Poincaré institute

Reinventing rational points

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Rational lines on cubic hypersurfaces

Julia Brandes

One of the most intensely studied question in the intersection of analytic number theory and algebraic geometry concerns the existence and distribution of rational points on cubic hypersurfaces, but the analogous question regarding lines or higher-dimensional linear spaces is far less understood. In this talk, we will show that every smooth cubic hypersurface of projective dimension at least 29 contains a rational line, superseding earlier bounds due to Dietmann and Wooley. This is joint work with Rainer Dietmann.