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 ⊂ X ⊂ 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 ⊂ X is unbalanced. The proof reveals a new geometric connection between the universal Brill-Noether variety W 1 9,6 and a moduli space F h of lattice polarized K3 surfaces (for a certain rank 3 lattice h). As a by-product we prove the unirationality of F h and show that W 1 9,6 is birational to a projective bundle over a moduli space of lattice polarized K3 surfaces F h ′ for a certain rank 4 lattice h ′ which contains h as a sublattice.