  Let F_{g,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 F_{g,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,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,n} has nonnegative Kodaira dimension, and compare this with an upper bound
where F_{g,n} is unirational or uniruled using Mukai models of K3 surfaces in
g<21. This reveals the exact transition point of Kodaira dimension in some g.
