Stable maps and singular curves on K3 surfaces

Abstract

In this thesis we study singular curves on K3 surfaces. Let $\mathcal{B}_g$ denote the stack of polarised K3 surfaces of genus $g$ and set $p(g,k)=k^2(g-1)+1$. There is a stack $ \mathcal{T}^n_{g,k} \to \mathcal{B}_g$ with fibre over the polarised surface $(X,L)$ parametrising all unramified morphisms $f: C \to X$, birational onto their image, with $C$ an integral smooth curve of genus $ p(g,k)-n$ and $f_*C \sim kL$. One can think of $ \mathcal{T}^n_{g,k}$ as parametrising all singular curves on K3 surfaces such that the normalisation map is unramified (or equivalently such that the curve has "immersed" singularities). The stack $ \mathcal{T}^n_{g,k}$ comes with a natural moduli map $$\eta \; : \;\mathcal{T}^n_{g,k} \to \mathcal{M}_{p(g,k)-n}$$ to the Deligne-Mumford stack of curves, defined by forgetting the map to the K3 surface. We first show that $\eta$ is generically finite (to its image) on at least one component of $\mathcal{T}^n_{g,k} $, in all but finitely many values of $p(g,k)-n$. We also consider related questions about the Brill-Noether theory of singular curves on K3 surfaces as well as the surjectivity of twisted Gaussian maps on normalisations of singular curves. Lastly, we apply the deformation theory of $\mathcal{T}^n_{g,k}$ to a seemingly unrelated problem, namely the Bloch-Beilinson conjectures on the Chow group of points of K3 surfaces with a symplectic involution.