[Paper Review] Stable maps and singular curves on K3 surfaces

Michael Kemeny|arXiv (Cornell University)|Jul 1, 2015

Algebraic Geometry and Number Theory96 references3 citations

TL;DR

This paper studies singular curves on K3 surfaces using stable maps and deformation theory, focusing on unramified morphisms from smooth curves to K3 surfaces. It establishes that the moduli map from the stack of such maps to the moduli space of curves is generically finite to its image in most cases, providing a new framework for Brill–Noether theory and applications to Chow groups and Gaussian maps on K3 surfaces with symplectic involutions.

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} o \mathcal{B}_g$ with fibre over the polarised surface $(X,L)$ parametrising all unramified morphisms $f: C o X$, 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 $$η\; : \;\mathcal{T}^n_{g,k} o \mathcal{M}_{p(g,k)-n}$$ to the Deligne-Mumford stack of curves, defined by forgetting the map to the K3 surface. We first show that $η$ 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.

Motivation & Objective

• To develop a new moduli-theoretic framework for studying singular curves on K3 surfaces using stable maps.
• To analyze the geometry of the moduli map from stable maps to the moduli space of curves.
• To investigate Brill–Noether theory for nodal and singular curves lying on K3 surfaces.
• To apply deformation theory of stable maps to the Bloch–Beilinson conjectures for K3 surfaces with symplectic involutions.
• To understand the surjectivity of twisted Gaussian maps on normalizations of singular curves on K3 surfaces.

Proposed method

• Constructs the stack $\mathcal{T}^{n}_{g,k}$ parametrizing unramified, birational morphisms from smooth curves of genus $p(g,k)-n$ to polarized K3 surfaces of genus $g$.
• Defines a natural moduli map $\eta: \mathcal{T}^{n}_{g,k} \to \mathcal{M}_{p(g,k)-n}$ sending each stable map to the normalization of its image curve.
• Applies deformation theory to analyze the fiber dimensions of $\eta$, proving generic finiteness under certain conditions.
• Uses induction on the Brill–Noether parameter $\rho(g,r,d)$, analyzing the base-point free part of line bundles via evaluation morphisms.
• Employs semi-stable reduction and flat families of torsion-free sheaves to control degenerations and fiber dimensions.
• Relies on the universal property of $\overline{W}^r_d(C)$ and identifies moduli spaces of effective line bundles on smooth loci to bound fiber dimensions.

Experimental results

Research questions

• RQ1 Is the moduli map $\eta: \mathcal{T}^{n}_{g,k} \to \mathcal{M}_{p(g,k)-n}$ generically finite to its image for at least one component of $\mathcal{T}^{n}_{g,k}$?
• RQ2 What is the Brill–Noether theory of singular curves on K3 surfaces, particularly for nodal models?
• RQ3 Under what conditions is the twisted Gaussian map on the normalization of a singular curve on a K3 surface surjective?
• RQ4 How can deformation theory of stable maps be applied to the Bloch–Beilinson conjectures for K3 surfaces with symplectic involutions?
• RQ5 Can the geometry of singular curves on K3 surfaces be studied without degenerating to unions of rational surfaces?

Key findings

• The moduli map $\eta: \mathcal{T}^{n}_{g,k} \to \mathcal{M}_{p(g,k)-n}$ is generically finite to its image on at least one component for all but finitely many $g$ and $k$.
• The proof of generic finiteness relies on controlling fiber dimensions via decomposition of the cokernel of the evaluation map into smooth and singular parts of the curve.
• For any curve $C$ in $|L|$ that is integral and nodal, the Brill–Noether theorem holds whenever $\rho(g,r,d) < 0$, and the result extends to such curves via the framework developed.
• The fiber dimension of the map $f: I^0 \to \overline{W}^{r'}_{d'}(C)$ is bounded by $d - d'$, which ensures $\dim(I) < \rho(g,r,d)$ when $r > 0$.
• The construction of the map $g: I^0 \to \overline{W}^{r'}_{d-e}(C)$ via twisting by line bundles of degree $d - d' - e$ allows control over fiber dimensions using the dimension of the symmetric product of the smooth locus.
• The results establish a bridge between stable map theory and classical questions in algebraic geometry, including the Bloch–Beilinson conjectures for K3 surfaces with Nikulin involutions.