5/30/2023 0 Comments The Singularity by Mark RodsethDu Val challenged him into making his claims explicit, which he did the next week! In his article Arf-1949 shows that the completion of the local ring at the singularity of the branch carries all the information necessary to obtain the multiplicity sequence. While Du Val was explaining his findings, a young mathematician in the audience, none other than Cahit Arf, observed and claimed that there was an efficient algebra behind these geometrical arguments. Later Du Val-1942, while he was in ˙ Istanbul University, showed that if the multiplicity sums up to certain geometrically significant steps in the successive blow-up process were known then the whole multiplicity sequence could be obtained from these sums through a modified version of the Jaco-bian algorithm. Among those who attacked this problem we can quote Semple-1938 who analyzed the geometry of successive blow-ups on a singular curve branch in 3-space. The simple question of understanding space curves with singularities, after the plane curves were relatively understood, proved to be mightier than the mortals who asked it. As soon as you pass the level of basic definitions you run into simple questions whose answers lie in deeper waters than one dares to dive. Study of curves is full of lessons in modesty. We also briefly mention current literature on the subject and some related research problems. In this note we summarize Arf's work on Arf rings and discuss its relevance in geometry.
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