Moduli of generalized line bundles on a ribbon

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.

Relative compactified Jacobiansof linear systemson Enriques surfaces

“…Let γ be the genus 3 of Γ. It was shown by Chen and Kass in [CK11], that all the components of the Simpson moduli space have dimension γ except, possibly, a (4γ ′ −3)-dimensional component, which exists when 4γ ′ −3 ≥ γ. This component parametrizes rank 2 semistable sheaves on Γ ′ .…”

Remarks on Nahm’s equations

“…Simpson's results on the moduli spaces of sheaves imply [5] that if det(w − φ) = 0 defines a ribbon X in the surface O(2), then the co-Higgs bundle is defined by the direct image of one of two types of sheaves:…”

On the irreducible components of the compactified Jacobian of a ribbon

“…Its degree is deg(I ) = χ(I ) − χ(O X ), while its index b(I ) is the length of the torsion part of its restriction to X red . This is the definition of index given for any pure coherent sheaf on a ribbon in [D, §6.3.7], which is equivalent to the more involved one given in [CK,Definition 2.7], that is specific for generalized line bundles. It is easy to check that b(I ) is a non-negative integer.…”

“…It is easy to check that b(I ) is a non-negative integer. Moreover, it holds also that deg(I ) − b(I ) is an even number; more precisely deg(I ) − b(I ) = 2 deg(I ), where I is the unique line bundle on X red such that I | X red = I ⊕ T with T a torsion sheaf (see [CK,Fact 2.8] and its reference; note that in the statement of the cited fact there is a typographical error: b(I ′ ) should be deg(I )).…”

“…Observe that M(O X , P 0 ) is a natural compactification of the Jacobian of X when the line bundles on X are stable, (i.e. when g ≥ 2ḡ, by [CK,Lemma 3.2]).…”