Everywhere local solubility for hypersurfaces in products of projective spaces

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 [25]. 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 $$87.4\%$$ . As in the case of plane cubics [2], 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.