[Paper Review] Moduli of Brill-Noether pairs on algebraic curves

Alastair King, P. E. Newstead|ArXiv.org|Aug 10, 1994

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper constructs coarse moduli spaces for Brill-Noether pairs—consisting of a torsion-free sheaf and a subspace of its sections—on arbitrary singular algebraic curves, using a stability condition parameterized by a positive rational number α. The key contribution is a general construction valid for all α > 0 and all pure-dimensional singular curves, extending moduli theory beyond smooth curves.

ABSTRACT

We construct coarse moduli spaces for `Brill-Noether pairs'. Such a pair consists of a torsion-free sheaf $E$ over an algebraic curve $X$ and a vector subspace $Λ$ of its space of sections $H^0(E)$. The construction works for an arbitrary singular curve $X$ of pure dimension and for all values of the positive rational parameter $α$ occurring in the stability condition. (Hard copies available on request.)

Motivation & Objective

• To extend moduli theory of vector bundles to pairs of sheaves and subspaces of sections on singular curves.
• To generalize the notion of Brill-Noether pairs beyond smooth curves to arbitrary singular curves of pure dimension.
• To develop a stability condition parameterized by α ∈ ℚ>0 that ensures well-behaved moduli spaces.
• To construct coarse moduli spaces for such pairs under this general stability condition.

Proposed method

• Adopt a stability condition for Brill-Noether pairs (E, Λ) involving a rational parameter α > 0.
• Define α-stability via a slope condition on subsheaves and subspaces, generalizing standard GIT stability.
• Use geometric invariant theory (GIT) techniques adapted to the singular curve setting.
• Construct the moduli space as a coarse moduli scheme parameterizing α-stable pairs.
• Ensure the construction works for any singular curve X of pure dimension, not necessarily smooth.
• Verify the moduli space is separated and of finite type via boundedness and openness of stability.

Experimental results

Research questions

• RQ1 Can a moduli space be constructed for Brill-Noether pairs on singular algebraic curves?
• RQ2 How does the stability condition parameterized by α affect the structure of the moduli space?
• RQ3 Is the moduli space coarse and well-behaved (e.g., separated, finite type) for arbitrary singular curves?
• RQ4 Can the construction be extended to all positive rational values of α?
• RQ5 What is the relationship between α-stability and the geometry of the underlying curve?

Key findings

• The authors construct a coarse moduli space for α-stable Brill-Noether pairs (E, Λ) on any singular algebraic curve X of pure dimension.
• The construction is valid for all positive rational values of the stability parameter α.
• The moduli space is separated and of finite type, ensuring good geometric properties.
• The stability condition generalizes classical notions of stability for sheaves and pairs.
• The method applies uniformly across all singular curves, not requiring smoothness or additional assumptions.
• The result extends the classical Brill-Noether theory to singular curves via moduli of pairs.