Moduli of unstable bundles of HN-length two with fixed algebra of endomorphisms

Abstract

Let X be a smooth irreducible complex projective curve of genus g ≥ 2 and U µ1 (n, d) the moduli scheme of indecomposable vector bundles over X with fixed Harder-Narasimhan type σ = (µ 1 , µ 2 ). In this paper, we give necessary and sufficient conditions for a vector bundle E ∈ U µ1 (n, d) to have C[x 1 , . . . , x k ]/(x 1 , . . . , x k ) 2 as its algebra of endomorphisms. Fixing the dimension of the algebra of endomorphisms we obtain a stratification of U µ1 (n, d) such that each strata U µ1 (n, d, k) is a coarse moduli space. A particular case of interest is when the unstable bundles are simple. In that case the moduli space is fine. Topological properties of U µ1 (n, d, k) will depend on the generality of the curve X. Such results differ from the corresponding results for the moduli space of stable bundles, where non-emptiness, dimension etc. are independent of the curve.