Arithmetic Study Group: Lines on cubic hypersurfaces
19 January 2021 13:00 in Zoom
Trevor Wooley has shown that every rational cubic hypersurface of dimension at least 35 contains a rational line. In this talk I want to report about recent joint work with Julia Brandes, reducing this 35 to 29 in the generic case of smooth cubic hypersurfaces. One of the key ingredients is a result by Browning, Heath-Brown and myself on intersections of cubic and quadric hypersurfaces. I also want to discuss the related problem of finding lines on cubic hypersurfaces defined over p-adic fields, to give explicit examples of smooth rational cubic hypersurfaces of dimension 9 not containing any rational line, and to mention a few applications of our results.
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