Moduli of lattice polarized K3 surfaces via relative canonical resolutions

2017/04/10 by Bopp, Christian, Hoff, Michael
#13D02 #14D20 #14H45 #14J28 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1704.02753

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⊂ ℙ⁸, 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 W¹_9,6 and a moduli space F^\mathfrakh of lattice polarized K3 surfaces (for a certain rank 3 lattice \mathfrakh). As a by-product we prove the unirationality of F^\mathfrakh and show that W¹_9,6 is birational to a projective bundle over a moduli space of lattice polarized K3 surfaces F^\mathfrakh' for a certain rank 4 lattice \mathfrakh' which contains \mathfrakh as a sublattice.

Related

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• The moduli of curves of genus 6 and K3 surfaces
• The Moduli of Singular Curves on K3 Surfaces
• On lattice-polarized K3 surfaces
• On the Brill-Noether theory for K3 surfaces