Motives of hypersurfaces of very small degree

Abstract:

We study the Chow motive (with rational coefficients) of a hypersurface $X$ in the projective space by using the variety $F(X)$ of $l$-dimensional planes contained in $X$. If the degree of $X$ is sufficiently small, we show that the primitive part of the motive of $X$ is the tensor product of a direct summand in the motive of a suitable complete intersection in $F(X)$ and the $l$-th twist $\mathbb {Q} (-l)$ of the Lefschetz motive.