arXiv subjects

Katerina Santicola

Publications and source records attributed to Katerina Santicola.

4 recordsLinked to original sources

arXiv2026

Refined obstructions to local-global principles for 0-cycles

We introduce new `refined' obstructions to local-global principles for 0-cycles on algebraic varieties over number fields. Assuming finiteness of relevant Tate--Shafarevich groups, we show that the Hasse principle and weak approximation for 0-cycles on generalised Kummer varieties and bielliptic surfaces are controlled by obstructions of this new type. As an additional application of our refined obstructions, we answer a question of Zhang about the relationship between the Brauer--Manin and connected descent obstructions for 0-cycles. We also show that a Corwin--Schlank style refined obstruction set coincides with the set of global 0-cycles, conditionally on the Section Conjecture.

math.AG↗

arXiv2026

Quartic del Pezzo surfaces over $\mathbb{F}_p(t)$ without quadratic points

We construct an infinite family of quartic del Pezzo surfaces over $\mathbb{F}_p(t)$ with no quadratic points, for all primes $p\neq 2$. This answers a question of Colliot--Thélène, Creutz and Viray in the negative, which asks whether every quartic del Pezzo surface has quadratic points over $C_2$ fields. We exhibit a Brauer--Manin obstruction on the variety parametrising lines associated to the quartic del Pezzo surface.

math.NT↗

arXiv2024

Curves with prescribed rational points

Given a smooth curve $C/\mathbb{Q}$ with genus $\geq 2$, we know by Faltings' Theorem that $C(\mathbb{Q})$ is finite. Here we ask the reverse question: given a finite set of rational points $S\subseteq \mathbb{P}^n(\mathbb{Q})$, does there exist a smooth curve $C/\mathbb{Q}$ contained in $\mathbb{P}^n$ such that $C(\mathbb{Q})=S$? We answer this question in the affirmative by providing an effective algorithm for constructing such a curve.

math.NT↗

arXiv2022

Reverse Engineered Diophantine Equations over $\mathbb{Q}$

Let $\mathscr{P}_\mathbb{Q}=\{ α^n \; : \; α\in \mathbb{Q}, \; n \ge 2\}$ be the set of rational perfect powers, and let $S \subseteq \mathscr{P}_\mathbb{Q}$ be a finite subset. We prove the existence of a polynomial $f_S \in \mathbb{Z}[X]$ such that $f(\mathbb{Q}) \cap \mathscr{P}_\mathbb{Q}=S$. This generalizes a recent theorem of Gajović who recently proved a similar theorem for finite subsets of integer perfect powers. Our approach makes use of the resolution of the generalized Fermat equation of signature $(2,4,n)$ due to Ellenberg and others, as well as the finiteness of perfect powers in non-degenerate binary recurrence sequences, proved by Pethő and by Shorey and Stewart.

math.NT↗