MUKAI MODELS AND BORCHERDS PRODUCTS

Let F-g,F- n be the moduli space of n -pointed K 3 surfaces of genus g with at worst rational double points. We establish an isomorphism between the ring of pluricanonical forms on F-g,F- n and the ring of certain orthogonal modular forms, and give applications to the birational type of F g,n . We prove that the Kodaira dimension of F-g,F- n stabilizes to 19 when n is sufficiently large. Then we use explicit Borcherds products to find a lower bound of n where F-g,F- n has nonnegative Kodaira dimension, and compare this with an upper bound where F-g,F- n is unirational or uniruled using Mukai models of K 3 surfaces in g <= 20. This reveals the exact transition point of Kodaira dimension in some g .