Various problems related to the classification in higher dimensional birational geometry

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We study extremal contraction morphisms of terminal threefolds such that the inverse image F of a point z is an irreducible curve, and try to classify the neighbourhood of F. They consist of 3 kinds, flipping contractions, divisorial contractions, and Q -conic bundles. Among them, flipping contractions had been classified, and we have found that the rest can be studied. There are at most two non-Gorenstein points, and the one-point case has been classified except for one case, and the treatment of the exceptional case is being written. Kawakita has proved that a divisorial contraction such that F is a surface is a weighted blow up. Kawanoue and Matsuki have introduced an invariant for a singular surface embedded in a smooth threefold which effectively improves in a sequence of blow ups specified by their algorithm which resolves the singularity of the surface.