Conics meeting eight lines over perfect fields
arXiv:2107.05543 · doi:10.1016/j.jalgebra.2023.04.024
Abstract
Over the complex numbers, there are 92 plane conics meeting 8 general lines in projective 3-space. Using the Euler class and local degree from motivic homotopy theory, we give an enriched version of this result over any perfect field. This provides a weighted count of the number of plane conics meeting 8 general lines, where the weight of each conic is determined the geometry of its intersections with the 8 given lines. As a corollary, real conics meeting 8 general lines come in two families of equal size.
20 pages. Revised version with various errors corrected. Final version, but comments still welcome!