Vector Bundles and Torsion Free Sheaves on Degenerations of Elliptic Curves

Abstract

In this paper we give a survey about the classification of vector bundles and torsion free sheaves on degenerations of elliptic curves. Coherent sheaves on singular curves of arithmetic genus one can be studied using the technique of matrix problems or via Fourier-Mukai transforms, both methods are discussed here. Moreover, we include new proofs of some classical results about vector bundles on elliptic curves.

Generic sheaves on elliptic curves

“…H 0 is uniserial, and decomposes into a coproduct x∈E U x of connected uniserial subcategories, whose associated quivers are homogeneous tubes, and the mouth of each homogeneous tube is a simple sheaf. [2,3].…”

Coherent systems on a nodal curve of genus one

“…We denote by \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\bf {F}_r, r\ge 1$\end{document}, the unique vector bundle (Atiyah bundle) defined inductively by the following conditions (see 7, Section 3, Lemma 2). \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\bf {F}_1$\end{document} is the trivial rank one line bundle. For r ≥ 2, the bundle \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\bf {F}_r$\end{document} is given by the nontrivial extension (unique upto a nonzero scalar) …”

A construction of the rational function sheaves on elliptic curves

“…Lemma 2.5 (see [1,2]) Any indecomposable coherent sheaf F on E is semistable. If two semistable coherent sheaves F , H ∈ cohE satisfy μ(F ) > μ(H), then Hom(F, H) = 0.…”

“…If two semistable coherent sheaves F , H ∈ cohE satisfy μ(F ) > μ(H), then Hom(F, H) = 0. Lemma 2.6 (see [2,4]) H has the following detailed description.…”