[Paper Review] Duality Construction of Moduli Spaces

Georg Hein|ArXiv.org|Aug 29, 1997

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper introduces a duality construction for the moduli space of rank-2 coherent sheaves on an algebraic surface, providing an alternative to the standard GIT approach. It establishes a finite Barth-type morphism, generalizing jumping lines, and proves the existence of a dual moduli space that enables intrinsic construction without GIT techniques.

ABSTRACT

We show for the moduli space of rank-2 coherent sheaves on an algebraic surface that there exists a 'dual' moduli space. This dual space allows a construction of the first one without using the GIT construction. Furthermore, we obtain a Barth-morphism, generalizing the concept of jumping lines. This morphism is by construction a finite morphism.

Motivation & Objective

• To construct the moduli space of rank-2 coherent sheaves on an algebraic surface without relying on Geometric Invariant Theory (GIT).
• To define and establish the existence of a 'dual' moduli space that provides an alternative parametrization of the same objects.
• To generalize the concept of jumping lines via a finite morphism, analogous to Barth's construction in the case of vector bundles.
• To provide a duality framework that reveals deeper structural properties of moduli spaces of coherent sheaves.
• To offer a new, intrinsic method for constructing moduli spaces using duality rather than quotient constructions.

Proposed method

• Utilizes duality theory in algebraic geometry to relate the moduli space of rank-2 coherent sheaves to its dual counterpart.
• Constructs a morphism from the moduli space to a projective space, generalizing the classical notion of jumping lines.
• Employs the notion of a Barth morphism, which is shown to be finite by construction.
• Applies techniques from coherent sheaf theory and duality on surfaces, particularly focusing on the dualizing sheaf and Ext-sheaves.
• Relies on the existence of a dual moduli space to bypass the need for GIT quotient constructions.
• Uses the structure of the derived category and Serre duality to relate the original and dual moduli spaces.

Experimental results

Research questions

• RQ1 Can the moduli space of rank-2 coherent sheaves on a surface be constructed without using Geometric Invariant Theory?
• RQ2 What is the nature of a dual moduli space for such sheaves, and how does it relate to the original space?
• RQ3 How can the classical notion of jumping lines be generalized to coherent sheaves via a morphism?
• RQ4 Is there a finite morphism associated with the moduli space that captures stability and degeneracy behavior?
• RQ5 What structural properties emerge from the duality between the original and dual moduli spaces?

Key findings

• A dual moduli space exists for the moduli space of rank-2 coherent sheaves on an algebraic surface.
• The dual moduli space allows for a construction of the original moduli space without invoking Geometric Invariant Theory.
• A Barth-type morphism is constructed that generalizes the concept of jumping lines and is finite by construction.
• The duality framework reveals a symmetric structure between the original and dual moduli spaces.
• The finite morphism provides a new tool for studying degenerations and stability conditions of sheaves.
• The results demonstrate that duality can serve as a foundational construction principle in moduli theory, independent of GIT.