Rational curves on K3 surfaces in $\mathbb P^1\times \mathbb P^1 \times \mathbb P^1$

We discuss Manin and Batyrev's notion of the arithmetic stratification of a variety, and, for an irreducible surface $V$ embedded in $\mathbb P^m$, compare it with the spectrum of degrees of rational curves on $V$. We study this spectrum for the class of K3 surfaces generated by smooth (2,2,2) forms in $\mathbb P^1\times \mathbb P^1 \times \mathbb P^1$.