Limites de curvas planas duais via séries de Puiseux

Abstract

This work addresses the limit of plane dual curves, a topic in algebraic geome- try that seeks to understand what happens to curves when they undergo degeneracy

processes. In such situations, the dual of the limiting curve does not always coincide with the limit of the duals, and therefore a more careful examination is required.

First, we introduce some fundamental concepts, such as Puiseux series, discri- minants of univariate polynomials, and intersections in the projective plane. Next, we

address the duality between points and lines in this setting and apply this concept to the study of the dual curve of a smooth projective plane curve. We then examine how this duality behaves within flat families of curves. The main result provides a precise formula for describing the limit of such dual curves, relating its components to the duals of each component of the special fiber of the family, as well as to certain discriminants. Finally, examples illustrate the theorem and demonstrate the practical utility of the theory developed. The research thus seeks to bring the reader closer to a more intuitive understanding of how curves and their duals interact in limiting situations.