Integral points of bounded height on a certain toric variety

Abstract

We determine an asymptotic formula for the number of integral points of bounded height on a certain toric variety, which is incompatible with part of a preprint by Chambert-Loir and Tschinkel. We provide an alternative interpretation of the asymptotic formula we get. To do so, we construct an analogue of Peyre's constant α and describe its relation to a new obstruction to the Zariski density of integral points in certain regions of varieties.

Integral Points on Singular Del Pezzo Surfaces

“…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.…”

“…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…”

“…These Tamagawa numbers are multiplied with rational numbers α i,A , where A is a maximal-dimensional face of the Clemens complex for D i , depending on the geometry of certain effective cones, as in equation (11). Following [25…”

“…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 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].…”