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 ${\mathcal{T}}_{g,k}^n\to {\mathcal{B}}_g$ 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 $f_{\ast }C\sim kL$. One can think of ${\mathcal{T}}_{g,k}^n$ 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 ${\mathcal{T}}_{g,k}^n$ comes with a natural moduli map $$\eta :{\mathcal{T}}_{g,k}^n\to {\mathcal{M}}_{p(g,k)-n}$$ 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 ${\mathcal{T}}_{g,k}^n$, 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 ${\mathcal{T}}_{g,k}^n$ to a seemingly unrelated problem, namely the Bloch-Beilinson conjectures on the Chow group of points of K3 surfaces with a symplectic involution.

arXiv commentPhD Thesis, defended 11.06.15. Chapter 2 contains an elementary exposition on the deformation theory of stable maps to K3 surfaces. Otherwise, all results have already appeared in arXiv:1202.4968 and arXiv:1401.1047

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