2020
DOI: 10.48550/arxiv.2001.09815
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Hyperbola method on toric varieties

Abstract: We develop a very general version of the hyperbola method which extends the known method by Blomer and Brüdern for products of projective spaces to a very large class of toric varieties. We use it to count Campana points of bounded log-anticanonical height on many split toric Q-varieties with torus invariant boundary. We apply the strong duality principle in linear programming to show the compatibility of our results with the conjectured asymptotic.

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Cited by 7 publications
(12 citation statements)
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References 14 publications
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“…In [8], based on Salberger's approach and an analytic method about multivariate Dirichlet series attached to certain arithmetic functions [9], la Bretèche proves a refinement of Manin's conjecture (1) with power-saving secondary terms. Subsequent work of Pieropan and Schindler [35] deals with the distribution of Campana points on toric varieties. On the other hand, by [42,Theorem] and [14,Theorem 1.3], smooth projective split toric varieties satisfy purity of strong approximation, i.e., for any such W as in Principle 1.2, W(k) is dense in W(A k ).…”
Section: Empiricism and Main Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…In [8], based on Salberger's approach and an analytic method about multivariate Dirichlet series attached to certain arithmetic functions [9], la Bretèche proves a refinement of Manin's conjecture (1) with power-saving secondary terms. Subsequent work of Pieropan and Schindler [35] deals with the distribution of Campana points on toric varieties. On the other hand, by [42,Theorem] and [14,Theorem 1.3], smooth projective split toric varieties satisfy purity of strong approximation, i.e., for any such W as in Principle 1.2, W(k) is dense in W(A k ).…”
Section: Empiricism and Main Resultsmentioning
confidence: 99%
“…Let us choose an admissible pairing σ (1) = {ρ r+1 , • • • , ρ r+d }, and let {n ∨ ρ r+1 , • • • , n ∨ ρ r+d } be the corresponding dual base. According to the exact sequence (35), the restriction of π : X 0 → X to the affine neighbourhood U σ ≃ A n with respect to the parametrisation given by σ is…”
Section: Construction and Liftingmentioning
confidence: 99%
“…We will apply Lemma 4.17 to the case when U is defined by inequalities of the type i x α i i ≤ T for α i ≥ 0. It that case it can be reinterpreted as a form of the hyperbola method on toric varieties [PS20] which only works if the function f (x 1 , . .…”
Section: (49)mentioning
confidence: 99%
“…In this section we will also prove a lemma which can be interpreted as a form of the hyperbola method for toric varieties [PS20] for functions f (x 1 , • • • , x n ) which can be written as a product of single-variable functions f i (x i ). The advantage of this new form is that it can also be applied when x i ≤T f i (x i ) is not of order of magnitude a power of T .…”
Section: Introductionmentioning
confidence: 99%
“…In [20], Pieropan and Schindler establish the Campana-Manin conjecture for complete smooth split toric varieties satisfying an additional technical assumption, by developing a very general version of the hyperbola method. In [29], Xiao treats the case of biequivariant compactifications of the Heisenberg group over Q, using the height zeta function method.…”
Section: Introductionmentioning
confidence: 99%