Regular and Singular Points of Algebraic Curves. Tangents

Abstract

A point on an algebraic curve is either “simple” or “singular.” At a simple point the curve is “smooth.” In general,a point on a curve is assigned a “multiplicity” that indicates how many times it has to be counted as a point of the curve. The “tangents” of a curve will also be explained.One can decide whether a point is simple or singular with the help of the local ring at the point.The facts from Appendix E on Noetherian rings and discrete valuation rings will play a role in this chapter.Toward the end, some theorems from Appendix F on integral ring extensions wil l al so be needed.