Everywhere local solubility for hypersurfaces in products of projective spaces

We prove that a positive proportion of hypersurfaces in products of projective spaces over 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 P¹× P¹ defined over 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 P¹× P¹ soluble over 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.