The Manin-Peyre conjecture for smooth spherical Fano threefolds

The Manin-Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. The case N features a number of structural novelties; most notably, one may lose regularity of the ambient toric variety, the height conditions may contain fractional exponents, and it may be necessary to exclude a thin subset with exceptionally many rational points from the count, as otherwise Manin's conjecture in its original form would turn out to be incorrect.


Introduction
Let G be a connected reductive group over Q.A normal G-variety X is called spherical if a Borel subgroup of G has a dense orbit in X. Spherical varieties are a very large and interesting class of varieties that admit a combinatorial description by spherical systems and colored fans [Lu, BP, LV] generalizing the combinatorial description of toric varieties.Indeed, if the acting group G has semi-simple rank 0, then G is a torus.
If G has semi-simple rank 1, we may assume that G = SL 2 × G r m for some r ≥ 0. Let G/H = (SL 2 × G r m )/H be the open orbit in X and define Then the homogeneous space SL 2 /H ′ is spherical, and there are three possible cases: • either H ′ is a maximal torus (the case T ), • or H ′ is the normalizer of a maximal torus in SL 2 (the case N ), • or the homogeneous space SL 2 /H ′ is horospherical, in which case X is isomorphic (as an abstract variety) to a toric variety.In a recent paper [BBDG], the authors initiated a program to establish Manin's conjecture for (split models over Q of) smooth spherical Fano varieties, based on (a) the combinatorial description of spherical varieties, (b) the universal torsor method, and (c) techniques from analytic number theory including the Hardy-Littlewood circle method and multiple Dirichlet series.In particular, in case T , we confirmed Manin's conjecture for threefolds as well as for some higher-dimensional varieties.
Here we relied on Hofscheier's classification [Ho] of smooth spherical Fano threefolds over Q, which identifies four such cases having natural split models over Q.
In this paper we fully resolve the harder case N .This is achieved by a further development of the methods employed in [BBDG], and we proceed to describe the major new ingredients.
The first point concerns a general reformulation of Manin's conjecture as a counting problem.Let X be a smooth split projective variety over Q with big and semiample anticanonical class ω ∨ X whose Picard group is free of finite rank.Assume that its Cox ring R(X) is finitely generated with precisely one relation with integral coefficients.This defines a canonical embedding of X into a (not necessarily complete) ambient toric variety Y • ; it can be completed to a projective toric variety Y such that the natural map Cl Y → Cl X = Pic X is an isomorphism and −K X is big and semiample on Y .Under the assumption that Y is regular, Sections 2 -4 in [BBDG] (culminating in [BBDG,Propositions 3.7 & 4.11]) provide a general scheme to parametrize the rational points on X in terms of the universal torsor and to express the Manin-Peyre constant in terms of the Cox coordinates.The corresponding counting problem is in many cases amenable to techniques from analytic number theory.The assumption that Y is regular holds for smooth Fano threefolds of type T , but may fail in the case of type N .Part 1 of the present paper generalizes the passage to the universal torsor to varieties X for which Y is not necessarily regular.This result is independent of the theory of spherical varieties and should therefore have applications elsewhere.
The second point is of analytic nature.The universal torsor of spherical varieties of semi-simple rank 1 and type T has a defining equation of the form x 11 x 12 − x 21 x 22 − some monomial = 0, which needs to be analyzed subject to rather complicated height conditions.The fact that we have a decoupled bilinear form in four variables is crucial for the method and allows in particular an auxiliary soft argument based on lattice considerations.For type N , the equation takes the form x 11 x 12 − x 2 21 − some monomial = 0. From an analytic perspective, this may be very delicate.A considerable portion of this paper is devoted to the investigation of the particular equation (1.1) x 11 x 12 − x 2 21 − x 31 x 32 x 2 33 = 0 with variables constrained to dyadic boxes Here X is large, and 1 ≤ Y ≤ X, and we need an asymptotic formula for the number of integer solutions where the error term saves a fixed power of min(Y, X/Y ).It is conceivable that a modern variant of the circle method (like [HB]) can handle this, but this is not straightforward.There are considerable uniformity issues, since we need to deal simultaneously with the cases when Y is small, say Y = exp((log log X) 2 ) (in which case the equation looks roughly like a sum of two squares and a product), and Y is large, say Y = X/ exp((log log X) 2 ) (in which case the equation looks roughly like a sum of a square and two products).To keep track of uniformity, we will not use the circle method directly but instead apply Poisson summation to selected variables depending on the ranges of parameters.This is ultimately more or less equivalent to the delta-symbol method of Duke-Friedlander-Iwaniec [DFI], but it is in this case a more convenient packaging.
The shape of equation (1.1) offers a new feature that was not present in [BBDG], and for which in fact very few examples are known.In the special case where −x 31 x 32 is a square, the equation (1.1) describes a split quaternary quadratic form over Q of signature (2, 2), i.e., the sum of two hyperbolic planes.In this case it is well-known (see e.g.[HB]) that the asymptotic formula contains an additional logarithm.In particular, Manin's conjecture in its original form turns out to be wrong, and instead we need to prove a "thin subset version" of Manin's conjecture: we first remove a certain portion from the variety with exceptionally many points and then confirm the conjecture for the remaining set.More precisely, there should be a thin subset T of the set of rational points X(Q) such that, for an anticanonical height H, as in Manin's conjecture with Peyre's constant c.We refer the reader to the general discussion of this phenomenon in [LST], and to the (to our knowledge) only example of a smooth Fano variety [BHB] for which a thin version of Manin's conjecture has appeared in print.Soon after this work has been circulated in manuscript form, Fano threefolds of Mori-Mukai type II.25 have been discussed in the preprint [BBH], providing yet another example where the thin set version of the Manin-Peyre conjecture is true.
We now describe our results in more detail.As explained in [BBDG,§11], there are three smooth spherical Fano threefolds over N over Q that are neither horospherical nor equivariant compactifications of G 3 a ; we construct split models X 1 , X 2 , X 3 over Q as in Table 1.
rk  1. Smooth Fano threefolds of type N that are spherical, but not horospherical More precisely, let X 1 the blow-up of the quadric Q = V(z 11 z 12 − z 2 21 − z 31 z 32 ) ⊂ P 4 Q in the conic C 33 = V(z 31 , z 32 ).This is a smooth Fano threefold of type II.29 in the Mori-Mukai classification.We have a fibration X 1 → P 1 that is defined in Cox coordinates by (x 11 : • • • : x 33 ) → (x 31 : x 32 ).As explained above, we must remove the thin subset Let W 2 = P 1 Q × P 2 Q with coordinates (z 01 : z 02 ) and (z 11 : z 12 : z 21 ).Let C 32 be the curve V(z 02 , z 11 z 12 − z 2 21 ) on W 2 and let X 2 be the blow-up of W 2 in C 32 .This is a smooth Fano threefold of type III.22.We will see later that the height conditions (cf.(12.2) below) contain fractional exponents α ν ij ; see also [BT].Let T 2 ⊂ X 2 (Q) be the set of rational points where at least one Cox coordinate is zero.
Let X 3 be the blow-up of the quadric Q = V(z 11 z 12 − z 2 21 − z 31 z 32 ) ⊂ P 4 Q in the points P 01 = V(z 11 , z 12 , z 21 , z 31 ) and P 02 = V(z 11 , z 12 , z 21 , z 32 ).Its type is III.19.As before, let T 3 ⊂ X 3 (Q) be the set of rational points where at least one Cox coordinate is zero.
In Sections 2.3 and 11, we will define natural anticanonical height functions H j : X j (Q) → R, j = 1, 2, 3, using the anticanonical monomials in their Cox rings.We write N j (B) = N Xj (Q)\Tj ,Hj (B).
Theorem 1.1.The Manin-Peyre conjecture holds for the smooth spherical Fano threefolds X 2 , X 3 of semisimple rank one and type N , and a thin version of the Manin-Peyre conjecture holds for X 1 .More precisely, there exist explicit constants C 1 , C 2 , C 3 such that for 1 ≤ j ≤ 3. The values of C j are the ones predicted by Peyre.
Together with previous results, this covers all types of smooth spherical Fano threefolds.
Acknowledgements.The authors thank the anonymous referee for useful remarks and suggestions.

Part 1. Metrics and heights via Cox rings and universal torsors
Universal torsors and Cox rings were introduced and studied by Colliot-Thélène and Sansuc [CTS1,CTS2] and Cox [Cox].
If a variety X has a finitely generated Cox ring with one relation, this gives a description of X as a hypersurface in a toric variety Y .The description of height and Tamagawa measures in [BBDG, Part 1] relies on the assumption [BBDG,(2.3)]that Y can be chosen to be regular, which does not hold in our examples X 2 , X 3 .Here, we describe one approach how to circumvent this problem; see also [BT].
Our constructions of metrizations, heights, and Tamagawa measures on universal torsors follow the work of Salberger [Sal]; see also [BBS] and [BBDG].This should be compared to the closely related work of Peyre on universal torsors for Manin's conjecture [Pey2,Pey3,Pey4].

Heights and parametrization
We start with a situation similar to (but in several respects more general than) [BBDG,Section 2].Let Y • be a smooth split toric variety over Q.We do not assume that Y • is complete, but we still assume the weaker property that Y • has only constant regular functions.Let be its Cox ring, where x 1 , . . ., x J correspond to the torus invariant prime divisors in Y • .Let 0 ̸ = Φ ∈ R(Y • ) be a homogeneous equation, and let X ⊆ Y • be the corresponding subvariety.
We assume that X is smooth and projective, with big and semiample anticanonical class −K X .Moreover, we assume that every torus orbit in Y • meets X.We also assume that the pullback map Pic Y • → Pic X sends a big and semiample divisor class L • to −lK X for some l ∈ Z ≥0 .Finally, we assume Φ ∈ Z[x 1 , . . ., x J ].
Remark 2.1.A situation as above can be naturally obtained starting from a smooth split projective variety over Q with big and semiample anticanonical class −K X and finitely generated Cox ring where x 1 , . . ., x J is a system of pairwise nonassociated Pic X-prime generators and Φ ̸ = 0.By [ADHL,3.2.5], there exists a canonical embedding into an ambient toric variety Y • , and it satisfies all the above assumptions.Moreover, the pullback map Pic Y • → Pic X is an isomorphism, and we may take L • = −K X under this identification.
If Σ is the fan of any toric variety, we write Σ max for the set of maximal cones and Σ(1) for the set of rays.
Let Σ • be the fan of Y • .The generators x 1 , . . ., x J ∈ R(Y • ) are in bijection to the rays ρ ∈ Σ • (1); we also write x ρ for x i corresponding to ρ.
Let Y be a completion of Y • such that the pullback map Cl Y → Cl Y • = Pic Y • is an isomorphism and L • is big and semiample on Y ; let Σ be the fan of this toric variety Y .We have Σ(1) = Σ • (1) and R(Y ) = R(Y • ).For example, we may choose the unique Y such that L • is ample on Y , which exists by [ADHL,Proposition 2.4.2.6]; we call this Y the standard small completion of Y • .
We do not assume that Y is regular (in contrast to [BBDG,Section 2]).Let ρ : Y ′′ → Y be a toric desingularization of Y that does not change the smooth locus of Y .Such a desingularization can be obtained by suitably subdividing the singular cones in the fan Σ of Y into a smooth fan Σ ′′ .
Let Y ′ ⊂ Y ′′ be a toric subvariety with Y • ⊂ Y ′ .This means that the fan Σ ′ of Y ′ is a subfan of Σ ′′ and contains Σ • .If we write ρ J+1 , . . ., ρ J ′ for the rays in Σ ′ (1) \ Σ(1), we can write for the Cox ring of Y ′ , again with the correspondence between rays ρ i and generators y i .
Since X is smooth, we can identify it with its strict transform under ρ : Y ′′ → Y ; it is a hypersurface in Y ′ defined by a homogeneous equation Φ ′ that is obtained from Φ by homogenizing (therefore, For simplicity, we assume that every cone in Σ ′ is the face of a maximal cone. Let We define the rational section 2.1.Universal torsors and models.For universal torsors and Cox rings of toric varieties, see [CTS1,§4], [Cox], [Sal,§8]. Let π ′ : Y ′ 0 → Y ′ be the universal torsor as in [Sal,§8].Then the restriction π ′ : X 0 → X to the preimage of X ⊂ Y ′ is a torsor, but not necessarily universal since the acting torus (dual to Pic Y ′ ) may be too large.The toric varieties 0 are defined by the vanishing of x ρ for all ρ ∈ S ′ j , where the primitive collections (2.2) S ′ 1 , . . ., S ′ r ′ ⊆ Σ ′ (1) are all sets with the following property: S ′ j ̸ ⊆ σ(1) for all σ ∈ Σ ′ , but for every proper subset S ′′ j of S ′ j , there is a σ ∈ Σ ′ with S ′′ j ⊆ σ(1).
From the fans and their maps, we may construct Z-models π again as in [Sal,§8].By our assumption that Φ has integral coefficients, we obtain Z-models π We assume: (2. 3) The toric variety Y ′ is chosen such that X is proper over Spec Z.
This is always possible since for Y ′ = Y ′′ the scheme Y ′ is projective over Spec Z.
Proposition 2.2.We have Proof.The proof is as in [BBDG,Proposition 2.2] by (2.3).The referee kindly pointed out that an argument is also contained in the work of Peyre [Pey2,Pey3,Pey4].□ 2.2.Metrization of ω −1 X via Poincaré residues.Let L be any big and semiample divisor class on Y ′ such that L| X = −lK X .Then there exists a uniquely determined divisor E with support supp E in the exceptional locus of ρ such that The following lemma shows that (after possibly enlarging l) such an L can always be found.
Lemma 2.3.After suitably enlarging l, there exists a big and semiample divisor class on Y ′ such that L| X = −lK X .
Proof.Since L • is ample, it is Q-Cartier on Y .Hence, after replacing l by a positive multiple, we may assume that L • is Cartier on Y .Let L ′′ be the pullback ρ * (L • ).Then L ′′ is big and semiample on Y ′′ , and moreover L ′′ | X = −lK X .The same is then true for L = L ′′ | Y ′ .□ As before, let S ⊂ Σ ′ (1) be such that {deg(y ρ ) | ρ / ∈ S} is a basis of Pic Y ′ .Hence there is a unique (not necessarily effective) Weil divisor . The characters defined by z S ρ for ρ ∈ S form a basis of M = Hom(U, G m ).By [CLS,Proposition 8.2.3], we have a global nowhere vanishing section s Y ′ of ω Y ′ (D 0 ) (defined up to sign, independent of S; we have s Y ′ = Ω 0 / ρ∈Σ ′ (1) y ρ for Ω 0 from [CLS,(8.2.3) With y D0 = ρ∈Σ ′ (1) y ρ , for each S as above, defines a nowhere vanishing global section.On U S , we have Let P l be a set of polynomials F ∈ Q[y 1 , . . ., y J ′ ] of degree L. For each polynomial F ∈ P l of degree L, let D(F ) be the effective divisor on Y ′ of class deg F = L defined by F (in Cox coordinates).If X ⊂ supp D(F ), then we remove F from our set P l ; clearly, this does not change the results that we want to prove.For X ̸ ⊂ supp D(F ), we define We have the Poincaré residue map Res : on the open subset of X S where ∂Φ ρ /∂z S ρ0 ̸ = 0, for any ρ 0 ∈ S. Furthermore, Res l : Lemma 2.4.The section Res(ϖ S ), Res l (ϖ F ) extends uniquely to a nowhere vanishing global section of ω X (D(S) ∩ X), ω l X (D(F ) ∩ X), respectively.Proof.For Res(ϖ S ), this is as in [BBDG,Lemma 2.3], i.e., similar to [BBS,Lemma 13].The computation for Res l (ϖ F ) is analogous, using , which we can also view as a global section of ω −1 X , ω −l X , respectively.
Lemma 2.5.The sections τ □ From now on, we assume: (2.5) P l only contains monic monomials (of degree L), and for each σ ∈ Σ ′ max there exists F ∈ P l such that supp div F does not meet U σ .
Then the set P l is in particular basepoint-free.
We define a v-adic norm/metric on ω −1 X by ∥τ (P )∥ v := min . Lemma 2.6.Let p be a prime such that X is smooth over Z p .On ω −l X , the p-adic norm ∥ • ∥ p defined by ∥τ (P )∥ p := min Proof.See [BBDG,Lemma 3.3].For such a τ not vanishing in P , let Q ∈ P l be such that which is positive by Lemma 2.5 and the fact that the set P l is basepoint-free.Hence τ Q does not vanish in P , and For each F ∈ P l , the section τ F extends to a global section τ F of ω −l

X/Zp
, and ω −l X/Zp is generated by the set of all these τ F as an O X -module.The reason is that everything required for the definition of τ F above can also be defined over Z p .For the existence of the Poincaré residue map in this case, we refer to [KK,Definition 4.1].
For every F ∈ P l , we have τ F τ Q (P ) p ≤ 1 as in the computation above.This implies τ F (P ) = a F τ ξ (P ) for some a F ∈ Z p in the Q p -module ω −1 X (P ), and hence also τ σ (P ) = a F τ Q (P ) in the Z p -module P * (ω −1

X/Zp
) is generated by τ Q (P ).Hence ∥τ Q (P )∥ * p = 1 by definition of the model norm.We conclude 3. Height functions.For P ∈ X(Q), we define (2.6) for a local section τ of ω −1 X not vanishing in P .Remark 2.7.Let F , F 0 be homogeneous elements of the same degree in the Cox ring of Y ′ .If F 0 does not vanish in P , then F/F 0 can be regarded as a rational function on X that can be evaluated in P ∈ X(Q).
Lemma 2.8.We have for any polynomial F 0 of degree L not vanishing in P .
Proof.We have P ∈ X S (Q) for some S as above.We can compute H(P ) with τ := τ S by Lemma 2.5.Applying the which is our claim in the case F 0 := y lD(S)+E .The general case follows using the product formula.□ We lift the height function H to X 0 by composing it with π : X 0 (Q) → X(Q), giving Lemma 2.9.For P 0 ∈ X 0 (Q), we have As in the proof of [BBDG,Lemma 3.5], for F 0 of degree L not vanishing in P and F ∈ P l , we have (F/F 0 )(P ) = F (P 0 )/F 0 (P 0 ) if we compute (F/F 0 )(P ) as in Remark 2.7 and also regard F, F 0 as regular functions on X 0 (Q) that can be evaluated in P 0 .We apply this to Lemma 2.8 to obtain Then we use the product formula.□ In its integral model, this simplifies as follows.
Corollary 2.10.For P 0 ∈ X 0 (Z), we have Proof.This is analogous to [Sal,Proposition 11.3] and [BBDG,Corollary 3.6].For a prime p, we have P 0 (mod p) in X 0 (F p ).There is a σ ∈ Σ ′ max such that y ρ (P 0 (mod p)) ̸ = 0 ∈ F p for all ρ / ∈ σ(1) since X 0 is defined by the irrelevant ideal in X 1 .Choose Q ∈ P l such that supp div Q does not meet U σ .Then we have Q(P 0 (mod p)) ̸ = 0 ∈ F p , and hence |Q(P 0 )| p = 1.Therefore, we have max F ∈P l |F (P 0 )| 1/l p = 1 and only the archimedean factor in Lemma 2.9 remains.□ 2.4.Counting problem.The following result parametrizes the set N X(Q)\T,H (B) in terms of integral points on the universal torsor of the ambient toric variety Y ′ (which is given by its Cox ring (2.1) and the primitive collections (2.2)), the equation Φ ′ , and the monomials in P l .The resulting counting problem is amenable to methods of analytic number theory.
Proposition 2.11.Let X be a variety as in Section 2, let Y ′ be a toric variety satisfying (2.3), let L be a divisor class as in Section 2.2, and let P l be a set of monomials satisfying (2.5).
Let T be an arbitrary subset of X(Q).Then using the notation (2.1) and (2.2).
Proof.This follows from Proposition 2.2 and Corollary 2.10.□

Peyre's constant
We keep the notation and assumptions of Section 2. In addition, we assume from now on that we are in the situation of Remark 2.1.In particular, the pullback map Pic Y • → Pic X is an isomorphism and X is split.
3.1.Tamagawa measures.By [Pey1,(2.2.1)] and [Sal,Theorem 1.10 is the natural projection and z S ρ0 is expressed in terms of the other coordinates using the implicit function for Φ S .Proof.This is analogous to the proof of [BBDG,Proposition 4.1].However, we work with ∥τ F0 (P )∥ v for F 0 = x lD(S)+E and use F 0 (z S ) = 1 in our affine coordinates on X S (Q v ).At the end, comparing the definitions of ϖ S and ϖ F shows ϖ F /(ϖ S ) l = y lD(S)+E /F , hence τ F Res(ϖ S ) l = τ F /(τ S ) l = F/y lD(S)+E , and hence the integrals in (3.1) are equal.□ Remark 3.2.Since we have assumed that every cone in Σ ′ is the face of a maximal cone, the open subvarieties X S for S = σ(1) with σ ∈ Σ ′ max cover X. Remark 3.3.If we are in the special case where X is covered by open subvarieties X S with S ⊂ Σ(1) for S = σ(1) with σ ∈ Σ ′ max , then (3.2) gives the same formula for the v-adic density as we would have obtained by working with R(X) directly (since the additional coordinates z S ρ are 1 for all ρ ∈ Σ ′ (1) \ Σ(1); up to the description of the height function defined via the monomials F ∈ P l ⊂ Q[y 1 , . . ., y J ′ ], in which the additional variables y J+1 , . . ., y J ′ corresponding to ρ ∈ Σ ′ (1) \ Σ(1) are set to z S ρ = 1).For example, this is the case if X is covered by the open subvarieties X S for S ∈ Σ • max .3.2.Measures on the torsor.Analogously to [BBDG,(4.4)],we obtain a v-adic measure m v on X 0 (Q v ), which is explicitly (for sufficiently small subsets M v ) , where π ρ0 is the projection to all coordinates y ρ with ρ ̸ = ρ 0 and where y ρ0 is expressed in terms of these coordinates using the implicit function theorem.

3.3.
Comparison to the number of points modulo p ℓ .As in [BBDG,§4.4],for any prime p and l ∈ Z >0 , we have In particular, the additional variables x ρ indexed by ρ ∈ Σ ′ (1)\Σ(1) (obtained via the desingularization of the ambient toric variety) appear here.
We now define c ∞ as in [BBDG,(4.11)].We work without the additional coordinates indexed by Σ ′ (1) \ Σ(1).Therefore, we can use the results from [BBDG,(4.11)],considering X to be embedded into the possibly singular toric variety Y .Since we are working with a more general height function, we change the definition of H ∞ to where the additional coordinates indexed by Σ ′ (1) \ Σ(1) are set to 1 on the right-hand side.
In order to proceed as in [BBDG,(4.11)],it remains to show that the monomials in P l with the additional coordinates set to 1 are homogeneous of degree −lK X ∈ Pic(X) = Cl(Y ′ ).

Lemma 3.6. If we write
In other words, it does not matter whether we compute the numbers α σ ρ in Pic X or Pic Y ′ .The same is true for the numbers b ρ,ρ ′ and b ρ ′ .Proof.For every ρ ∈ Σ ′ (1), we have We can therefore directly compute the pullbacks to X of all D ρ .
For every ρ ∈ Σ(1), the pullback of deg( Finally, the pullback of l −1 • L is −K X .By pulling back the defining equations for (α ′ ) σ , the result follows.The argument for b ρ,ρ ′ and b ρ ′ is the same.□ Lemma 3.7.Let F ∈ P l , and let F Y be the corresponding monomial where the additional coordinates indexed by Proof.We write y kρ ρ and, since the monomial F is of degree L, we obtain The proof now proceeds exactly as in the proof of Lemma 3.6.□ Proposition 3.8.In the notation above and assuming (3.3), we have Proof.First, we note that [BBDG,Lemma 4.8] is still valid.Moreover, we observe that the additional variables z S ρ for ρ ∈ Σ ′ (1) \ Σ(1) are 1 in the expression (3.2) for µ ∞ (X(R)) for S = σ(1).Therefore, the expected real density ω ∞ = µ ∞ (X(R)) has the same description as in [BBDG,(4.10)].Taking into account Lemma 3.7, the statement and proof of [BBDG,Proposition 4.10] stay the same.□

Part 2. Analysis of a diophantine equation
The counting problem in Corollary 11.1(a) corresponding to the variety X 1 is rather delicate and not covered by the general method of [BBDG].This part of the paper is devoted to a detailed investigation.

Elementary bounds
For ξ ∈ Z \ {0} we consider the equation (4.1) x 11 x 12 + x 2 21 + ξ 2 x 31 x 32 x 2 33 = 0, x 31 x 32 ̸ = −□ where all variables are non-zero integers.For X = (X ij ) with In many cases it will improve the readability considerably to relabel the variables, and it will be convenient to refer to (4.1) in the form Consequently, we will write X = (A, B, C, Y, W, Z).We generally write Using this notation, we start with some elementary bounds.Proposition 4.1.We have follows from a simple divisor argument.Alternatively, we can fix y, w, z so that which then determines a, b up to a divisor function.The equation for c defines an interval of length The claim follows now from (4.2) and ( 4.3) after taking suitable geometric means, namely min(AB

□
We refine this argument a bit as follows.For 0 < ∆ ≤ 1 and X = (X ij ) let N * ξ (X, ∆) be the set of solutions to (4.1) with the same size conditions as N ξ (X) except that for one index (ij) ∈ {( 21), (31), ( 32), ( 33 We also define N ξ (X, ∆) to the be the same size conditions as N ξ (X) except that 1 2 X 11 ≤ |x 11 | ≤ X 11 is replaced with X 11 ≤ |x 11 | ≤ X 11 (1 + ∆).The following proposition investigates N * ξ (X, ∆), while N ξ (X, ∆) comes up in Proposition 6.2.Proposition 4.2.We have We distinguish two cases.If it is not the c-variable that is restricted, then by the same argument as above we have and the claim follows.
If the restricted variable is c, then we have similarly This completes the proof.□ Remarks: 1) It will be convenient to introduce the following short-hand notation for expressions like those in Proposition 4.2.For With this notation, the bounds in Propositions 4.1 and 4.2 involve X ( 1 2 , 1 4 , 1 4 ) .2) In order for N ξ (X) to be non-zero, we must have

Character sums
We consider the following two character sums.For a, c, z, ξ Let τ denote the divisor function.
Lemma 5.1.We have Proof.We have The w-sum vanishes unless a(z 2 ξ 2 ,a2) a2 | h 2 , so we obtain , a 2 where S(., ., .) is the Kloosterman sum.If h 1 = h 2 = 0, then the claim follows by the formula In general we use Weil's bound Lemma 5.2.We have where d 2 runs over all discriminants (positive or negative).If a is odd, then Let us first consider the case k 1 = k 2 = 0. We split the modulus d 2 = u2 ρ into an odd part and a power of two.The α, c, z-sum becomes * By the well-known evaluation of the Gauß sum, the first c, z-sum equals Summing this over α, we see that only the contribution of (x, u) = □ survives, and the first α, c, z-sum For the second α, c, z-sum modulo powers of 2 we argue similarly, but we have to distinguish a few cases.Recall first that for odd α we have This vanishes, unless (x, 2 ρ ) = □ in which case it equals .
This is equivalent to the formula given in the lemma.We now turn to the estimation for general k 1 , k 2 and for simplicity restrict ourselves to odd a, as in the statement of the lemma.In this case we can evaluate the two Gauß sums in c, z simply by completing the square, and we obtain .
Let δ denote the conductor of χ (x,d2) * and write Then by the well-known evaluation of quadratic character sums, the α-sum is bounded by .
We decompose uniquely (x, d 2 ) = t 1 t 2 such that t 1 is the largest square coprime to t 2 .Then δ = rad(t 2 ), δ 1 = t 2 /rad(t 2 ), δ 2 = t 1 , so that we obtain the upper bound and the claim follows.□ We also recall the following standard estimate.
where the sum runs over all discriminants.The implied constants depend only on ε.
Proof.This is standard by Mellin inversion and the convexity bound for Dirichlet L-functions (with Euler factors at primes dividing g removed) Suffice it to say that the (two-sided) Mellin transform of V is entire and satisfies for all A > 0. The second bound follows from quadratic reciprocity as follows: By quadratic reciprocity we have χ ∆ (n) = χ ñ(∆) for n > 0 where ñ is the discriminant computed as follows: write n = 2 a y with y odd, and let y * denote the discriminant satisfying |y * | = y.Let a ′ = a + 2 if a = 2 and a ′ = a otherwise.Then ñ = 2 a ′ y * .If n < 0 then χ ∆ (n) = χ ñ(∆)χ ∆ (−1).In this way, the second bound follows from the first by detecting the condition ∆ ≡ 0, 1 (mod 4) by characters.□ Remark: Using the strongest available uniform subconvexity bounds, the exponents 1/4 can be replaced with 1/6.

Upper bound estimates
For X = (A, B, C, Y, W, Z) and ξ ∈ Z \ {0} we recall the definition of N ξ (X) and N ξ (X, ∆) from Section 4, and we now define a smooth version of the former.Let Ω ≥ 3 be a parameter.We choose an even, smooth, non-negative test function (Strictly speaking, Ñξ (X, Ω) depends on V and not only on Ω, but this small abuse of notation is convenient.)Fix a small constant λ ≤ 10 −6 and consider X, ξ satisfying (6.1) We will see later that these are the critical size conditions.
Our aim in this section is to establish the following three estimates that we will prove simultaneously.Recall the notation (4.4).Proposition 6.1.For X, ξ satisfying (6.1) we have Proposition 6.2.For X, ξ satisfying (6.1) and 0 < ∆ < 1 we have Proposition 6.3.For X, ξ satisfying (6.1) and Ω ≥ 3 we have Y where M 2 is given by (6.4) below.We also have where M3 is defined in (6.12) below.
The rest of this section is devoted to the proof.We will occasionally use the following notation: for a positive integer n let √ n + denote the smallest integer whose square is a multiple of n.
We have trivially N ξ (X) ≤ Ñξ (X, Ω) for every Ω ≥ 3 and We apply two different strategies.We first apply Poisson summation in y, w and then Poisson summation in c, z.In order to obtain Proposition 6.2, we will occasionally replace V (a/A) with the characteristic function on A ≤ |a| ≤ A(1 + ∆).In the interest of a reasonably compact presentation, we will not introduce extra notation for this.
6.1.Poisson summation in y, w.We first add the contribution of wy = −□ to (6.2).As in the proof of Proposition 4.1, we see by a divisor argument that this infers an error of at most O(C(Y W ) 1/2 Z∥X∥ ε ) with an implied constant depending only on ε.Then we apply Poisson summation y, w to obtain with S ξ as in (5.1) and Let for arbitrary N > 0 and for arbitrary j ∈ N 3 0 .
Proof.We observe that the volume of the (y, w)-region defined by The first claim follows now by repeated integration by parts, the second by differentiating under the integral sign.There is an important subtlety: we combine each application of the operator z∂ z with a partial integration in y, i.e.
This completes the proof.□ We now write the right hand side of (6.3) as M 1 + M 2 where M 1 is the off-diagonal contribution (h 1 , h 2 ) ̸ = (0, 0) and M 2 is the diagonal contribution h 1 = h 2 = 0.By Lemma 5.1 we have (6.4) We postpone the analysis of M 2 and investigate first M 1 .By Lemma 5.1 and Lemma 6.4 we have By several applications of (6.1), we see that Note that up until now we have not used any property of the weight V (a/A) except that it restricts a ≪ A. In particular, (6.5) continues to hold with any Ω if V (a/A) is replaced by the characteristic function on A ≤ |a| ≤ A(1 + ∆).
We can now complete the proof of the first half of Proposition 6.3 by recalling (6.3) and noting that (6.1) implies As in the proof of Proposition 4.2 we reduce the power of C using (4.5), which completes the proof of the first half of Proposition 6.3.We now turn to M 2 as given in (6.4).We will evaluate this asymptotically later, but for now content ourselves with an upper bound given by Together with (6.5) for Ω = 3 we obtain (6.7) After this interlude we now return to (6.6) and estimate the right hand side by For notational simplicity let us write d ξ = d/(d, ξ 2 ).Then we can continue to estimate The d-sum is Combining this with the off-diagonal contribution (6.5) and choosing Ω = 3, we have shown our first important bound 6.2.Poisson summation in c, z.We now return to (6.2) and apply Poisson summation in c, z getting with T as in (5.2) and Let Lemma 6.5.For |a| ≍ A we have for arbitrary N, j 1 , j 2 ∈ N 0 .
Proof.As in Lemma 6.4 we observe that the volume of the (c, z)-region defined by |c| ≍ C, |z| ≍ Z, |c 2 + xz 2 | ≍ AB is trivially bounded by O(CZ), but also by O(AB/|x| 1/2 ).Indeed, if x > 0, this is the volume of the ellipse c 2 + xz 2 ≪ AB, while for x < 0 we have c = |x|z 2 + O(AB) for fixed |z| ≍ Z, which has volume ≪ AB/(|x| 1/2 Z).Taking the geometric mean, we bound the (c, z)-volume by O((CZAB) 1/2 /|x| 1/4 ).The claims follow now by repeated partial integration and differentiation under the integral sign.As in the proof of Lemma 6.4, each application of x∂ x is coupled with an integration by parts in z. □ As before we decompose where M1 is the off-diagonal contribution (k 1 , k 2 ) ̸ = 0 and M2 is the diagonal contribution k 1 = k 2 = 0.For notational simplicity let .
In the following we write (a, b ∞ ) := max n (a, b n ) and [d 1 , d 2 ] for the least common multiple of d 1 and d 2 .Using Lemma 5.2 and Lemma 6.5 we obtain We write k 2 1 ξ 2 yw = −k 2 2 + αd 2 .Since yw ̸ = −□ and (k 1 , k 2 ) ̸ = (0, 0), we have α ̸ = 0, and moreover α ≡ −k 2 2 (mod d 1 ).Once α and k 2 are chosen, the variables y, w, k 1 are determined up to a divisor function.We conclude the upper bound We now invoke (6.1) several times to conclude that Thus we obtain the simplified bound (6.9) Note that in order to derive this bound we did not use any property of the weight V (a/A), except that it bounds a ≪ A. In particular, (6.9) continues to hold for all Ω, if V (a/A) is replaced with the characteristic function on A ≤ |a| ≤ A(1 + ∆).
On the other hand, again by Lemma 5.2 we have where d 2 runs over all (positive or negative) discriminants and the factor 2 comes from the fact that V is even and we have used the decomposition |a| = d 1 |d 2 | from Lemma 5.2.
Before we manipulate this further, we complete the proof of Proposition 6.2.Replacing V (d 1 d 2 /A) in the previous display by the characteristic function on A ≤ |d 1 d 2 | ≤ A(1 + ∆), we obtain by a simple divisor estimate the upper bound for the right hand side of (6.10).By (6.1), the factor τ (ξ 2 ) can be absorbed into ∥X∥ ε .Combining this with (6.9), we obtain Together with (6.7) we obtain Using (4.5), we replace the second appearance of C 1/2 with (AB) . By several applications of (6.1) we have which completes the proof of Proposition 6.2.

We further decompose M2 = M3 + M4
where M3 is the contribution d 2 = □ and M4 is the contribution d 2 ̸ = □.We have (6.12) We will later evaluate this asymptotically, but for now we content ourselves with the upper bound To bound M4 , we insert a smooth partition of unity localizing |d 2 | ≍ D 2 , say, so that where D 2 ≪ A runs over powers of 2 and for a suitable smooth compactly supported function W .We estimate M4 (D 2 ) in two ways.First we re-insert the contribution of yw = −□ at the cost of an error and then sum over y, w with Lemma 5.3 and 6.5 getting a bound (6.14) Obviously this majorizes the previous error term.
Alternatively, we can also re-insert the contribution of d 2 = □ at the cost of an error and then sum over d 2 with Lemma 5.3 and 6.5 getting a bound (6.15) which again majorizes the previous error.
On the other hand, choosing Ω = 3, we also invoke (6.13) to obtain our second important bound We now combine this with (6.8) to conclude The proof of Proposition 6.1 is now completed by (6.11).

An asymptotic formula
We now upgrade the previous upper bound to an asymptotic formula for N ξ (X).The main term features the singular series and the singular integral that would follow from a formal application of the circle method.With this in mind we define (7.1) For later purposes we compute this as an Euler product.We have c,y,w,z (mod q) e d(c 2 + ξ 2 wyz 2 q = q 1 q 4 * d (mod q) c (mod q) ξ 2 yz 2 ≡0 (mod q) e dc 2 q .
If p is a prime and n ≥ 1, then a simple combinatorial argument shows that the number of pairs y, z For a prime p let r p = v p (ξ) denote the p-adic valuation of ξ.Evaluating geometric sums, we finally conclude In particular, the Euler product is absolutely convergent and satisfies We also define the singular integral for a tuple X = (X 11 , X 12 , X 22 , X 31 , X 32 , X 33 ) with X ij ≥ 1 as That the α-integral is absolutely convergent follows from the estimates For future reference we state the similar bounds Our aim in this section is to prove the following asymptotic formula.Let 3) = (5/8, 1/8, 1/4).(7.5) Proposition 7.1.The exists δ > 0 with the following property.For X = (X ij ), ξ satisfying (6.1) and any ε > 0 we have The rest of this section is devoted to the proof.As a first step we estimate the effect of smoothing.To this end we recall the definition of N ξ (X, ∆) and N * ξ (X, ∆): the former restricts one of the variables c, y, w, z to a small interval, the latter the variable a.By symmetry and (6.1), the bound in Proposition 6.2 holds also when b is restricted to a small interval.Combining Propositions 4.2 and 6.2, we obtain ) .
An evaluation of N ξ (X, Ω) is given in Proposition 6.3, and we proceed to evaluate the main terms M 2 and M3 defined in (6.4) and (6.12).

Computation of M
where By symmetry we can restrict a, c, z to be positive at the cost of a factor 8. We denote by the Mellin transform of F ξ .It is holomorphic in all three variables (since V has compact support), and by Lemma 6.4 and partial integration we have for all N > 0 where By (6.1) we have S, U ≪ Ω∥X∥ 10λ .Let us also define By Mellin inversion we have By a long, cumbersome and uninspiring, but completely straightforward computation based on geometric series we can compute L ξ (s, u, v) as an Euler product.If v p (ξ) = r p , then where the p-Euler factor of H ξ (s, u, v) is given by For p ∤ ξ (i.e.r p = 0) this simplifies considerably as In particular, we see that L ξ (s, u, v) is holomorphic in Shifting contours to the left in (7.7) we conclude that again (6.1) and the now familiar device based on (4.5), we can write the error term as (7.9) By definition and using symmetry again to remove the factor 8, we have By Fourier inversion we have ) db dy dw da dz dc dα.
It remains to remove the smoothing and quantify the error from replacing V with the characteristic function on [1/2, 1].By (7.3) and (7.4), we see that (7.10) Combining this with (7.9), (7.6), the first part of Proposition 6.3 and choosing Ω = min(A, C, Z) 1/50 , we have shown

Computation of M3 .
The argument for M3 is similar.Recall from (6.12) that where Ψ satisfies the bounds of Lemma 6.5.Recall that all variables run over positive integers, except for y, w that run over positive and negative integers.We first add back the contribution wy = −□ at the cost of an error (7.12) ≪ τ (ξ) by estimating trivially the contribution of all variables.
Since V is even, we can rewrite M3 , up to the error (7.12), as 4 ) where now all variables run over positive integers. Let As before, we denote by G ξ (s, u, v) the Mellin transform of G ξ (a, y, w); it is entire in all three variables and by Lemma 6.5 satisfies which is absolutely convergent in ℜu, ℜv > 1, ℜs > −1/2.Then by Mellin inversion we have M3 = (2) (2) (2) As before, we analyze Lξ (s, u, v) by computing its Euler product expansion.A similarly long and cumbersome computation yields where H ξ (s, u, v) ≪ τ (ξ) is holomorphic in the same region (7.8) and H ξ (0, 1, 1) = E ξ .Shifting contours, we conclude as before which also contains the error term from (7.13).Unraveling the definitions and using symmetry to remove the factor 4 and the ±-sign, we see that G ξ (0, 1, 1) = F ξ (0, 1, 1), so that by (7.10) we get Combining this with (7.6) and the second part of Proposition 6.3 and choosing Ω = min(Y, W, A) 1/50 , we have shown It remains to combine (7.11) and (7.14).As in (6.11) we conclude This completes the proof of Proposition 7.1.

Introducing the height conditions
Let P > 1 be a large parameter.Let as before X = (A, B, C, Y, W, Z), where all entries are restricted to powers of 2. The condition that the entries of X are powers of 2 will remain in force throughout this section.Let x = (a, b, c, y, w, z) ∈ N 6 , g = (η, ξ) ∈ N 2 .For given x, g let X = X (P, g, x) be the set of tuples X satisfying max(aA, bB, cC) 2 max(yY, wW Fix some sufficiently small λ > 0. We call a pair (X, g = (η, ξ)) bad if (6.1) is violated (i.e., one of these inequalities does not hold), otherwise we call it good.We call it very good if the following stronger version of (6.1) holds: Clearly there are at most O((log P ) 6 ) tuples X (the entries being powers of 2) satisfying (8.1).Moreover, it is easy to see that there are at most (8.3) ≪ log H(log log P ) 6 such tuples satisfying (8.1), (8.2) and Let X bad (P, g, x) be the set of X ∈ X (P, g, x) such that (X, g) is bad.Let X H (P, g, x) be the set of X ∈ X (P, g, x) such that (8.4) holds.Finally let X * (P, g, x) = X bad (P, g, x) ∪ X H (P, g, x).
Both X * (P, g, x) and N (P, g, x) depend on H, but this is not displayed in the notation.Our main result in this section is Proposition 8.1.For 1 ≪ H ≤ P and 0 < λ < 1 we have Proof.We first claim that (8.6) Let δ = λ/100 and suppose that (8.7) N ξ (X) ≥ P 1−δ+ε .
We show that this implies that (6.1) holds, so X is good, and since there are only O(P ε ) tuples X, this implies the claim.From Proposition 4.1 and (8.1) we have Contemplating this sequence of inequalities, we conclude from (8.7) that (8.8) which also implies that all of the three blocks AB, C 2 , Y W Z 2 must be within B 20δ .This shows our claim that X must satisfy (6.1) with λ = δ/100.This establishes (8.6).We complement this with a second bound.From Proposition 4.1 and (8.1) we have Combining this with (8.6), we conclude This is acceptable for (8.5).
For the contribution of X ∈ X H (P, g, x) \ X bad (P, g, x) we observe that Proposition 6.1 is available.We note that upon using (8.1).By (8.3), this is acceptable for the very good tuples.For tuples that are good, but not very good, the previous inequality along with the same argument as leading to (8.8) shows for such X.Since there are at most O((log P ) 6 ) such tuples, the proof is complete.□ Part 3. Proof of Theorem 1.1

Geometry
Table 2 contains the nine smooth spherical Fano threefolds over Q that are not horospherical (since the horospherical smooth Fano threefolds are all either toric or flag varieties; see [Ho,§6.3]).The notation T and N in [Ho,Table 6.5] and in our Table 2 refers to the cases described at the beginning of the introduction (Section 1) and in [BBDG,§10.2We proceed to describe the three N cases X 1 , X 2 , X 3 in Table 2 that are not equivariant G 3 acompactifications [HM] in more detail.From the description in the Mori-Mukai classification, we can construct a split form over Q in each case.We then recall from Hofscheier's list the description using the Luna-Vust theory of spherical embeddings.
The three varieties will be equipped with an action of G = SL 2 × G m .Let ε 1 ∈ X(B) always be a primitive character of G m composed with the natural inclusion X(G m ) → X(B).9.1.X 1 of type II.29.Consider P 4 Q with coordinates (z 11 : z 12 : z 21 : z 31 : z 32 ) and the hypersurface Let X 1 be the blow-up of Q in C 33 .This is a smooth Fano threefold of type II.29.We may define an action of which turns Q into a spherical variety.The following description using the Luna-Vust theory of spherical embeddings can be easily verified.The lattice M has basis (α + ε 1 , α − ε 1 ).We denote the corresponding dual basis of the lattice N by (d 1 , d 2 ).Then there is one color with valuation d = d 1 + d 2 , and the valuation cone is given by the variety X 1 is a spherical G-variety and the blow-up morphism X 1 → Q can be described by a map of colored fans.The following figure illustrates this.We obtain a projective ambient toric variety Y 1 .From the description of Σ max in [BBDG,§10.3],we deduce that Y 1 is smooth and that −K X1 is ample on Y 1 .Hence assumption [BBDG,(2.3)]holds, and we work with Q with an additional variable q, and let Q ′ = V(z 11 z 12 − z 2 21 − z 31 z 32 , q 2 − z 31 z 32 ) ⊂ P 5 Q .The covering map Q ′ → Q given by forgetting q induces a covering map of blow-ups X ′ 1 → X 1 .The image of the last map is the set which is therefore thin; in particular the set T 1 from the introduction is also thin.(2α, ε).We denote the corresponding dual basis of the lattice N by (d, ε * ).Then there is one color with valuation 2d = 1 2 α ∨ , and the valuation cone is given by V = {v ∈ N Q : ⟨v, α⟩ ≤ 0}.Since the curve C 32 is G-invariant, the variety X 2 is a spherical G-variety, and the blow-up morphism X 2 → W can be described by a map of colored fans.The right-hand arrow in the following figure illustrates this.The dotted circles in the colored fan of X 2 specify the standard small completion Y 2 of the ambient toric variety Y • 2 .We note that Y 2 is singular and that it is not possible to obtain a smooth small completion with the construction from [BBDG,§10.3]Here, u 31 = −d 1 and u 32 = −d 2 are the valuations of V(z 31 ) and V(z 32 ), respectively, and u 01 and u 02 are the valuations of the exceptional divisors E 01 over P 01 and E 02 over P 02 , respectively.

Counting problems
Applying the first part of this paper, we obtain the following counting problems, in which T j is always the subset of X j (Q) where all Cox coordinates are nonzero and, in case of X 1 , where x 31 x 33 ̸ = −□.For simplicity, we write N j (B) for N Xj (Q)\Tj ,Hj (B) as in the introduction, and we write {x, y} to mean x or y.Proof.For X 1 , we argue as in [BBDG] since the ambient toric variety Y 1 is regular.
For X 2 , we apply Proposition 2.11 and obtain the counting problem .
For X 3 , we similarly obtain The height condition is given by the monomials In this counting problem, we observe that this equation together with gcd(z 1 z 2 , x 31 x 32 ) = 1 implies z 1 = ±1 and z 2 = ±1.The torsor equation also allows us to simplify the coprimality conditions.□ Remark 11.2.The varieties X 1 , X 2 and X 3 are as in Remark 2.1, and in each case we have chosen L as in the proof of Lemma 2.3.After having eliminated the additional variables in the proof of Corollary 11.1, we have obtained the same monomials P 1 (x), P 2 (x), P 3 (x) as if we had directly applied [BBDG] (disregarding that Y is singular).In this case, [BBDG,Lemmas 3.8,3.9,and 3.10] still apply, with the difference that the vertices of the polytopes are not necessarily integral.Moreover, [BBDG,Lemma 4.7] is also valid without the assumption that Y is smooth.
12. Application: Proof of Theorem 1.1 12.1.The analytic machinery.Theorems 8.4, 9.2 and 10.1 in [BBDG] provide an asymptotic formula for counting problems as in Corollary 11.1 under various assumptions and show in addition that the shape of the asymptotic formula agrees with the Manin-Peyre prediction.We need a small variation of these results that we state in full detail for the reader's convenience.Suppose that we are given a diophantine equation (12.1) for certain nonnegative exponents α ν ij whose variables are restricted by coprimality conditions (12.3) denote the number of integral solutions to (12.1) with nonzero variables x j subject to (12.2) and (12.3).With these data, we define the following quantities.For g ∈ as in [BBDG,(8.11),(8.14)].
As in [BBDG,(5.1)-(5.6)], for b ∈ N k define the (formal) singular series the (absolutely convergent) singular integral [BBDG,(3.6),(7.1) -(7.3)] define the block matrix where [BBDG,(7.5)].Let J be a set of subsets of pairs (i, j) with 0 ≤ i ≤ k, 1 ≤ j ≤ J i .For H ≥ 1, some small fixed constant 0 < λ < 1 and b, y ∈ N J let N b,y (B, H, J ) be the number of solutions x ∈ (Z \ {0}) J satisfying the conditions (12.4) and at least one of the conditions min (12.5) Let S y (B, H, J ) denote the set of all x ∈ [1, ∞) J that satisfy (12.5) and the M inequalities in the second part of (12.4).We choose a maximal linearly independent set of R rows Z 1 , . . ., Z R of the matrix (A 1 A 2 ).Let Z R+1 , . . ., Z J be the remaining rows of (A 1 A 2 ).As in [BBDG,(8.23),(8.24)]let B = (b kl ) ∈ R (J−R)×R be the unique matrix with Under Hypothesis 2 below (cf.(12.10)), the last row (A 3 A 4 ) of A can be written as a linear combination of Z Suppose these R rows are indexed by a set I of pairs (i, j) with 0 ≤ i ≤ k, 1 ≤ j ≤ J i with |I| = R.As in [BBDG,(9.1) and let F be the affine (R−1)-dimensional hypersurface Φ * (t) = 0 over R. Let χ I be the characteristic function on the set and define the surface integral cf.[BBDG,(9.3),(8.36),(8.34)].
For any vector ζ satisfying (12.7),where we allow more generally ζ i ≥ 0, and for arbitrary ζ 0 > 0, we also assume that the system of J + 1 linear equations 12.16) in M variables has a solution σ ∈ R M >0 .
Let σ ∈ Σ max be the cone generated by the rays corresponding to x 11 , x 21 , x 31 , x 32 , let ρ 0 be the ray corresponding to x 11 , and let ρ 1 be the ray corresponding to x 02 ; then conditions (3.3) are satisfied.The correct shape of the leading constant now follows from Proposition 3.5 and Proposition 3.8.
1 (mod 4).For each discriminant we denote by χ D = (D/.) the corresponding quadratic character.It is primitive if and only if D is a fundamental discriminant.If d ∈ N is odd we write d * for the unique discriminant with |d * | = d.For an odd number d we write ϵ d = χ −4 (d) ∈ {1, i}.

Table 2 .
]. Smooth Fano threefolds that are spherical, but not horospherical 9.2.X 2 of type III.22.Let W = P 1 Q × P 2 Q with coordinates (z 01 : z 02 ) and (z 11 : z 12 : z 21 ).Let C 32 be the curve V(z 02 , z 11 z 12 − z 2 21 ) on W .Let X 2 be the blow-up of W in C 32 .This is a smooth Fano threefold of type III.22.We may define an action of G = SL 2 × G m on W by in this case.We may, however, construct a resolution of singularities Y ′′ 2 → Y 2 which does not affect X 2 .This is illustrated by the left-hand arrow in the figure above.9.3.X 3 of type III.19.Consider P 4 Q with coordinates (z 11 : z 12 : z 21 : z 31 : z 32 ) and the hypersurface Q = V(z 11 z 12 − z 2 21 − z 31 z 32 ) ⊂ P 4 Q .It contains the points P 01 = V(z 11 , z 12 , z 21 , z 31 ), P 02 = V(z 11 , z 12 , z 21 , z 32 ).Let X 3 be the blow-up of Q in P 01 and P 02 .This is a smooth Fano threefold of type III.19.Since P 01 and P 02 are G-invariant, the variety X 3 is a spherical G-variety and the blow-up morphism X 3 → Q can be described by a map of colored fans.The right-hand arrow in the following figure illustrates this.