Vector bundles on a neighborhood of a ruled surface in a threefold

Abstract

Let S be a ruled surface inside a smooth 3-fold W. Choose a neighborhood V of S in W and let Ŝ be the formal completion of S in V. Let E be a vector bundle over Ŝ. We show that (under suitable conditions) the local deformation space of E is finite dimensional and smooth. Moreover, we show that E is a flat limit of a flat family of vector bundles whose general element we describe explicitly.

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