Everywhere local solubility for hypersurfaces in products of projective spaces

TL;DR

This paper demonstrates that a positive proportion of hypersurfaces in products of projective spaces over rationals are locally soluble everywhere, extending previous results and providing explicit proportions for certain genus 1 curves.

Contribution

It generalizes a theorem of Poonen and Voloch to a broader class of hypersurfaces and studies the local solubility of genus 1 curves in  imes , including explicit proportions.

Findings

A positive proportion of hypersurfaces are everywhere locally soluble.

Approximately 87.4% of genus 1 curves of bidegree (2,2) in  imes  are locally soluble.

The local solubility proportion over _p is a rational function of p.

Abstract

We prove that a positive proportion of hypersurfaces in products of projective spaces over $\mathbb{Q}$ are everywhere locally soluble, for almost all multidegrees and dimensions, as a generalization of a theorem of Poonen and Voloch. We also study the specific case of genus $1$ curves in $\mathbb{P}^1 \times \mathbb{P}^1$ defined over $\mathbb{Q}$, represented as bidegree $(2,2)$-forms, and show that the proportion of everywhere locally soluble such curves is approximately $87.4\%$. The proportion of these curves in $\mathbb{P}^1 \times \mathbb{P}^1$ soluble over $\mathbb{Q}_p$ is a rational function of $p$ for each finite prime $p$. Finally, we include some experimental data on the Hasse principle for these curves.