Divisorial Extractions from Singular Curves in Smooth 3-Folds, I

Consider a singular curve $\Gamma$ contained in a smooth 3-fold $X$. Assuming the general elephant conjecture, the general hypersurface section $\Gamma\subset S\subset X$ is Du Val. Under that assumption, this paper describes the construction of a divisorial extraction from $\Gamma$ by Kustin--Miller unprojection. Terminal extractions from $\Gamma\subset X$ are proved not to exist if $S$ is of type $D_{2k}, E_7$ or $E_8$ and are classified if $S$ is of type $A_1,A_2$ or $E_6$. The $A_n$ and $D_{2k+1}$ cases shall be considered in a further paper.