Generalizations of the hexagramme mystique

Abstract

The history of projective geometry is a very complex one. Most of the more formal developments on the subject were made in the 19th century as a result of the movement away from the geometry of Euclid. If one digs a little deeper, however, one can see that the basic concepts upon which this branch of geometry is based can be traced back as far as the fourth century, where a theorem of Pappus of Alexandria appears as Proposition 139 of Book VII of the Mathematical Collection. These very early discoveries along with Euclid’s Elements are the building blocks for the foundations that were laid down by the projective geometers of the 17th century. It is here that the history of the subject becomes more interesting. Great strids were made in the 17th century, but for some reason projective geometry did not become popular among mathematicians until the 19th century. From this moment, very important results on this subject were made by great mathematicians as Max Noether or David Hilbert. In particular, the base of these notes is the study of the theory of plane algebraic curves. Willing to know more about the geometry behind the plane algebraic curves, I began to work with the Algebraic Curves of William Fulton [1]. Introducing myself with the algebraic sets and its ideals, and with its properties as well, I venture on the theory of intersection of plane algebraic curves, studying them on the affine plane and on the projective plane. To doing so, I had to apprehend so importants results such that the intersection number at points on curves, the Bézout’s Theorem or the Max Noether Fundamental Theorem. As an application, I proved some problems of the algebraic geometry, from the classics to the most contemporary, begining with the Pappu’s Theorem and ending with the addition on the Elliptic Law. Moreover, I state some ideas of plane algebraic curves from a more modern point of view, talking about the divisors on smooth curves and the concepts that derive from them.