Hilbert problem 15 and Ritt-Wu method. I
The article under review provides a proof of a theorem in H. Schubert's classical text [Calcül der abzählenden Geometrie. Leipzig: Teubner (1879; JFM 11.0460.01)] on the enumerative relations among six special lines associated to a given planar cuspidal cubic curve and a given point in the plane. A cuspidal cubic curve in the projective complex plane has three special points: its cusp point, its inflection point, and the intersection point of its inflection tangent line with the tangent line at its cusp. Given such a curve and a general point in the plane, there is six special lines passing through the point, namely the three lines which connect the given point with the three special points plus the three lines which are tangent to the curve and pass through the given point. Schubert's theorem states that, fixing any four of these six special lines, there are only finitely many possibilities for the other two. This theorem also gives the exact numbers of possibilities, depending on which four of the six lines have been fixed. To rephrase the theorem, one can consider the map which, for a fixed point in the plane, sends a planar cuspidal cubic curve to the six special lines associated to the curve and the fixed point. Its image lives in a product of six projective lines, each of these six factors consisting of all lines in the plane that pass through the fixed point. Schubert's theorem says that this image is four-dimensional and it states its multidegree. As Schubert did not provide a full formal proof of his theorem, the author of the article at hand shows that Schubert's theorem is correct. The proof combines theoretical insights with computations and makes heavy use of a Maple implementation of the Ritt-Wu method. In addition, the author proves that the singular locus of the four-dimensional image of the map described above is an irreducible surface.
scientific article; zbMATH DE number 7032427