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 ${\mathcal{W}}_{9,6}^1$ 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 ${\mathcal{W}}_{9,6}^1$ is birational to a projective bundle over a moduli space of lattice polarized $K3$ surfaces ${\mathcal{F}}^{{\mathfrak{h}}^{\prime }}$ for a certain rank $4$ lattice ${\mathfrak{h}}^{\prime }$ which contains $\mathfrak{h}$ as a sublattice.

arXiv comment22 pages, final version, accepted in Journal of Pure and Applied Algebra