Related papers: Uniform boundedness for rational points

The strong Bombieri-Lang conjecture postulates that, for every variety $X$ of general type over a field $k$ finitely generated over $\mathbb{Q}$, there exists an open subset $U\subset X$ such that $U(K)$ is finite for every finitely…

Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a…

We establish a locally uniform a priori bound on the dynamics of a rational function $f$ of degree $>1$ on the Berkovich projective line over an algebraically closed field of any characteristic that is complete with respect to a non-trivial…

We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves can be placed on a closed surface of genus g such that any two of the curves intersects at most once. Although the gap is…

We prove a global uniform Artin-Rees lemma type theorem for sections of ample line bundles over smooth projective varieties. This result is used to prove an Artin-Rees lemma for the polynomial ring with uniform degree bounds. The proof is…

In this paper we show the lower bound of the set of non-zero $-K^2$ for normal surface singularities establishing that this set has no accumulation points from above. We also prove that every accumulation point from below is a rational…

We prove that for any of a wide class of elliptic surfaces $X$ defined over a number field $k$, if there is an algebraic point on $X$ that lies on only finitely many rational curves, then there is an algebraic point on $X$ that lies on no…

There are no known failures of Bounded Negativity in characteristic 0. In the light of recent work showing the Bounded Negativity Conjecture fails in positive characteristics for rational surfaces, we propose new characteristic free…

Let $X$ be a smooth projective hypersurface defined over $\mathbb{Q}$. We provide new bounds for rational points of bounded height on $X$. In particular, we show that if $X$ is a smooth projective hypersurface in $\mathbb{P}^n$ with $n\geq…

Let $k$ be a rational congruence function field and consider an arbitrary finite separable extension $K/k$. If for each prime in $k$ ramified in $K$ we have that at least one ramification index is not divided by the characteristic of $K$,…

We consider the problem of deciding if a set of quantum one-qudit gates $\mathcal{S}=\{g_1,\ldots,g_n\}\subset G$ is universal, i.e if the closure $\overline{<\mathcal{S}>}$ is equal to $G$, where $G$ is either the special unitary or the…

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic zero, prove almost all cases in positive characteristic and improve the proofs of the previously…

We prove that there exist rational but not uniformly rational smooth algebraic varieties. The proof is based on computing a certain numerical obstruction developed in the case of compactifications of affine spaces. We show that for some…

In 1998, Bremner conjectured that elliptic curves over the rationals having long sequences of distinct rational points whose $x$-coordinates are in arithmetic progression, have large rank. This was proved some years ago in a strong form as…

A telegraphic survey of some of the standard results and conjectures about the set $C({\bf Q})$ of rational points on a smooth projective absolutely connected curve $C$ over ${\bf Q}$.

Let N_q(g) denote the maximal number of F_q-rational points on any curve of genus g over the finite field F_q. Ihara (for square q) and Serre (for general q) proved that limsup_{g-->infinity} N_q(g)/g > 0 for any fixed q. In their proofs…

We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a $\C^*$-action. For varieties with an isolated singularity, we show that the presence of sufficiently many rational curves outside the…

We prove uniform boundedness statements for semistable pure sheaves on projective manifolds. For example, we prove that the set of isomorphism classes of pure sheaves of dimension 2 that are slope semistable with respect to ample classes…

Given d in IN, we prove that all smooth K3 surfaces (over any field of characteristic p other than 2,3) of degree greater than 84d^2 contain at most 24 rational curves of degree at most d. In the exceptional characteristics, the same bounds…

In this short note we show that the uniform abc-conjecture over number fields puts strong restrictions on the coordinates of rational points on elliptic curves. For the proof we use a variant of the uniform abc-conjecture over number fields…