4 - Divisorial Contractions to Curves

Summary

Every threefold divisorial contraction that contracts the divisor to a curve is the usual blow-up about the generic point of the curve. It is uniquely described as the symbolic blow-up as far as it exists. The general elephant conjecture is settled by Kollár and Mori when the fibre is irreducible. On the assumption of this conjecture, the symbolic blow-up always exists as a contraction from a canonical threefold. We want to determine whether it is further terminal. Tziolas analysed the case when the extraction is from a smooth curve in a Gorenstein terminal threefold, and Ducat did when it is from a singular curve in a smooth threefold. They follow the same division into cases based upon the divisor class of the curve in the Du Val section. Tziolas describes the symbolic blow-up as a certain weighted blow-up, whilst Ducat realises it by serial unprojections. The unprojection is an operation to construct a new Gorenstein variety from a simpler one. The contraction can be regarded as a one-parameter deformation of the birational morphism of surfaces cut out by a hyperplane section. In reverse, one can construct a threefold contraction by deforming an appropriate surface morphism.