Hilbert Problem 15 and Ritt-Wu Method (II)

Abstract

This paper proves three statements of Schubert about cuspal cubic curves in a plane by using the concept of generic point of Van der Waerden and Weil and Ritt-Wu methods. They are relations of some special lines: 1) For a given point, all the curves containing this point are considered. For any such curve, there are five lines. Two of them are the tangent lines of the curve passing through the given point. The other three are the lines connecting the given point with the cusp, the inflexion point and the intersection point of the tangent line at the cusp and the inflexion line. 2) For a given point, the curves whose tangent line at the cusp passes through this point are considered. For any such curve, there are four lines. Three of them are the tangent lines passing through this point and the other is the line connect the given point and the inflexion point. 3) For a given point, the curves whose cusp, inflexion point and the given point are collinear are considered. For any such curve, there are five lines. Three of them are tangent lines passing through the given point. The other two are the lines connecting the given point with the cusp and the intersection point of the tangent line at the cusp and the inflexion line.