Graded algebras of vector bundle maps over an elliptic curve (Q1327765)

We study here a kind of homogeneous coordinate rings of matrix algebras over an elliptic curve. Let \(X\) be an elliptic curve over an algebraically closed field \(k\) with \(\text{char} (k) \neq 2\). Choose a point \(P \in X\) and let \({\mathcal L} = {\mathcal L} (P)\) be the invertible \({\mathcal O}_ X\)-module associated to the divisor \(P\). For a positive integer \(n\) let \({\mathcal E}_ n\) be an indecomposable locally free \({\mathcal O}_ X\)-module of rank \(n\) which is a successive extension of \({\mathcal O}_ X\). Such a module exists uniquely up to isomorphism. We form the \({\mathcal O}_ X\)-algebra \({\mathcal E} \text{\textit{nd}} ({\mathcal E}_ n)\), the sheaf of local endomorphisms of \({\mathcal E}_ n\), and then form a graded \(k\)-algebra \[ \Lambda (n) = \bigoplus_{i \geq 0} \Gamma (X, {\mathcal E} \mathit{nd} ({\mathcal E}_ n) \bigotimes {\mathcal L}^{\otimes_ i}) = \bigoplus_{i \geq 0} \Hom ({\mathcal E}_ n, {\mathcal E}_ n \bigotimes {\mathcal L}^{\otimes_ i}). \] In this paper we give an explicit description of the algebra \(\Lambda (n)\). Details and proofs will appear elsewhere.