Everywhere local solubility for hypersurfaces in products of projective spaces

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.

Added a case to Conjecture 1.2 (and 2.4), and a few other minor changes. To appear in Research in Number Theory