Mukai models and Borcherds products

Let be the moduli space of n-pointed $K3$ surfaces of genus g with at worst rational double points. We establish an isomorphism between the ring of pluricanonical forms on and the ring of certain orthogonal modular forms, and give applications to the birational type of . We prove that the Kodaira dimension of stabilizes to $19$ when n is sufficiently large. Then we use explicit Borcherds products to find a lower bound of n where has nonnegative Kodaira dimension, and compare this with an upper bound where is unirational or uniruled using Mukai models of $K3$ surfaces in g≤ 20. This reveals the exact transition point of Kodaira dimension in some~g.