Contraction criteria for reducible rational curves with components of length one in smooth complex threefolds

Abstract

Let X be a smooth complex threefold and C a linear chain of n smooth rational curves in X, each intersecting the canonical sheaf κ x trivially, and each having length 1, where the length is Kollar's invariant. Formal criteria will be given to determine when C contracts, when C deforms, and when C neither contracts or deforms in X, the formal completion of X. It is shown precisely, using the curve C, its components, and their defining ideals, how the behavior of C coincides with the deformation theory of the compound An singularity.