Families of rational curves on holomorphic symplectic varieties and applications to 0-cycles

We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of K3[ⁿ]-type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed n, we show that there are only finitely many polarization types of holomorphic symplectic variety of K3[ⁿ]-type that do not contain such a uniruled divisor. As an application we provide a generalization of a result due to Beauville-Voisin on the Chow group of 0-cycles on such varieties.