The Moduli of Singular Curves on K3 Surfaces

2014/01/06 by Kemeny, Michael
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1401.1047

Abstract

In this article we consider moduli properties of singular curves on K3 surfaces. Let B_g denote the stack of primitively polarized K3 surfaces (X,L) of genus g and let Tⁿ_g,k → B_g be the stack parametrizing tuples [(f: C → X, L)] with f an unramified morphism which is birational onto its image, C a smooth curve of genus p(g,k)-n and f_*C ∈ |kL|. We show that the forgetful morphism η : Tⁿ_g,k → M_p(g,k)-n is generically finite on one component, for all but finitely many values of p(g,k)-n. We further study the Brill--Noether theory of those curves parametrized by the image of η, and find a Wahl-type obstruction for a smooth curve with an unordered marking to have a nodal model on a K3 surface in such a way that the marking is the divisor over the nodes.

Related

• Stable maps and singular curves on K3 surfaces
• Moduli of nodal curves on K3 surfaces
• Nodal curves with general moduli on K3 surfaces
• The Curves of Elliptic Nodes on K3 Surfaces
• Moduli of lattice polarized K3 surfaces via relative canonical resolutions