Moduli of lattice polarized K3 surfaces via relative canonical resolutions

For a smooth canonically embedded curve C of genus 9 together with pencil |L| degree 6, we study the relative canonical resolution C⊂X⊂P8, where X is scroll swept out by |L|. We show that second syzygy bundle in this C⊂X unbalanced. The proof reveals new geometric connection between universal Brill–Noether variety W9,61 and moduli space Fh lattice polarized K3 surfaces (for certain rank 3 h). As by-product prove unirationality birational to projective over Fh′ for 4 h′ which contains h as sublattice.