INTEGRAL POINTS ON SINGULAR DEL PEZZO SURFACES

Abstract In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type 
$\mathbf {A}_1+\mathbf {A}_3$
 and prove an analogue of Manin’s conjecture for integral points with respect to its singularities and its lines.


U. Derenthal and F. Wilsch
further references), in most cases using universal torsors, often combined with advanced analytic techniques.
In recent years, a conjectural framework for the density of integral points has emerged in the work of Chambert-Loir and Tschinkel [7]. The purpose of this paper is to initiate a systematic investigation of integral points of bounded height on del Pezzo surfaces. Only a few of them are covered by general results for equivariant compactifications of vector groups [8] or the incomplete work on toric varieties [9] (see also [25]); most del Pezzo surfaces are out of reach of this harmonic analysis approach since they are not equivariant compactifications of algebraic groups [13,14]. Del Pezzo surfaces are inaccessible to the circle method, which gives asymptotic formulas for integral points only on high-dimensional complete intersections [3], [7, § 5.4]. Therefore, we adapt the universal torsor method to integral points in order to confirm new cases of an integral analogue of Manin's conjecture. See also [24] for a three-dimensional example.
As rational and integral points coincide on a projective variety X, the study of the latter becomes interesting on its own on an integral model of the complement X \ Z of an appropriate boundary Z. Our first result (Theorem 10 in Section 2) is a general treatment of possible boundaries on singular del Pezzo surfaces of low degree. For singular cubic surfaces, Z must be an A-singularity; for singular quartic del Pezzo surfaces, Z must be an A-singularity or a line passing only through A-singularities. Furthermore, A 1singularities behave differently than other A-singularities.
Therefore, a good starting point seems to be a quartic del Pezzo surface that contains an A 1 -and an A 3 -singularity and three lines, which is neither toric [12,Remark 6] nor a compactification of G 2 a [13]. For each boundary Z admissible in the sense of Theorem 10, we get an associated counting problem and prove an asymptotic formula of the shape cB(log B) b−1 (Theorem 1), encountering a range of different phenomena when dealing with the different types of boundary. These asymptotic formulas admit a geometric interpretation (Theorem 2). In particular, the leading constant c consists of Tamagawa numbers as defined in [7] and combinatorial constants (analogous to the constant α defined by Peyre for rational points) as defined in [9] for toric varieties and studied in greater generality in [25]; this is the first result applying this combinatorial construction in a nontoric setting.

The counting problem
Let S ⊂ P 4 Q be the quartic del Pezzo surface defined by over Q, with an A 1 -singularity Q 1 = (0 : 1 : 0 : 0 : 0) and an A 3 -singularity Q 2 = (0 : 0 : 0 : 0 : 1). Let S ⊂ P 4 Z be its integral model defined by the same equations over Z. The closure of every rational point P ∈ S(Q) is an integral point P ∈ S(Z); both are represented (uniquely up to sign) by coprime (x 0 , . . . ,x 4 ) ∈ Z 5 \ {0} satisfying the defining equations (1). Recall that studying integral points becomes interesting only when we choose a boundary Z to consider integral points on S \Z, and that the types of boundaries Integral Points on Singular Del Pezzo Surfaces 3 in Theorem 10 for our case are the singularities and the lines; we start with the former. To do so, let Z 1 = Q 1 , Z 2 = Q 2 ; in addition to these, we study the boundary Z 3 = Q 1 ∪ Q 2 , which goes beyond the setting of weak del Pezzo pairs described in the beginning of the following section. Let Z i = Z i and U i = S \ Z i . Hence, P lies in U 3 (Z), say, if and only if it is does not reduce to one of the singularities modulo any prime p. In other words, a representative (x 0 , . . . ,x 4 Since the sets U i (Z) of integral points are clearly infinite, we consider integral points of bounded height. We work with the height functions H 1 (P ) = max{|x 0 |, |x 2 |, |x 3 |, |x 4 |}, H 2 (P ) = max{|x 0 |, |x 1 |, |x 2 |, |x 3

|}, and
because they can be interpreted as log-anticanonical heights on a minimal desingularization, as we shall see below ( Lemma 14). It turns out that the number of integral points of bounded height is dominated by the integral points on the three lines (4) in fact, there are infinitely many integral points of height 1 on some of them. Therefore, we count integral points only in their complement V = S \ {x 2 = 0}. Hence, we are interested in the asymptotic behavior of the number of integral points of bounded log-anticanonical height that are not contained in the lines. Explicitly, this is Recall that the second type of boundary is a line, resulting in Z 4 = L 1 ,Z 5 = L 2 ,Z 6 = L 3 with the notation in equation (4). Let Z i = Z i in S and U i = S \ Z i for i = 4,5,6. Analogously to the first three cases, a point (x 0 : · · · : x 4 ) ∈ S with coprime x 0 , . . . ,x 4

U. Derenthal and F. Wilsch
We work with the heights H 4 (P ) = max{|x 0 |,|x 2 |,|x 3 |}, which will again turn out to be log-anticanonical on a minimal desingularization. Let N i (B) for i = 4,5,6 be defined as in equation (5). They satisfy descriptions as in equation (6), with the integrality condition (2) replaced by condition (7). Our second result consists of asymptotic formulas for these counting problems.
Cases 5 and 6 are symmetric: the involutive automorphism of S exchanges the lines L 2 and L 3 and the height functions H 5 and H 6 , while leaving

The expected asymptotic formula
Similarly to the case of rational points [2,22], our asymptotic formulas for the number of integral points of bounded height should be interpreted on a desingularization ρ : S → S. Here, S is a weak del Pezzo surface, that is, a smooth projective surface whose anticanonical bundle ω ∨ S is big and nef (but not ample in our case). To interpret the number of points on is a reduced effective divisor with strict normal crossings. In the context of integral points, the log-anticanonical bundle ω S (D i ) ∨ assumes the role of the anticanonical bundle. From this point of view, Theorem 1 can be interpreted in the framework described in [7].

5
The minimal desingularization ρ : S → S is an iterated blowup of P 2 Q in five points. The analogous blowup of P 2 Z results in an integral model ρ : S → S (see section Section 3 for more details). Then D 1 , D 2 are the divisors above Q 1 ,Q 2 , respectively, and D 3 = D 1 + D 2 is the one over both; see Figure 1 for their dual graph (Dynkin diagram). Our discussion is simplified by the fact that the pairs ( S,D i ) are split, in the sense that Pic S → Pic S Q is an isomorphism and [18, Definition 1.6] holds, and by the fact that we are working over Q.
The preimage of the complement V of the lines on S is the complemenent V of all negative curves on S.
This leads to the reinterpretation of our counting problem as on the minimal desingularization, and we prove in Lemma 14 that H i • ρ is a loganticanonical height function on U i (Z) ∩ V (Q). Note that the log-anticanonical bundle ω S (D i ) ∨ is big and nef for i = 1,2,4,5,6 but big and not nef for i = 3 (Lemma 12); the unusual shape of H 3 is clearly related to this. From the shape of asymptotic formulas in previous results [9,8,23,24] and the study of volume asymptotics in [7], we expect that where the leading constant can be decomposed into a finite part c i,fin and an Archimedean part c i,∞ that we shall describe and determine in Section 6 precisely.
The finite part which behaves similarly as in the case of rational points, is defined as an Euler product of convergence factors and p-adic Tamagawa numbers. We compute the latter as p-adic integrals over U i (Z p ) (Lemma 24); they turn out to be simply # U i (F p )/p dim S . This reflects the fact that integral points should be distributed evenly in the set U i (Z p ), which has positive and finite volume with respect to the modified Tamagawa measure τ ( S,Di),p defined in [7]. (However, we do not prove such an equidistribution result here.) On the other hand, 100% of the integral points are arbitrarily close to the boundary with respect to the real-analytic topology, ordered by height. This makes the analysis of c i,∞ much more delicate than for rational points. More precisely, the points close to the minimal strata of the boundary-that is, the intersection of a maximal set of intersecting components of D i -should dominate the counting function. These strata are encoded in the (analytic) Clemens complex C an R (D i ). For a split surface, the vertices of this Clemens complex correspond to the irreducible components of the boundary divisor D i , and there is an edge for each intersection point of two divisors. The Archimedean constant 6 Figure 1. The Clemens complex of D 3 is the disjoint union of those of D 1 (left) and D 2 (right). It is the Dynkin diagram of the A 1 -and A 3 -singularities Q 1 ,Q 2 .

Figure 2.
Integral points on U 1 of height ≤ 90. The boundary divisor is the central vertical line. Some horizontal and diagonal lines look accumulating, but in fact are not: They contain ∼ c B points, which is less than the cB(log B) 5 points on U ; the constants c can however be up to 2, while the constant c in our main theorem is numerically ≈ 0.0003.
is a sum over the faces A of maximal dimension of the Clemens complex, which correspond to the minimal strata D A of D i . For each maximal-dimensional face A, we have a product of a rational factor α i,A and an Archimedean Tamagawa number τ i,DA,∞ (D A (R)) coming from a residue measure as defined in [7]. This measure can be interpreted as a real density, which is supported on In the first case, the Clemens complex consists of only one vertex corresponding to the boundary divisor above the A 1 -singularity Q 1 , and integral points accumulate near it ( Figure 2). In the second and third case, the maximal-dimensional faces A 1 ,A 2 of the Clemens complex correspond to the two intersection points D A1 ,D A2 of the divisors above the A 3 -singularity, and 'most' integral points are very close to these two intersection points ( Figure 3). Correspondingly, the Archimedean Tamagawa number is the volume of the boundary divisor in the first case, and it is the volume of the two intersection points in the second and third cases. In the remaining cases, it similarly is a volume of intersection points (Lemma 25). The rational factor α i,A is particularly interesting in our examples. It is introduced in [9] for toric varieties and generalized in [25] to be where U i,A is the subvariety consisting of U i and the divisors corresponding to A. For vector groups [8] and wonderful compactifications [23], the effective cone is generated by the boundary divisors and simplicial, which makes the treatment of this factor easy. In [24], it behaves similarly as Peyre's α for projective varieties since the boundary has just one component; it is also much simpler since the Picard number is 2. Our second and following cases behave in a different way since the Clemens complex is not a simplex, providing the first nontrivial treatment of this factor for a nontoric variety. Here, it turns out that the resulting polytopes for the different maximal faces fit together to one polytope whose volume appears in the leading constant of the counting problem (Lemma 28). In case 4, one of the polytopes has volume 0, making this an example for the obstruction [25, Theorem 2.4.1 (i)] to the existence of integral points near the corresponding minimal stratum of the boundary (Remark 27). The exponent of logB is expected to be Here, dim C an R (D i ) + 1 is the maximal number of components of the boundary divisor D i that meet in the same point, and Q[U i ] × = Q × in each case. While the obstruction described in [25] can lead to this number being smaller than expected if it affects all maximal-dimensional faces of the Clemens complex, this does not happen in our fourth case as there are three unobstructed faces remaining.
We can reformulate Theorem 1 as follows.
as B → ∞, where the constants c i,∞ , c i,fin and b i are as in equations (9), (10) and (12), respectively.

Strategy of the proof
In Section 2, we define and classify weak del Pezzo pairs ( S,D), which have big and nef log-anticanonical bundle ω S (D) ∨ (Theorem 10).
In Section 3, we describe a universal torsor on the minimal desingularization of S, we show that our height functions are log-anticanonical, and we describe them in terms of Cox coordinates. This leads to a completely explicit counting problem on the universal torsor (Lemma 15), with a (2 rk Pic Ui : 1)-map to our set of integral points of bounded height: roughly, the torsor variables corresponding to the boundary divisors must be ±1, and in the case of a big and base point free (whence nef) log-anticanonical class, the height function H i is given by monomials in the Cox ring of log-anticanonical degree. The third case seems to be one of the first examples of the universal torsor method with respect to a height for a divisor class that is big and not nef.
In Section 4, we estimate the number of points in our counting problem on the universal torsor using analytic techniques. Here, we approximate summations over the torsor variables by real integrals V i,0 (B); the coprimality conditions lead to an Euler product that agrees with c i,fin (Lemmas 16 and 17). This step is similar to the case of rational points treated in [11]; hence, we shall be very brief.
In Section 5, to complete the proof of Theorem 1, our goal is to transform V i,0 (B) into 2 rk Pic Ui C i B(log B) bi−1 , where C i is the product of the volume of a polytope (which turns out to be α i,A ) and a real density (which agrees with the Archimedean Tamagawa numbers τ i,DA,∞ (D A (R))), up to a negligible error term. In the first case, there is a complication due to an inhomogeneous expression (with respect to the grading by the Picard group) in the domain of V 1,0 (Lemma 18 and more importantly Lemma 19); here, a subtle estimation is necessary. In the third case, we modify the height function H 3 to H 3 (which coincides essentially with H 2 ) as in Lemma 20. These extra complications have never appeared in the universal torsor method for rational points; we believe that they are typical for integral points and nonnef heights.
In Section 6, we prove Theorem 2 by explicitly computing the expected constants discussed in Section 1.2.

Classification of weak del Pezzo pairs
For us, a weak del Pezzo pair ( S,D) consists of a smooth projective surface S with a reduced effective divisor D with strict normal crossings such that the log-anticanonical bundle ω S (D) ∨ is big and nef. The aim of this section is to study the possible choices of divisors D on a weak del Pezzo surface S that render the pair ( S,D) weak del Pezzo in this sense.

Remark 3.
Considering pairs (X,D) is standard when studying integral points: While rational and integral points coincide on complete varieties as a consequence of the valuative criterion for properness, the study of integral points becomes a distinct problem on an integral model U of a noncomplete variety U. Then one passes to a compactification, more precisely, a smooth projective variety X containing U such that the boundary D = X \ U is a reduced effective divisor with strict normal crossings. In particular, the pair (X,D) is smooth and divisorially log terminal.
The goal is then to count the number of points on U of bounded log-anticanonical height (that is, with respect to ω X (D) ∨ ), excluding any strict subvarieties (or, more generally, thin subsets) whose points would contribute to the main term. Setting D = 0 then recovers the setting of Manin's conjecture on rational points.

Remark 4.
In its original form [16,21], Manin's conjecture makes a prediction about the number of rational points on smooth Fano varieties: smooth projective varieties whose anticanonical bundle is ample. These conditions can be relaxed-for example, only requiring that the anticanonical be big and nef, viz. to weak Fano varieties and the two-dimensional varieties thereof, weak del Pezzo surfaces. Weak del Pezzo surfaces S are precisely the smooth del Pezzo surfaces S = S and the minimal desingularizations ρ : S → S of del Pezzo surfaces with only ADE-singularities [10].
Since ρ is a crepant resolution-that is, ω S = ρ * ω S -counting points on S of bounded anticanonical height amounts to counting points on S of bounded anticanonical height after excluding points on the exceptional locus. By [2,22], an asymptotic formula for the number of rational points on S should be interpreted in terms of its minimal desingularization S; for example, the Picard rank ρ of S appears in the expected asymptotic formula. The number of rational points of bounded height has been shown to conform to the same prediction as in Manin's conjecture for many weak del Pezzo surfaces (see the references in [1, § 6.4.1]).
Generalizing the question even further, it suffices to assume that the anticanonical bundle is big to guarantee that the number of rational points of bounded anticanonical height outside a suitable divisor is finite. Adding some conditions that make Peyre's constant well-defined leads to the notion of an almost Fano variety [22, Définition 3.1], for which it makes sense to ask whether Manin's conjecture holds. While this is known to be the case for some of them, Lehmann, Sengupta, and Tanimoto showed that one cannot expect the conjecture to be true in general in this widest setting [19, Remark 1.1, Example 5.17].
To simplify the exposition, let S be a weak del Pezzo surface whose degree d is at most 7. Let D = α∈A D α ⊂ S be a reduced and effective divisor with strict normal crossings and irreducible components D α .
Proof. Recall that a divisor is nef if its intersection with all negative curves is nonnegative.
and this number is nonnegative if and only if (i) and (iii) hold for E. If E is a (−1)-curve, then and this number is nonnegative if and only if (ii) and (iii) hold for E.

Remark 6.
If ρ : S → S is the minimal desingularization of a singular del Pezzo surface, then Lemma 5 shows: If one of the (−2)-curves above a singularity Q ∈ S is in D, then by Lemma 5 (i) all curves above this singularity must be in D. Similarly, if a (−1)-curve whose image in S contains a singularity Q is in D, then all (−2)-curves above Q must be in D. By Lemma 5 (iii), Q must be an A-singularity in both cases.
The surface S can be described by a sequence of r = 9 − deg S blowups where π i is the blowup in a point p i that does not lie on a (−2)-curve on S (i−1) . Let π : S → P 2 be their composition. Let 0 = π * , where is the class of a line on P 2 , and Then the Picard group of S is freely generated by the classes 0 , . . . , r . The intersection form is given by i . j = 0 for i = j, 2 0 = 1, and 2 i = −1 for i ≥ 1. Let P be the image of an exceptional divisor of one of the the blowups in P 2 and n P be the number of exceptional curves mapped to P. Then these negative curves form a chain, the first n P − 1 of which are (−2)-curves whose classes have the form i1 − i2 , . . ., is−1 − is followed by a (−1)-curve whose class has the form is . The anticanonical class is 3 0 − 1 − · · · − r , and we fix an anticanonical divisor −K. Denote by [F ] the class of a divisor or line bundle F in the Picard group. For L 1 , Proof. For the first statement, we just have to note that −ε 0 + 1≤j≤k a j j is not effective for any ε > 0. Turning to the second statement, assume for contradiction that L is big and nef. Note that 0 − i has nonnegative intersection with all (−1)-curves.
If 0 − i has (strictly) negative intersection with a (−2)-curve E, then this curve needs to have class The only negative curves that could have negative intersection with 0 − j have class j − k . As curves of classes i − j , j − k , etc., are contracted to a single point by π, we can eventually find an i with L ≤ 0 − i and such that 0 − i has nonnegative intersection with all negative curves. Then 0 − i is nef. But ( 0 − i ) 2 = 0, whence it cannot be big.

big and nef, then D is contained in the union of all negative curves.
Proof. Assume for contradiction that D contains a nonnegative curve C, but that −K −D is big and nef. In particular, −K − D is big for all D ⊂ D. Since C is nonnegative, it is the strict transform of a curve C 0 on P 2 . Then We first reduce to the case of C 0 being a line. If d ≥ 3, then [−K − C] ≤ 1≤i≤r a i i , which is not big by Lemma 7 (i). If C 0 is a nondegenerate conic, then a 1 , . . . ,a r ≤ 1 since C 0 has multiplicity ≤ 1 in all images of the exceptional divisors. Moreover, since C 2 ≥ 0, at most four of the a i are nonzero. It follows that Let C 0 be a line. As the self-intersection of C is nonnegative, [C] = 0 − j for some j or [C] = 0 . In the first case, C 0 contains the center P = π 1 · · · π j (p j ) of a blowup. If n P > 1, then π −1 (P ) contains (−2)-curves. Appealing to Lemma 5 (i), the first (−2)-curve must be contained in D, as must the remaining (−2)-curves by repeated applications. Let E 0 be the sum of these (−2)-curves.
j is the class of the final (−1)-curve in the chain. If n P = 1 or [C] = 0 , set E 0 = 0 and C = C; in the first case, set j = j; in the latter case, fix an arbitrary j and note that [C] ≤ 0 − j . Then C satisfies the conditions in Lemma 5 for all negative curves in the preimage of P by this construction, and it does the same for all other curves contracted by π as it does not meet them. For what remains, we distinguish three cases. Case 1. The curve C does not meet any of the remaining (−2)-curves, and C.E ≤ 1 for all remaining (−1)-curves. Then (−K − C ) is nef by Lemma 5. But (−K − C ) 2 ≤ 4 − (r − 1) ≤ 0, so it cannot be big.

U. Derenthal and F. Wilsch
Case 2. The curve C meets one of the remaining (−2)-curves E. Then E ⊂ D by Lemma 5 (i). Since E is the strict transform of a curve in P 2 , its class satisfies As E is the strict transform of a curve on P 2 , its class verifies [E] ≥ [F ] for a (−2)-class [F ] of the same shape as in the previous case; hence, −K − D is not big or not nef.

Remark 9.
The assumption deg S ≤ 4 in Proposition 8 is necessary: Let S be a smooth del Pezzo surface of degree at least 5 that is a blowup of P 2 in at most 4 points in general position. Then the strict transform D of a line that meets precisely one of these points is an example of a nonnegative curve such that ω S (D) ∨ is big and nef. Proof. Let D be a reduced effective divisor such that −K − D is big and nef. By Proposition 8, D = E i has to be supported on negative curves. Consider the complete subgraph G of the Dynkin diagram on the vertices corresponding to components of D. By Lemma 5 (iii), each of its connected components is a path or a cycle. Let N 1 be the number of (−1)-curves in D, and N 2 be the number of (−2)-curves. Then v = N 1 + N 2 is the number of vertices of G, and denote by e its number of edges.
The self-intersection of the log-anticanonical divisor is As −K.E is zero for (−2)-curves and 1 for (−1)-curves, we get Since −K − D is big and nef, this self-intersection must be positive. If G is connected and not a cycle, then e = v − 1, so d − 2 − N 1 > 0. In case d = 4, this leaves us with N 1 ≤ 1, in case d = 3 with N 1 = 0, and in case d ≤ 2 with an immediate contradiction. In each case, the resulting divisors satisfy the asserted description using Remark 6 and that the graph is a path.
It remains to prove that G has to be connected and not a cycle. If G is not connected and does not contain a cycle, then (−K − D) 2  For N 1 = 0, we note that only Dynkin diagrams of type A, D, and E appear as intersection graphs of (−2)-curves, and these do not contain double edges nor more general cycles.
If If N 1 = 2, then d ≥ 3. In this case, the anticanonical model contracts (−2)-curves and maps (−1)-curves to lines. The resulting two lines then need to intersect with multiplicity 2, an impossibility.
Finally, if N 1 = 3, then d = 4. In this case, the anticanonical model φ : S → S = Q 1 ∩ Q 2 ⊂ P 4 is a (possibly singular) intersection of two quadrics, still contracting all (−2)curves and mapping all (−1)-curves to lines. The resulting three lines need to intersect pairwise. If they were contained in a plane P, this plane would intersect Q 1 in three lines, an impossibility. So the three lines intersect in a point Q. The tangent space at Q needs to contain each plane containing two of these lines, whence Q is singular. Then S → S factors through the blowup Y of P 4 in Q. The strict transforms of the lines do not intersect on Y, and thus the (−1)-curves on S do not intersect. It follows that each of them intersects a (−2)-curve above Q. Hence, the (−2)-curves are contained in D, and at least one of them needs to intersect three other negative curves in D. Now, Lemma 5 (iii) implies that −K − D cannot be nef.
Conversely, if D is one of the divisors in the statement, then it is nef by Lemma 5, and its self-intersection is positive by equation (14); hence, it is also big.

Passage to a universal torsor
As in the introduction, let S ⊂ P 4 Q be the singular quartic del Pezzo surface defined by the equations (1). By [12,15] (but using the notation and numbering of [11, Section 8]), a Cox ring of its minimal desingularization S is with grading for a certain basis 0 , . . . , 5 of Pic S. See Figure 4 for the dual graph of the divisors E i corresponding to η i . The minimal desingularization S can be described as a certain sequence of five iterated blowups of P 2 Q in rational points [11]: first blow up three points P 1 , P 2 , P 4 on a line l 3 , resulting in exceptional curves E 1 , E 2 and E 4 ; then blow up the intersection of E 4 with a line l 7 , resulting in an exceptional curve E 6 ; then blow up the intersection of E 6 with the strict transform of l 7 , resulting in an exceptional curve E 5 . With this description, E 3 is the strict transform of l 3 , E 4 that of E 4 , E 6 that of E 6 , E 7 that of l 7 , E 8 that of a general line through P 2 and E 9 that of a line through P 1 such that E 7 ,E 8 ,E 9 meet in one point, recovering the above grading using a basis as before Lemma 7.

U. Derenthal and F. Wilsch
With a point of view coming from S, the divisor D 1 = E 7 is the (−2)-curve on S above the singularity Q 1 on S, the divisor D 2 = E 3 + E 4 + E 6 is the sum of the (−2)-curves above Q 2 , the divisor D 3 = D 1 +D 2 = E 3 +E 4 +E 6 +E 7 is the sum of all (−2)-curves, and E 5 ,E 2 ,E 1 are the (−1)-curves that are the strict transforms of the three lines L 1 ,L 2 ,L 3 on S as in equation (4), respectively, while E 8 and E 9 correspond to the two further generators of the Cox ring. The divisors lie above the lines L 1 , L 2 and L 3 , respectively. Since V ⊂ S is the complement of the lines, which contain the singularities, its preimage V ⊂ S is the complement of the negative curves E 1 , . . . ,E 7 .
The irrelevant ideal of R is I irr = (η i ,η j ), where the product runs over all pairs i < j such that there is no edge between E i and E j in Figure 4. The sections m,Z -torsor. This morphism induces a 2 6 -to-1-correspondence (19), (20) hold, η 1 · · · η 7 = 0,}, where η 1 η 9 + η 2 η 8 + η 4 η 3 5 η 2 6 η 7 = 0, gcd(η i ,η j ) = 1 if E i and E j do not share an edge in Figure 4, and Proof. Since π is a G 6 m,Z -torsor so are its restrictions to the open subschemes U i . Integral points η ∈ Spec R Z are lattice points (η 1 , . . . ,η 9 ) ∈ Z 9 satisfying the equation in the Cox ring. Such a point is integral on the complement of V(I irr ) = V(η i ,η j )-the union running over all i,j which do not share an edge in Figure 4-if it does not reduce to any of the V(η i ,η j ) for any prime, that is, if the gcd-condition (19) holds.
We now turn to studying the log-anticanonical bundles and the height functions associated with them. Recall that the case D 6 can be reduced to D 5 by symmetry as in (8).

Lemma 12. The only nonzero reduced effective divisors D ⊂ S such that ω S (D) ∨ is big and nef are D i for i ∈ {1,2,4,5,6}. Consider the sets
of monomials in the Cox ring R of degree ω S (D i ) ∨ for i = 1,2,4,5, respectively. For η ∈ Z 9 satisfying equation (19), none of these sets can vanish simultaneously modulo a prime p. The respective log-anticanonical bundles are base point free.
The log-anticanonical bundle ω S (D 3 ) ∨ is big, but not nef, whence not base point free. It has a representation ω S (D 3 ) ∨ ∼ = L 1 ⊗ L ∨ 2 as a quotient of the nef bundles L 1 and L 2 whose sections are elements of degree 4 0 − 1 − 2 and 3 0 − 1 − 2 − 3 in the Cox ring,

respectively. Consider the sets of Cox ring elements
Then neither L 1 nor L 2 can vanish simultaneously modulo a prime p.
Proof. The first statement is a special case of Theorem 10.
For the first set, assume that p | η 8 η 9 for a prime p. Then p η 3 · · · η 6 , since the corresponding divisors E 3 , . . . ,E 6 share an edge with neither E 8 nor E 9 in Figure 4, while at most one of η 1 ,η 2 ,η 7 can be divisible by p. Hence, the second or third section is not divisible by p.
By the same arguments (replacing vanishing modulo p by vanishing over Q), the loganticanonical bundles ω S (D i ) ∨ for i = 1,2,4,5 and the bundles L 1 and L 2 are base point free, whence nef. On the other hand, ω S (D 3 ) ∨ is not nef since its intersection number with E 5 is −1.
By Lemma 12, these gcds are one, and so the claim follows for i ∈ {1,2}. Since U 3 and U 3 are the intersections of the respective open subschemes within the first two cases, the assertion follows for i = 3. The cases i = 4 and i = 5 can be proved using an analogous reformulation of the first two conditions in equation (7).  For η = (η 1 , . . . ,η 9 ) ∈ R 9 satisfying (18) and (20), let (3) and H i is the log-anticanonical height on S(Q) induced by the sections in Lemma 12.
The case i = 3 is more complicated. The log-anticanonical height function H 3 induced by the metrics on L 1 and L 2 satisfies by an analogous argument as in the previous cases. Now note that, for η ∈ Y 3 (Z) ∩ π −1 ( V )(Q), we can simplify this to Indeed, η 3 ,η 4 ,η 6 ,η 7 have absolute value 1 and the remaining variables absolute value at least one. Then
where #D i denotes the number of irreducible components of D i .
Proof. We combine Lemma 11 and Lemma 14. The #D i coordinates η j belonging to irreducible components E j ⊂ D i satisfy |η j | = 1. By symmetry, we can further assume that η j = 1, making the 2 6 -to-1-correspondence from Lemma 11 a 2 6−#Di -to-1correspondence.

Counting
In our counting process, we treat η 9 as a dependent variable using the torsor equation from (15), which we regard as a congruence modulo the coefficient η 1 of η 9 . First, we sum over η 8 and then over the remaining variables. Since this is similar to the case of rational points in [11], we shall be brief.
In the second case, using the second and the third height conditions, it is η1,η2,η5,η7 In the third case, using the second height condition, it is η1,η2,η5 The remaining cases are very similar.

Proof.
In the first case, by equation (18), the last height condition is

U. Derenthal and F. Wilsch
Hence, by [11, Lemma 5.1(4)], 1 (η 1 , . . . ,η 6 ; B) In the second case, we use In the third case, we use Therefore, [11,Proposition 4.3,Corollary 7.10] gives the result in the first three cases. The final cases are similar to the second and third cases.

Volume asymptotics
We must show that the real integrals V i,0 (B) in Lemma 17 grow of order B(log B) bi−1 .
To determine the integral over the first one, we remove the condition |x 0 (x 0 + x 3 )| ≤ 1, introducing an error of at most by using the symmetry in the signs of x 0 and x 3 . The last inequality implies that x 3 has a distance of at least 1/|x 0 | (which is ≥ 1) from −x 0 . Since x 0 > 0 and x 3 > −1, it cannot 22

U. Derenthal and F. Wilsch
be smaller, and thus −x 0 + 1/x 0 ≤ x 3 ≤ 1 holds. We thus get We can now integrate the first term in equation (25) over x 0 and get To treat the second term, we begin with a change of variables x 0 = x 0 + x 3 and add the condition |x 0 | ≤ 1, introducing an error of at most again using the symmetry of the integral. The third condition implies |x 3

and thus we get
(For the second inequality, note that x 0 − 1/x 0 ≤ 1 implies x 0 ≤ 2.) Thus, the second term of equation (25) is The condition |x 0 (x 0 − x 3 )| ≤ 1 is implied by the second and third condition, so we can remove it. Removing |x 0 − x 3 | ≤ 1 introduces an error of at most by the symmetry of the integral. The conditions imply −1 ≤ x 3 ≤ x 0 − 1 and thus Thus, the integral of the second summand of equation (25) is Since |x0|,|x2|,|x 2 0 /x2|≤1 adding equations (26) and (27) yields the desired result.

Lemma 21.
We have

U. Derenthal and F. Wilsch
Proof. Again, we apply the coordinate change (24), which shows that The integral over x 0 ,x 2 is 8 by equation (28). Restricting to positive η i introduces a factor of 2 5 . Substituting η i = B ti turns dη i /η i into log B dt i , and we thus arrive at This integral can be interpreted as the volume of a polytope, which we compute using Magma.
For the second case, using the first height condition yields We proceed as in the first case; here, we can compute the volume by hand. The final two cases are analogous.
For the third case, we observe that |η 1 η 2 η 2 5 | can be ignored in the definition of H 3 since it is the geometric average of |η 2 1 η 2 2 | and |η 4 5 |. Now the computation is very similar to the second case.
Plugging this into Lemma 17 (after applying Lemma 18 and Lemma 20 in the first and third cases) completes the proof of Theorem 1.

The leading constant
We show that Theorem 1 can be abstractly formulated as Theorem 2. Part of the leading constants (9) are p-adic Tamagawa volumes τ ( S,Di),p ( U i (Z p )) as defined in [7, § § 2.1. 10, 2.4.3]. These measures are similar to the usual Tamagawa volumes studied in the context of rational points, except for factors 1 Di p that are constant and equal to 1 on the set of p-adic integral points at almost all places (in fact, at all finite places in our cases). Over the reals, the analogous volumes, when evaluated on the full space of real points, would be infinite. Instead, residue measures τ i,DA,∞ supported on minimal strata D A (R) of the boundary divisors appear in the leading constant (10), cf. [7, § 2.1.12]. These can be interpreted as a density function for the set of integral points (100% of which are in arbitrarily small real-analytic neighborhoods of the boundary; hence, a density function has to be supported on the boundary), cf. [8, 3.5.8], or the leading constant of an asymptotic expansion of the volume of height balls with respect to τ ( S,Di),∞ , cf. [7,Theorem 4.7].
In addition, we have to compute factors α i,A as in equation (11) (cf. [25]), similar to Peyre's in the case of rational points [22]. Again, there is one of these factors associated with any minimal stratum A of the boundary.
The third set consists of points that are integral with respect to both Q 1 and Q 2 . Therefore, we obtain it as the intersection of the previous two sets. For the description of U 3 as a disjoint union, we start with the one of U 2 and intersect each set with U 1 . Here, |y| ≤ 1 implies |x| ≥ 1 since otherwise |xy 2 | < 1. Furthermore, |y| > 1 and |xy 2 | ≤ 1 implies |x| < 1; hence, |xy 2 | ≥ 1 must hold. Finally, |y| > 1 and |x + y| ≤ 1 implies |x| = |y| > 1.
The final two cases are analogous.
The final two cases are similar: for i = 4, the integrals over the two disjoint sets in Lemma 22 are 1 and p −1 , respectively, while, for i = 5, the integrals over the three disjoint sets are 1, p −1 and p −1 , respectively.
The remaining parts of the constant are associated with maximal faces of the Clemens complex. Recall from Section 1.2 and Figure 1 that the Clemens complex of the geometrically irreducible divisor D 1 consists of just one vertex E 7 . For D 2 , we have three vertices corresponding to its components, and two 1-simplices A 1 = {E 3 ,E 4 } and A 2 = {E 4 ,E 6 } between the intersecting exceptional curves (Figures 1 and 4). The Clemens complex for D 3 = D 1 + D 2 is the disjoint union of the previous two cases; its maximaldimensional faces are again A 1 and A 2 . For D 4 , they are A 1 , . . . ,A 4 , and for D 5 , they are A 1 ,A 2 ,A 5 (Figure 4).
For a face A of the Clemens complexes associated with D i , we set D A = E∈A E and Δ i,A = D i − E∈A E. For a maximal-dimensional face A of a Clemens complex, the adjunction isomorphism and a metric on the log-canonical bundle ω S (D i ) induce a metric on the bundle ω DA ⊗ O S (Δ i,A )| DA on D A . Since A is maximal, the canonical section 1 Δi,A does not have a pole on D A , so since D A (R) is compact, the norm 1 Δi,A | DA O S (Δi,A)|D A ,∞ is bounded on D A (R) for any metric. Hence, Again evaluating equation (30) in the image of (x,y), we get the volume |x| |x| max{1,|x|,|y|,|y(y + x)|} dy = R 1 max{1,|y 2 |} dy = 4, which we normalize by multiplying with c R = 2.
We conclude by noting that the classes of E 3 ,E 4 ,E 6 ,E 7 in Pic S are linearly independent; hence, rk Pic U i = rk Pic S −#D i (with #D i as in Lemma 15). This observation, Lemma 24 and Lemma 28 allow us to reformulate Theorem 1 as Theorem 2 for i ∈ {1, . . . ,5}. For the final case, we equip the log-anticanonical bundle ω S (D 6 ) ∨ with the metric pulled back from ω S (D 5 ) ∨ along the isomorphism 8; since all constructions in this section are invariant under metric-preserving isomorphisms, the theorem follows for i = 6.