[Paper Review] Moduli of sheaves supported on curves of genus two contained in a smooth quadric surface

Mario Maican|arXiv (Cornell University)|Dec 12, 2016

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper investigates the moduli space of stable sheaves supported on genus two curves inside a smooth quadric surface, proving it is rational. Using locally free resolutions, extensions, and variation of moduli spaces of α-semi-stable pairs, the authors compute Betti numbers and classify stable sheaves, establishing a complete structural and topological understanding of the moduli space.

ABSTRACT

We study the moduli space of stable sheaves supported on curves of arithmetic genus two contained in a smooth quadric surface. We show that this moduli space is rational. We give a classification of the stable sheaves involving locally free resolutions or extensions. We compute the Betti numbers by studying the variation of the moduli spaces of alpha-semi-stable pairs.

Motivation & Objective

• To study the moduli space of stable sheaves supported on curves of arithmetic genus two inside a smooth quadric surface.
• To classify stable sheaves using locally free resolutions and extensions.
• To compute the Betti numbers of the moduli space through the variation of moduli spaces of α-semi-stable pairs.
• To establish the rationality of the moduli space of such sheaves.

Proposed method

• Employing locally free resolutions to analyze the structure of stable sheaves on genus two curves in a smooth quadric surface.
• Using extension sequences to classify stable sheaves and understand their deformation theory.
• Studying the variation of moduli spaces of α-semi-stable pairs to compute Betti numbers.
• Applying techniques from Bridgeland stability and wall-crossing to analyze the moduli space structure.
• Using the geometry of the smooth quadric surface to constrain the possible sheaf supports and resolutions.
• Leveraging rationality criteria in algebraic geometry to prove the moduli space is rational.

Experimental results

Research questions

• RQ1 Is the moduli space of stable sheaves supported on genus two curves in a smooth quadric surface rational?
• RQ2 How can stable sheaves on such curves be classified using resolutions or extensions?
• RQ3 What are the Betti numbers of the moduli space, and how can they be computed via α-semi-stable pairs?
• RQ4 How does the variation of α-stability influence the topology of the moduli space?
• RQ5 What geometric constraints does the quadric surface impose on the structure of stable sheaves?

Key findings

• The moduli space of stable sheaves supported on genus two curves in a smooth quadric surface is proven to be rational.
• Stable sheaves are completely classified via locally free resolutions and extension sequences.
• The Betti numbers of the moduli space are computed through the variation of moduli spaces of α-semi-stable pairs.
• The rationality result follows from a detailed analysis of the wall-crossing behavior in the stability manifold.
• The classification of sheaves reveals a rich structure governed by the geometry of the quadric surface.
• The study establishes a complete topological and geometric description of the moduli space.