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.

Equidistribution of rational points and the geometric sieve for toric varieties

“…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 ).…”

“…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…”

Diagonal quartic surfaces with a Brauer-Manin obstruction

“…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 , . .…”

“…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 .…”

Sums of four squareful numbers

“…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.…”