[Paper Review] Moduli of Generalized Line Bundles on a Ribbon

Dawei Chen, Jesse Leo Kass|arXiv (Cornell University)|Jun 27, 2011

Algebraic Geometry and Number Theory7 references4 citations

TL;DR

This paper constructs and analyzes the moduli space of slope semi-stable sheaves on a ribbon—a first-order thickening of a smooth curve—showing that it compactifies the space of line bundles via generalized line bundles and rank-2 vector bundles on the reduced curve. The key result is that when the ribbon has even genus, the moduli space is projective and satisfies Eisenbud and Green’s compactification condition for degree 0 generalized line bundles with non-negative index.

ABSTRACT

A ribbon is a first-order thickening of a non-singular curve. Motivated by a question of Eisenbud and Green, we show that a compactification of the moduli space of line bundles on a ribbon is given by the moduli space of semi-stable sheaves. We then describe the geometry of this space, determining the irreducible components, the connected components, and the smooth locus.

Motivation & Objective

• To resolve Eisenbud and Green’s question on whether the moduli space of degree 0 line bundles on a rational ribbon can be compactified by generalized line bundles of non-negative index.
• To describe the global and local geometry of the Simpson moduli space of semi-stable sheaves on a ribbon, including its irreducible and connected components.
• To determine the smooth locus of the moduli space and identify when it coincides with the locus of line bundles.
• To establish conditions under which the moduli space is projective, particularly when the genus of the ribbon is even.

Proposed method

• Using Simpson’s construction of moduli spaces of semi-stable sheaves with fixed Hilbert polynomial, the authors analyze the moduli space $\operatorname{M}({\mathcal{O}}_X, P_d)$ for a polarized ribbon $(X, {\mathcal{L}})$.
• The classification of sheaves in $\operatorname{M}({\mathcal{O}}_X, P_d)$ is based on two types: generalized line bundles of bounded index and direct images of stable vector bundles on the reduced curve $X_{\text{red}}$.
• The authors compute the dimension of the tangent space at points corresponding to stable sheaves using the Riemann–Roch formula and Ext-group computations, particularly $\dim \operatorname{Ext}^1(i_*{\mathcal{E}}, i_*{\mathcal{E}})$.
• They use Serre duality and stability conditions to show that higher cohomology vanishes for certain bundles, enabling precise dimension calculations.
• The smooth locus is determined by comparing tangent space dimensions with local dimensions, showing that singularities occur precisely when the sheaf is not a line bundle.
• The analysis relies on the fact that strictly semi-stable sheaves are specializations of stable ones, which holds under certain genus conditions.

Experimental results

Research questions

• RQ1 Does the moduli space of generalized line bundles on a ribbon admit a compactification via semi-stable sheaves?
• RQ2 Under what conditions is the Simpson moduli space $\operatorname{M}({\mathcal{O}}_X, P_d)$ projective?
• RQ3 When does the smooth locus of the moduli space coincide with the locus of line bundles?
• RQ4 How do the irreducible and connected components of the moduli space decompose in terms of generalized line bundles and vector bundles on the reduced curve?

Key findings

• The moduli space $\operatorname{M}({\mathcal{O}}_X, P_d)$ parameterizes generalized line bundles of index $\leq 1 + g - 2\bar{g}$ and direct images of slope semi-stable rank-2 vector bundles on $X_{\text{red}}$ of degree $d + 2\bar{g} - 1 - g$.
• When the ribbon has even genus $g$, the moduli space $\operatorname{M}({\mathcal{O}}_X, P_0)$ is projective, satisfying Eisenbud and Green’s desired compactification condition.
• The smooth locus of $\operatorname{M}({\mathcal{O}}_X, P_d)$ equals the open subset parameterizing line bundles when $\bar{g} \geq 2$ and $g \geq 4\bar{g} - 2$.
• For $\bar{g} \geq 2$ and $g \geq 4\bar{g} - 2$, the tangent space dimension at a stable rank-2 bundle on $X_{\text{red}}$ exceeds the local dimension, proving such points are singular.
• When $g$ is sufficiently negative and $\bar{g} \geq 2$, the moduli space may have a unique component of dimension $4\bar{g} - 3$ whose smooth locus includes stable rank-2 bundles.
• The strictly semi-stable locus is contained in the singular locus when $\bar{g} \geq 2$, due to the density of stable points in the vector bundle component.