[Paper Review] Geometric invariant theory and moduli spaces of maps

David Swinarski|ArXiv.org|Feb 2, 2004

Algebraic Geometry and Number Theory47 references3 citations

TL;DR

This paper develops a geometric invariant theory (GIT) framework for constructing moduli spaces of stable maps, leveraging quotient constructions to parameterize holomorphic maps from curves to target varieties. The key contribution is a systematic GIT approach that generalizes earlier constructions and provides a foundational method for studying moduli of maps in algebraic geometry.

ABSTRACT

The results of this paper have been subsumed by the paper "A geometric invariant theory construction of spaces of stable maps," Elizabeth Baldwin and David Swinarski, arXiv:0706.1381

Motivation & Objective

• To establish a geometric invariant theory (GIT) construction for moduli spaces of stable maps.
• To generalize existing moduli space constructions using GIT quotient techniques.
• To provide a systematic framework for parameterizing stable maps from algebraic curves to projective varieties.
• To lay the groundwork for studying stability conditions and compactifications in moduli problems.

Proposed method

• Applies geometric invariant theory (GIT) to construct moduli spaces as quotients of parameter spaces under group actions.
• Uses the Hilbert-Mumford criterion to analyze stability of maps in the GIT setting.
• Constructs the moduli space as a projective quotient of a parameter space of maps with fixed degree and target.
• Employs linearizations of line bundles to define stable and semistable loci in the parameter space.
• Applies the theory of quotients in algebraic geometry to ensure the resulting moduli space is a separated, projective scheme.
• Relies on the existence of a compactification via GIT, ensuring properness and good geometric properties.

Experimental results

Research questions

• RQ1 How can geometric invariant theory be used to construct moduli spaces of stable maps?
• RQ2 What conditions ensure the existence of a projective moduli space via GIT quotient?
• RQ3 How do stability conditions in GIT correspond to geometric stability of maps?
• RQ4 What is the relationship between GIT constructions and existing compactifications of moduli of maps?

Key findings

• The paper establishes a GIT construction for moduli spaces of stable maps, providing a new method for their compactification.
• The resulting moduli space is shown to be a projective scheme, ensuring good geometric and topological properties.
• Stability in the GIT sense corresponds to geometric stability of maps, validating the construction.
• The framework generalizes previous approaches and offers a uniform method applicable to various target varieties.
• The construction is compatible with the notion of stable maps in algebraic geometry, aligning with established moduli theory.
• The method provides a foundation for further study of invariants and enumerative geometry of maps.