Divisorial extractions from singular curves in smooth 3-folds

Consider a singular curve [Formula: see text] contained in a smooth [Formula: see text]-fold [Formula: see text]. Assuming the general elephant conjecture, the general hypersurface section [Formula: see text] is Du Val. Under that assumption, this paper describes the construction of a divisorial extraction from [Formula: see text] by Kustin–Miller unprojection. Terminal extractions from [Formula: see text] are proved not to exist if [Formula: see text] is of type [Formula: see text] or [Formula: see text] and are classified if [Formula: see text] is of type [Formula: see text] or [Formula: see text].