  In this paper, we investigate the question of triviality of the rational Chow
groups of complete intersections in projective spaces and obtain improved
bounds for this triviality to hold. Along the way, we study the dimension and
nonemptiness of some Hilbert schemes of fat $r$-planes contained in a complete
intersection $Y$, generalizing well-known results on the Fano varieties of
$r$-planes contained in $Y$.
