Rational points and non-anticanonical height functions

Anticanonical Rational Surfaces

A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven over an algebraically closed field of arbitrary characteristic. Applications, to be treated in separate papers, include questions involving: points in good pos...

On Uniform Bounds for Rational Points on Non-rational Curves

We show that the number of rational points of height ≤ H on a non-rational plane curve of degree d is Od(H 2/d−δ), for some δ > 0 depending only on d. The implicit constant depends only on d. This improves a result of Heath-Brown, who proved the bound O (H2/d+ ). We also show that one can take δ = 1/450 in the case d = 3.

Seshadri Constants on Rational Surfaces with Anticanonical Pencils

We provide an explicit formula for Seshadri constants of any polarizations on rational surfaces X such that dim |−KX | ≥ 1. As an application, we discuss relationship between singularities of log del Pezzo surfaces and Seshadri constants of their anticanonical divisors. We also give some remarks on higher order embeddings of del Pezzo surfaces.

Non-archimedean Yomdin–gromov Parametrizations and Points of Bounded Height

We prove an analog of the Yomdin–Gromov lemma for p-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of p-adic subanalytic sets, and to ...

Distribution of Rational Points: A Survey