[Paper Review] Nodal curves with general moduli on K3 surfaces

Flaminio Flamini, Andreas Leopold Knutsen+2 more|ArXiv.org|Jul 2, 2007

Algebraic Geometry and Number Theory1 references4 citations

TL;DR

This paper establishes that for $K3$ surfaces of degree $2p-2$ with $3 \leq p \leq 11$, nodal curves of geometric genus $g = p - \delta$ with $\delta \leq p-2$ dominate the moduli space $\mathcal{M}_g$ under normalization. Using deformation theory of pairs $(S,X)$, the authors prove that the map from the stack of such pairs to $\mathcal{M}_g$ is dominant, showing that general curves of genus $g \leq 10$ arise as normalizations of nodal curves on $K3$ surfaces, even when smooth curves of the same genus do not embed in such surfaces.

ABSTRACT

We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a d-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p-2, for p any integer between 3 and 11 and g = p - d between 2 and p. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs (S,X) to the moduli space of curves of genus g that associates to X the isomorphism class [C] of its normalization.

Motivation & Objective

• To investigate the modular properties of nodal curves on $K3$ surfaces, particularly whether their normalizations dominate the moduli space $\mathcal{M}_g$.
• To extend Beauville’s deformation-theoretic approach from smooth curves to nodal curves on $K3$ surfaces.
• To resolve the contrast between the non-dominance of smooth curves of genus 10 on $K3$ surfaces and the dominance of nodal curves of genus 9 in the same setting.
• To analyze the deformation-theoretic behavior of the normalization map $c_{p,\delta}$ from pairs $(S,X)$ to $\mathcal{M}_g$.

Proposed method

• Use of the stack $\mathcal{V}_{p,\delta}$ parametrizing pairs $(S,X)$, where $S$ is a $K3$ surface with primitive polarization $H$ of sectional genus $p$, and $X \in |H|$ is an irreducible $\delta$-nodal curve.
• Study of the sheaf $\mathcal{T}_S\langle X\rangle$ of tangent vectors to $S$ tangent to $X$, whose cohomology controls deformations of the pair $(S,X)$.
• Blow-up of $S$ at the nodes of $X$ to analyze the local deformation theory of the normalization map $c_{p,\delta}$.
• Reduction of the dominance of $c_{p,\delta}$ to the vanishing of $H^0(\mathcal{I}_N/S \otimes \Omega_S^1(H))$, where $N = \operatorname{Sing}(X)$.
• Application of cohomological criteria: dominance follows if $H^0(\mathcal{I}_N/S \otimes \Omega_S^1(H)) = 0$, which is shown via global sections argument on $H^0(\Omega_S^1(H)) \cong \mathbb{C}$.
• Use of exact sequences and evaluation maps on the blow-up to control the differential of $c_{p,\delta}$ and ensure smoothness or unramifiedness.

Experimental results

Research questions

• RQ1 Does the normalization map $c_{p,\delta}$ from pairs $(S,X)$ to $\mathcal{M}_g$ dominate the moduli space for nodal curves on $K3$ surfaces?
• RQ2 Why does $V_{11,1}$ dominate $\mathcal{M}_{10}$, while $V_{10,0}$ does not, despite the same genus?
• RQ3 Can a Wahl-type obstruction be defined for nodal curves to lie on $K3$ surfaces, analogous to the one for smooth curves?
• RQ4 Does the image of $c_{p,\delta}$ meet the hyperelliptic locus $\mathcal{H}_g$ in $\mathcal{M}_g$, and what is the dimension of such loci?
• RQ5 How do the results extend to $p \geq 12$, and what is the behavior of $c_{p,\delta}$ for $g \geq 12$?

Key findings

• For $3 \leq p \leq 11$ and $0 \leq \delta \leq p-2$, the normalization map $c_{p,\delta}|_V$ is dominant for any irreducible component $V$ of $V_{p,\delta}$, so the general curve of genus $g = p - \delta$ is the normalization of a nodal curve on a $K3$ surface.
• The fiber dimension of $c_{p,\delta}$ is $22 - 2g$, matching the expected dimension, confirming that the map is generically smooth and dominant.
• In the case $p = 11, \delta = 1$, every irreducible component of $V_{11,1}$ dominates $\mathcal{M}_{10}$, which is surprising since $V_{10,0}$ does not dominate $\mathcal{M}_{10}$.
• For $p = 10, \delta = 1$, the nodal curves in $V_{10,1}$ only fill a divisor in the boundary $\partial\overline{\mathcal{M}}_{10}$, despite their normalizations being general curves of genus 9.
• The locus of effective divisors $P+Q$ on a general genus 9 curve $C$ such that $X = C/(P=Q)$ lies on a $K3$ surface is a 1-dimensional cycle $\Gamma \subset C^{(2)}$.
• For $p \geq 12$, the paper conjectures that $c_{p,\delta}|_V$ is dominant for $g \leq 11$ and generically finite for $12 \leq g < p$, based on cohomological conditions on $H^0(\mathcal{I}_N/S \otimes \Omega_S^1(H))$ and $H^1(\mathcal{I}_N/S \otimes \Omega_S^1(H))$.