O problema de Steiner e variações do tipo ponto reta e cônica

Abstract

How many conics are tangent to five given conics? In the complex projective plane, the number 3264 presents itself as solution to this problem. However, finding this solution does not seem to be so simple. In this work, we dedicate ourselves to studying the concepts and tools necessary for solving problems similar to the one presented by Steiner. If P + L + Q = 5, how many conics pass through P points and are tangent to L lines and Q conics? Throughout the text, we explore classical results from Intersection Theory, particularly Bézout’s Theorem and an important example of a birational map, the Blow Up. Some results on the Chow Ring will help us understand how this count relates to the Zariski topology.