On the frontiers of polynomial computations in tropical geometry

We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as counting the number of connected components. We characterize the borderline between tractable and hard computations by proving $\mathcal{NP}$-hardness and #$\mathcal{P}$-hardness results under various strong restrictions of the input data, as well as providing polynomial time algorithms for various other restrictions.Comment: 17 pages, 5 figures. To appear in Journal of Symbolic Computatio

On the density of rational and integral points on algebraic varieties

We prove a conjecture of Heath-Brown on the number of rational points of bounded height for a large class of projective varieties.Comment: 25 page

Birational geometry of algebraic varieties, fibred into Fano double spaces

We develop the quadratic technique of proving birational rigidity of Fano-Mori fibre spaces over a higher-dimensional base. As an application, we prove birational rigidity of generic fibrations into Fano double spaces of dimension $M\geqslant 4$ and index one over a rationally connected base of dimension at most $\frac12 (M-2)(M-1)$. An estimate for the codimension of the subset of hypersurfaces of a given degree in the projective space with a positive-dimensional singular set is obtained, which is close to the optimal one.Comment: 30 pages, the final versio

Varieties with too many rational points

We investigate Fano varieties defined over a number field that contain subvarieties whose number of rational points of bounded height is comparable to the total number on the variety.Comment: 23 page

The density of rational points on non-singular hypersurfaces, II

For any integers $d,n \geq 2$, let $X \subset \mathbb{P}^{n}$ be a non-singular hypersurface of degree $d$ that is defined over $\mathbb{Q}$. The main result in this paper is a proof that the number $N_X(B)$ of $\mathbb{Q}$-rational points on $X$ which have height at most $B$ satisfies $N_X(B)=O_{d,\varepsilon,n}(B^{n-1+\varepsilon}),$ for any $\varepsilon>0$. The implied constant in this estimate depends at most upon $d, \varepsilon$ and $n$