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.