Seminar in Algebraic Geometry and Number Theory

Abstract: We show that any rationally defined cubic hypersurface of dimension at least 29 contains a rational line. The main ingredient is a result on linear spaces on quadratic forms over suitable imaginary quadratic field extensions of Q, in combination with a recent result of Bernert and Hochfilzer concerning the solubility of cubic forms over imaginary quadratic number fields. This is joint work with Rainer Dietmann.