Related papers: Manin's conjecture on toric varieties with differe…

We prove under some assumptions that the Tate conjecture holds for products of Fermat varieties of different degrees.

We use perfectoid spaces associated to abelian varieties and Siegel moduli spaces to study torsion points and ordinary CM points. We reprove the Manin-Mumford conjecture i.e. Raynaud's theorem. We also prove the Tate-Voloch conjecture for a…

The Torsion Anomalous Conjecture states that an irreducible variety $V$ embedded in a semi-abelian variety contains only finitely many maximal $V$-torsion anomalous varieties. In this paper we consider an irreducible variety embedded in a…

Given a nonsingular quartic del Pezzo surface, a conjecture of Manin predicts the density of rational points on the open subset of the surface formed by deleting the lines. We prove that this prediction is of the correct order of magnitude…

We investigate the geometrical structures of multipartite states based on construction of toric varieties. In particular, we describe pure quantum systems in terms of affine toric varieties and projective embedding of these varieties in…

For an optimal modular parametrization $J_0(n) \twoheadrightarrow E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, Manin conjectured the agreement of two natural $\mathbb{Z}$-lattices in the $\mathbb{Q}$-vector space $H^0(E,…

We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on…

The main result of the work ``The nilpotence conjecture in K-theory of toric varieties'' is extended to all coefficient fields of characteristic 0, thus covering the class of genuine toric varieties.

The purpose of this note is to prove Grothendieck's standard conjectures for the Fano variety of lines on a smooth cubic hypersurface in projective space.

We establish Manin's conjecture for a cubic surface split over Q and whose singularity type is 2A_2+A_1. For this, we make use of a deep result about the equidistribution of the values of a certain restricted divisor function in three…

We associate to any complete spherical variety $X$ a certain nonnegative rational number $\wp(X)$, which we conjecture to satisfy the inequality $\wp(X) \le \operatorname{dim} X - \operatorname{rank} X$ with equality holding if and only if…

Let X be a complete toric variety and let Y be a smooth projective variety with Picard number one. We prove that, if there exists a surjective morphism from X to Y, then Y is a projective space.

We discuss Manin's conjecture concerning the distribution of rational points of bounded height on Del Pezzo surfaces, and its refinement by Peyre, and explain applications of universal torsors to counting problems. To illustrate the method,…

A well-known conjecture of Orlov asks whether the existence of a full exceptional collection implies rationality of the underlying variety. We prove this conjecture for arithmetic toric varieties over general fields. We also investigate a…

Artin's conjecture is established for all forms that can be realised as a diagonal form on an hyperplane.

We say that a complete nonsingular toric variety (called a toric manifold in this paper) is over $P$ if its quotient by the compact torus is homeomorphic to $P$ as a manifold with corners. Bott manifolds (or Bott towers) are toric manifolds…

In this paper we determine the canonical arithmetic volume of hypersurfaces in smooth projective toric varieties. As a consequence, we prove a generalized Hodge index theorem on hypersurfaces in smooth projective toric varieties.

For a cycle of codimension 1 in a toric variety, its degree with respect to a nef toric divisor can be understood in terms of the mixed volume of the polytopes associated to the divisor and to the cycle. We prove here that an analogous…

Let A be an ample line bundle on a projective toric variety X of dimension n. We show that if l>=n-1+p, then A^l satisfies the property N_p. Applying similar methods, we obtain a combinatorial theorem: For a given lattice polytope P we give…

In this note we prove that any toric Fano manifold with nef tangent bundle is a product of projective spaces. In particular, it implies that Campana-Peternell conjecture hold for toric manifolds.