  We use the cut and paste relation $[Y^{[2]}] = [Y][\mathbb{P}^m] +
\mathbb{L}^2 [F(Y)]$ in $K_0(\text{Var}_k)$ of Galkin--Shinder for cubic
hypersurfaces arising from projective geometry to characterize cubic
hypersurfaces of sufficiently high dimension under certain numerical or
genericity conditions. Removing the conditions involving the middle Betti
number from the numerical conditions used extends the possible $Y$ to cubic
hypersurfaces, complete intersections of two quadric hypersurfaces, or complete
intersections of two quartic hypersurfaces. The same method also gives a family
of other cut and paste relations that can only possibly be satisfied by cubic
hypersurfaces.
