Motives of hypersurfaces of very small degree

TL;DR

This paper investigates the Chow motive of low-degree hypersurfaces in projective space, revealing a decomposition involving the variety of linear spaces contained in the hypersurface and a Lefschetz twist.

Contribution

It provides a new decomposition of the Chow motive for hypersurfaces of very small degree using the geometry of linear spaces contained in them.

Findings

Primitive part of the motive decomposes into a tensor product involving F(X)

Shows the motive is related to a complete intersection in F(X)

Establishes a link between hypersurface motives and linear space varieties

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 Q(-l) of the Lefschetz motive.