MUKAI MODELS AND BORCHERDS PRODUCTS.

Let Fg,n 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 Fg,n and the ring of certain orthogonal modular forms, and give applications to the birational type of Fg,n. We prove that the Kodaira dimension of Fg,n stabilizes to 19 when n is sufficiently large. Then we use explicit Borcherds products to find a lower bound of n where Fg,n has nonnegative Kodaira dimension, and compare this with an upper bound where Fg,n 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. [ABSTRACT FROM AUTHOR]