effective cone. Furthermore, in many cases it can be proved that the asymptotic properties of N(U, B) are intimately related to the structure of this cone.
Let P (x) = P (x 0 , ..., x r ) be a homogeneous polynomial of degree n in r+1 variables. A standard heuristic in number theory predicts that the number N P (B) := x | max(|x j |) ≤ B, P (x) = 0, and x ∈ Z r+1 of integral solutions of the equation P (x) = 0 of "height" ≤ B grows asymptotically like B r+1−n as B → ∞. When the number of variables is ≫ 2 n , the affine variety V P defined by P = 0 is smooth and there are no local obstructions, an asymptotic formula can be established using the classical circle method in analytic number theory (see [3], [25] and the references therein). Of course, there may be difficulties when the number of variables is small or the variety V P is singular.
The following example appeared in the paper by Duke, Rudnick and Sarnak [6]. Consider the vector space of binary forms of degree n The algebraic group SL 2 acts on this space by coordinate substitutions. When n = 3, the discriminant form generates the ring of SL 2 -invariants. Then there exists a constant c > 0 so that as B → ∞. Note that the exponent 2/3 is larger than what is predicted by the standard heuristic. More generally, one has the Theorem 0.1 [6] Fix a generic binary form f of degree n ≥ 3 with integral coefficients. Let N(B) be the number of binary forms SL 2 (Z)-equivalent to f with coefficients bounded by B. Then there exists a c > 0 such that We give a geometric interpretation of the exponent 2/n in Theorem 0.1. To this end, we refine the heuristics for counting integral points to take into account singularities of the relevant varieties (see Conjecture 1.6). We verify that Conjecture 1.6 is consistent with Theorem 0.1 in Theorem 2.1. Its proof involves the computations of effective cones alluded to above.

Singularities of pairs and effective cones
We work over a field of characteristic zero. Let X be a normal projective variety with canonical class K X and let D be a reduced effective Weil divisor of X.
consists of a smooth projective variety X and a strict normal crossings divisor D in X. This means that all irreducible components of D are smooth and intersect transversally.
Let (X, D) be a good pair and let Λ eff (X) denote the closed cone of effective divisors classes of X; a divisor is big exactly when its class is in the interior of this cone. Define where we identify line bundles and their divisor classes. Note that a(L, D) is a positive real number whenever −(K X + D) is big. The constant −a(L, D) is called the log-Kodaira energy of L (see [10]).
If (X, D) is not good then resolution of singularities implies the existence of a good resolution ρ : (X,D) → (X, D). Precisely, (X,D) is a good pair, ρ a birational projective morphism, andD is the union of the exceptional divisors of ρ and the proper transform of D. Recall that (X, D) is log- where the E j are the exceptional divisors of ρ and d j ≥ 0 for all j.
Example 1.2 When X is a smooth surface, (X, D) is log-canonical only when the curve D is smooth or nodal. If X is smooth of arbitrary dimension, D must have at worse nodes in codimension one.
If L is a line bundle on X put where D t ⊂X is the total transform of D. Note that a(L, D) is computed onX.

Proposition 1.3
Let (X, D) be a log-canonical pair and assume that X − D has canonical singularities. If L is a big line bundle on X then In particular, a(L, D) does not depend on the choice of a desingularization.
Proof. Choose a good resolution ρ : (X,D) → (X, D), so that In particular, each exceptional divisor not contained in the total transform D t has log discrepancy ≥ 1. Therefore, we have Proposition 1.4 Let (X 1 , D 1 ) and (X 2 , D 2 ) be log-canonical pairs, so that X 1 −D 1 and X 2 −D 2 have canonical singularities. Assume that π : X 1 → X 2 is a finite dominant morphism so that Let L be a big divisor on X 2 . Then a(L, D 2 ) = a(π * (L), D 1 ).
Proof. Given a finite dominant morphism π : X 1 → X 2 and a Q-Cartier divisor M on X 2 , M is effective iff π * (M) is effective. Indeed, the divisor π * π * M is defined and equal to deg(π)M. Combining this with Proposition 1.3 gives the result.
Remark 1.5 Let (X, D) be a log terminal pair so that X − D has singularities which are not canonical. Then our definition of the Kodaira energy differs slightly from Fujita's [10]. In applications to integral points, we are interested in invariants of the open variety X − D. In Fujita's definition, on passing from (X, D) to a good resolution, any exceptional divisors over X −D with negative discrepancy must be added to the boundary. This changes the open variety.

Integral points
Retain the notation from the previous section and assume that X and D are defined over a number field F .
as B → ∞, at least after a suitable finite extension of F and S.
The statement is independent of the choice of S and the choice of a metrization on L.
Many precise results about asymptotics of rational and integral points are currently available (see, for example, [9,2,4,21,22,6,7,8] and the references therein). As far as we know, Conjecture 1.6 is compatible with all of them. However, to actually check this compatibility one has to compute the geometric invariants of (some resolution of) the pair (X, D). In particular, one has to determine the effective cone. This can be a formidable task even for rational varieties, e.g., like the moduli space of pointed rational curves M 0,n (see [14]).

Computing effective cones
Let X be a nonsingular projective variety, perhaps with an action by a finite group G. We review strategies for computing the G-invariant effective cone Λ eff (X) G and thus the effective cone of the quotient X/G (cf. [16] . A family of curves passing through the generic point of X is automatically nef. Indeed, consider a family C → B of integral projective curves in X and an irreducible codimension-one subvariety D ⊂ X. If, for Fix a collection of effective divisors which we expect to generate Λ eff (X) G . To prove that Γ generates the (Ginvariant) effective cone, it suffices to find a collection of nef (G-invariant) curve classes Ξ = {C 1 , . . . , C ℓ } so that the cone generated by Γ contains the dual to the cone generated by Ξ.
In section 4, we shall use a refinement of this method (see [5], [24]). A divisor D ∈ Λ eff (X) is moving relative to Γ if some multiple of D contains no element of Γ as a fixed component. Every effective divisor is a sum where M is moving relative to Γ. To prove that Γ generates the effective cone, it suffices to show that M is an effective sum of the A i .
A curve class is nef relative to Γ if [C].M ≥ 0 for each M which is moving relative to Γ. Any family of curves passing through the generic point of some A i is nef relative to Γ. Consequently, to show that Γ generates the effective cone, it suffices to find a collection Ξ of curve classes, nef relative to Γ, so that the cone generated by Γ contains the dual to the cone generated by Ξ.

Construction of resolutions 2.1 Binary forms and SL -orbit closures
Let V be a two-dimensional vector space with coordinates z and w, equipped with the standard SL 2 -action. Let Sym n V * be the space of binary forms of It carries an induced action of SL 2 by substitution.
Associating to each form f = 0 its roots α 1 , . . . , α n yields a map The discriminant of a polynomial f is a homogeneous form in its coefficients x 0 , . . . , x n and defines a divisor D ⊂ X = P(Sym n V * ). Now we may state our main result: Theorem 2.1 (Computation of Kodaira Energy) Let f be a generic bilinear form of degree n, X f ⊂ P(Sym n V * ) the closure of the SL 2 -orbit through f , D f the intersection of the discriminant with X f , and L the restriction of the standard polarization to X f . Then we have a(L, D f ) = 2/n.
In particular, Conjecture 1.6 is consistent with Theorem 0.1.
To prove this, we require a resolution (i.e., a partial desingularization) of (X f , D f ) on which we may evaluate a(L, D f ) using Proposition 1.3. This resolution will be induced by a natural resolution of (X, D).
Remark 2.2 Example 1.2 shows that (X, D) is far from being log-canonical. When n = 3, the discriminant has cusps in codimension one: a transverse slice z 3 + bzw 2 + cw 3 intersects the discriminant in the cuspidal curve Our resolution of (X, D) will be a S n -quotient of a natural desingularization for (P(V ) n , ∆), where ∆ is the diagonal, i.e., the points lying over the discriminant. Both admit interpretations as moduli spaces of stable maps.

Moduli spaces
Fix an integer n ≥ 3. Let M 0,n denote the Knudsen-Mumford moduli space of stable curves of genus zero with n marked points [18] (C, p 1 , . . . , p n ).
Let M 0,n (P 1 , 1) denote the Kontsevich moduli space of stable maps of degree one from genus-zero curves with n marked points to P 1 [19,20,12] (C, p 1 , . . . , p n , µ : C → P 1 ). This is naturally isomorphic to the Fulton-MacPherson [11] configuration space P 1 [n] for n points in P 1 (see [12] §0). However, for our purposes it is convenient to use the moduli space notation.
Finally, we enumerate the boundary divisors of these moduli spaces. For each partition consider stable curves which form a divisor δ S,S ′ ⊂ M 0,n . The union of these is denoted δ. Note that the S n -orbits of {δ S,S ′ } correspond to the integers |S| = 2, . . . , ⌊n/2⌋.  1), B) is log-canonical.

Resolution for the full moduli space
We obtain a good resolution (X,D) of (X, D) using the above formalism. Consider the quotient map q : M 0,n (P 1 , 1) →X := M 0,n (P 1 , 1)/S n .

LetD[s] andD be the images of B[s]
and B under this map. Take S n -quotients of the point map to obtain a birational map ̺ :X → P(Sym n V * ), assigning to p 1 , ..., p n ∈ P 1 a polynomial vanishing at these points. The boundary divisorD [2] is the proper transform of the discriminant D under ̺. The boundary divisorsD[s] (for s ≥ 3) are the exceptional divisors for ̺.

Resolution of the generic orbit
Let α := (α 1 , ..., α n ) be a set of distinct complex numbers and f = f α the binary form of degree n with roots α j . Let C α ∈ M 0,n be the corresponding pointed rational curve and µ α ∈ M 0,n (P 1 ) the corresponding map. The fiber contains µ α . LetX f be the image of Y α under the quotient map q andD f its intersection with the boundaryD. This coincides with the general fiber of the map ψ ′ :X = M 0,n (P 1 , 1) → M 0,n /S n .
The map ̺ induces a resolution To describe the Y α explicitly, we use the tower  1), which is isomorphic to the product (P 1 ) 3 blown up along the small diagonal ∆ small . The boundary divisors correspond to the following stable maps In the above pictures the collapsed components are represented by vertical lines. Note that the normal bundle so that the exceptional divisor E = B[3] ≃ P 1 × P 1 . Let be the projection to the cross ratio of the marked points and the node and the projection onto the image of the collapsed curve.
The divisor B [2] is the proper transform of ∆, the large diagonal.
Case 1: Over the open subset of Y α 1 ,...,αn corresponding to the first two cases, φ n induces an isomorphism between Y α 1 ,...,αn and Y α 1 ,...,α n−1 . In the third case, we forget the image of the n-th marked point. The map φ n blows up the locus in Y α 1 ,...,α n−1 where α 1 , ..., α n−1 are on the collapsed component and α n coincides with the node (of attachment). This is a curve isomorphic to We summarize the above discussion in the following Proposition 2.6 Let α 1 , ...α n be distinct complex numbers. The forgetting maps induce a sequence of birational morphisms The moduli space of stable maps M 0,3 (P 1 , 1) is isomorphic to (P 1 ) 3 blown up along the small diagonal with exceptional divisor E ≃ P 1 × P 1 . The map φ j blows up the proper transform of π −1 1 (α j ). In particular, Y α 1 ,...,αn is smooth and its boundary has strict normal crossings, contained in B[n − 1] ∪ B[n].

Remark 2.7
We are blowing up along disjoint curves, so the order of the blow-up does not matter. Proposition 2.8 Let f be a generic binary form of degree n ≥ 3 with roots α 1 , ..., α n . Then the restriction of q to X f is ramified only along the boundary B ∩ X f . At generic points of (B[n] ∪ B[n − 1]) ∩ Y α , the restriction of q is unramified. We have the formula and (X f ,D f ) is log-canonical.
Proof. The argument is similar to the one in Proposition 2.5, and is omitted.

Verification of exponents
By Proposition 2.6, Y α is obtained by blowing up the (n − 3) sections of Let F 4 , ..., F n denote the corresponding exceptional divisors and identify E and its proper transform. Relabel so that S n acts on the F k , k = 1, ..., n, in the obvious way. Note that E and the F k generate Pic(Y α ).

Proposition 3.1 The S n -stable boundary divisors
3.2 Computation of the effective cone We may assume that the points α 1 , ..., α n are not contained in the fixed point locus of ρ t . Any singular element in the orbit closure is:  for some c ∈ N. Since the claim follows.
We have and by definition Hence Lemma 3.2 yields Thus a(L, D f ) = 2/n, as desired! 4 The S n -invariant effective cone of the full moduli space In this section, we compute the S n -invariant part of the effective cone of M 0,n (P 1 , 1), its canonical class, and the Kodaira energy of the line bundle We will also compute the Kodaira energy of H := ρ * O P n (+1).
The cohomology H * (M 0,n (P 1 , 1)) is generated by the classes L a and B S , subject to the relations denote an S n -invariant divisor class with no boundary divisors as fixed components.
Recall the description of the boundary divisor B S : Take s ≥ 3 and let C s ⊂ B S be the class of the generic fiber of the map forgetting the attaching point. Since C s passes through the generic point of B S , averaging C s over S n yields a curve class which is nef relative to Γ. In particular, for each S n -invariant divisor M = n j=2 d j B[j], moving relative to Γ, we have C s · M ≥ 0.
We compute intersections of C s with the various elements of Γ. First, the map β blows down the divisors B S for |S| = 2; the data of the collapsed component is lost completely. It follows that L · C s = 0. A simple combinatorial analysis gives which means that C s · B[s − 1] = s. Relation 1 gives . To summarize, we have Using this information, we extract inequalities on the coefficients of M. The condition M · C s ≥ 0 yields sd s−1 ≥ (s − 2)d s , so we get a chain of inequalities: If some d s < 0 then d j < 0 for each j ≥ s.
We consider another curve class in B S to get inequalities in the reverse direction. Fix s ≥ 2 and let R s denote the class of the generic fiber of induced by forgetting τ , one of the n+1−s points not contained in S. Again, R s passes through the generic point of B S , so averaging over S n yields a curve class such that R s · M ≥ 0.
We compute intersections as before. The map β sends R s to a line in P n , i.e., the linear forms with n − 1 fixed roots and one varying root. It follows that L · R s = 1. The line R s intersects B T properly in the following cases Summing over S n -orbits gives Applying Relation 1, we find gives d n ≥ n 2 − 5n + 8 2 d 2 .

Final remarks
A) The orbit closure X f depends on the form [f ]. We get equivariant compactifications of PGL 2 depending on moduli. This dependence is made abundantly clear in the blow-up description of Proposition 2.6.
B) The pair (M 0,n , δ) is of log general type: K M 0,n + δ is ample and log canonical (see, for example, §7.1 of [13]). The map ψ : M 0,n (P 1 , 1) → M 0,n is a log-Fano fibration onto a log-variety of general type (or the point, when n = 3).
C) The pair (M 0,n , δ) satisfies Vojta's conjecture. We realize M 0,n as an open subset of an algebraic torus with explicit complement. Fix n − 1 points in P n−3 in general position. Consider the set H of 1 2 (n−1)(n−2) hyperplanes spanned by n − 3 of the fixed points. Kapranov [15] has shown that M 0,n ≃ P n−3 − ∪ H∈H H. The torus is obtained by excising the n − 2 hyperplanes spanned by subsets of the first n − 2 of the points.