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 K_X trivially, and each having length 1, where the length is Kollár’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 A_n singularity.

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