Tom Fisher's Home Page

joint with Wei Ho and Jennifer Park

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 [PV04]. We also study the specific case of genus 1 curves in P¹ x 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 [BCF16], the proportion of these curves in P¹ x 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.

Everywhere local solubility for hypersurfaces in products of projective spaces   (24 pages)          

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We also make available the data files for the experiments described in Section 6, with H = 10, 30, 100, 300 and 1000. A Magma file carrying out some basic checks on this data is available here.