Rational approximations on toric varieties

Equidistribution of rational points and the geometric sieve for toric varieties

“…In contrast to the harmonic analysis technique which allows to handle toric varieties over general number fields for (1), the universal torsor method has the advantage that the parametrisation of rational points is made in a totally explicit way that strongly ties to the combinatorial data of the structural fan. This method also proves powerful (and works over arbitrary number fields) in [25], in which the author studied Diophantine approximation of rational points, a problem of completely different nature. Salberger's universal torsor method [38] is the main ingredient of our approach to proving an effective version of Principle 1.1 -Theorem 1.3 and the geometric sieve -Theorem 1.4, which together with Principle 1.1 imply Principle 1.2, and some further applications will be exhibited in §1.3.…”

Erratum to “Approximating rational points on toric varieties”

“…This conjecture is known in some special cases, primarily in dimension 2: it was shown for split rational surfaces of Picard rank at most four in [McK07], cubic surfaces in [MR16], and blow-ups of the n-th Hirzebruch surface at special configurations of at most 2n points in [San19]. The conjecture was verified for special classes of toric varieties in [Hua21], namely when p is in the open torus and the pseudo-effective cone Eff(X) is simplicial. This conjecture was also recently proved by McKinnon and the second-named author [MS21] for smooth projective split toric surfaces X assuming Vojta's Main Conjecture [Voj87, Conjecture 3.4.3]; higher dimensional analogues of the conjecture were also shown [MS21].…”