Mukai models and Borcherds products

abstract:

Let ${\Fgn}$ 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 ${\Fgn}$ and the ring of certain orthogonal modular forms, and give applications to the birational type of ${\Fgn}$. We prove that the Kodaira dimension of ${\Fgn}$ stabilizes to $19$ when $n$ is sufficiently large. Then we use explicit Borcherds products to find a lower bound of $n$ where ${\Fgn}$ has nonnegative Kodaira dimension, and compare this with an upper bound where ${\Fgn}$ is unirational or uniruled using Mukai models of $K3$ surfaces in $g\leq 20$. This reveals the exact transition point of Kodaira dimension in some~$g$.