[Paper Review] Picard groups of moduli of K3 surfaces of low degree K3 surfaces

Zhiyuan Li, Zhiyu Tian|arXiv (Cornell University)|Apr 11, 2013

Algebraic Geometry and Number Theory24 references5 citations

TL;DR

This paper studies the moduli spaces of quasi-polarized K3 surfaces of degree 6 and 8 using geometric invariant theory, verifying the Noether-Lefschetz conjecture in these cases. It establishes the structure of Picard groups for these moduli spaces, providing foundational results for the general case of low-degree K3 surfaces.

ABSTRACT

We study the moduli space of quasi-polarized K3 surfaces of degree 6 and 8 via geometric invariant theory. In particular, we verify the Noether-Lefschetz conjecture in these two cases. The general case is discussed at the end of the paper.

Motivation & Objective

• To investigate the moduli spaces of quasi-polarized K3 surfaces of degree 6 and 8 using geometric invariant theory.
• To verify the Noether-Lefschetz conjecture for these specific degrees.
• To determine the structure of the Picard groups of the moduli spaces of such K3 surfaces.
• To lay groundwork for the general case of low-degree K3 surfaces.

Proposed method

• Employing geometric invariant theory to construct and analyze the moduli spaces of quasi-polarized K3 surfaces of degree 6 and 8.
• Analyzing the stability conditions and quotient constructions to describe the moduli stacks.
• Using the geometry of linear systems and line bundles on K3 surfaces to study Picard groups.
• Applying known results on Noether-Lefschetz loci to verify the conjecture in the specified cases.
• Relating the moduli problem to GIT quotients of Hilbert schemes or related parameter spaces.
• Extending the analysis to the general case of low-degree K3 surfaces at the end of the paper.

Experimental results

Research questions

• RQ1 What is the structure of the Picard group of the moduli space of quasi-polarized K3 surfaces of degree 6?
• RQ2 How does geometric invariant theory describe the moduli space of degree 8 K3 surfaces?
• RQ3 Does the Noether-Lefschetz conjecture hold for K3 surfaces of degree 6 and 8?
• RQ4 What is the relationship between the Noether-Lefschetz locus and the Picard group in these cases?
• RQ5 How do the results for degree 6 and 8 K3 surfaces generalize to other low-degree cases?

Key findings

• The Noether-Lefschetz conjecture is verified for K3 surfaces of degree 6.
• The Noether-Lefschetz conjecture is verified for K3 surfaces of degree 8.
• The Picard group of the moduli space of degree 6 K3 surfaces is computed via GIT methods.
• The Picard group of the moduli space of degree 8 K3 surfaces is determined through geometric analysis.
• The results provide a foundation for the general case of low-degree K3 surfaces.
• The study establishes a framework for extending these results to other degrees using similar techniques.