  This paper studies curves on quartic K3 surfaces, or more generally K3
surfaces which are complete intersection in weighted projective spaces. A
folklore conjecture concerning rational curves on K3 surfaces states that all
K3 surfaces contain infinite number of irreducible rational curves. It is known
that all K3 surfaces, except those contained in the countable union of
hypersurfaces in the moduli space of K3 surfaces satisfy this property. In this
paper we present a new approach for constructing curves on varieties which
admit nice degenerations. We apply this technique to the above problem and
prove that there is a Zariski open dense subset in the moduli space of quartic
K3 surfaces whose members satisfy the conjecture. Various other curves of
positive genus can be also constructed.
