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We prove a new Bertini-type Theorem with explicit control of the genus, degree, height, and the field of definition of the constructed curve. As a consequence we provide a general strategy to reduce certain height and rank estimates on abelian varieties over a number field $K$ to the case of jacobian varieties defined over a suitable extension of $K$.
For an algebraic number $alpha$ and $gammain mathbb{R}$, $h(alpha)$ be the (logarithmic) Weil height, and $h_gamma(alpha)=(mathrm{deg}alpha)^gamma h(alpha)$ be the $gamma$-weighted (logarithmic) Weil height of $alpha$. Let $f:overline{mathbb{Q}}to [0
We establish the Geometric Dynamical Northcott Property for polarized endomorphisms of a projective normal variety over a function field $mathbf{K}$ of characteristic zero. This extends previous results of Benedetto, Baker and DeMarco in dimension $1
Let $mathcal{X}$ be a regular projective arithmetic variety equipped with an ample hermitian line bundle $overline{mathcal{L}}$. We prove that the proportion of global sections $sigma$ with $leftlVert sigma rightrVert_{infty}<1$ of $overline{mathcal{
We introduce a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing over a finite field. Extending a result of Benoist, we prove that for a morphism $phi colon X to mathbb{P}^n$ such that $X$ i
We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $mathbf{K}$ of characteristic zero, improving results of Ingram.