Rational equivalence for line configurations on cubic hypersurfaces in P^5

Let $k$ be an uncountable algebraically closed field of characteristic zero, and let $\mathscr X$ be a nonsingular cubic hypersurface in $\mathbb P^5$ over $k$. We prove that, for a very general hyperplane section $\mathscr Y$ of the cubic $\mathscr X$, there exists a countable set $\Xi $ of closed points on the Prymian of $\mathscr Y$, such that, if $\Sigma $ and $\Sigma '$ are two linear combinations of lines of the same degree on $\mathscr Y$, then $\Sigma $ is rationally equivalent to $\Sigma '$ on $\mathscr X$ if and only if the cycle class of $\Sigma -\Sigma '$, as a point on the Prymian, is an element of $\Xi $.