indicated that Kronecker may have anticipated Noether’s own argument using quadratic transformations to show how to obtain a curve having only ordinary singularities – those whose branches have distinct tangents.

Since Noether was very familiar with the mapping in [@Clebsch1866], he would have realized immediately how it can be used to desingularize a given mutliple point. Clebsch’s mapping takes six fundamental points in general position in the plane and blows these up into six of the 27 lines on a cubic surface $F_3$ (Fig. 2). Each set of five fundamental points determines a conic, and these six conics also go over into six lines. The remaining 15 are the images of the lines connecting pairs of the six fundamental points. If one then starts with a plane algebraic curve $C_n$ with singular point $P\in C_n$ of multiplicity $m$, then by letting $P$ coincide with one of the six fundamental points of a Clebsch mapping has the effect of blowing up this singularity into $m$ points on the image line. The curve $C_n$ will then pass over to a space curve $C'$ lying on $F_3$. The remainder of the argument then involves carefully finding a point $Q\in F_3$ so that the 