Moduli of lattice polarized K3 surfaces via relative canonical resolutions

Abstract

For a smooth canonically embedded curve $C$ of genus $9$ together with a pencil $|L|$ of degree $6$, we study the relative canonical resolution of $C\subset X\subset \mathbb{P}^8$, where $X$ is the scroll swept out by the pencil $|L|$. We show that the second syzygy bundle in this resolution of $C\subset X$ is unbalanced. The proof reveals a new geometric connection between the universal Brill--Noether variety $\mathcal{W}^1_{9,6}$ and a moduli space $\mathcal{F}^\mathfrak{h}$ of lattice polarized $K3$ surfaces (for a certain rank $3$ lattice $\mathfrak{h}$). As a by-product we prove the unirationality of $\mathcal{F}^\mathfrak{h}$ and show that $\mathcal{W}^1_{9,6}$ is birational to a projective bundle over a moduli space of lattice polarized $K3$ surfaces $\mathcal{F}^{\mathfrak{h}'}$ for a certain rank $4$ lattice $\mathfrak{h}'$ which contains $\mathfrak{h}$ as a sublattice.