Pairs of points and lines on cubics

I'll tell about a joint work with Evgeny Shinder. In Grothendieck ring of varieties there is a linear relation between classes of cubic hypersurface, its singular locus, its variety of lines, and its symmetric square (set of unordered pairs of points). The proof is surprisingly simple, but specialization of the relation to known motivic measures allows to find easily various invariants of variety of lines, such as number of lines on (possibly singular) cubic surface defined over various fields (of complex or real numbers, over number fields), Hodge numbers of variety of lines and the relation between Hodge structures of the variety of lines and of the cubic, etc. Besides, this computation provides yet another evidence to our conjecture that relates derived categories of coherent sheaves on the cubic and on its variety of lines.