Everywhere local solubility for hypersurfaces in products of projective spaces

We prove that a positive proportion of hypersurfaces in products projective spaces over $${\mathbb {Q}}$$ are everywhere locally soluble, for almost all multidegrees and dimensions, as generalization theorem Poonen Voloch [25]. also study the specific case genus 1 curves {P}}^1 \times {\mathbb {P}}^1$$ defined , represented bidegree (2, 2)-forms, show soluble such is approximately $$87.4\%$$ . As plane cubics [2], these {Q}}_p$$ rational function p each finite prime p. Finally, we include some experimental data on Hasse principle curves.