Conics meeting eight lines over perfect fields

TL;DR

This paper extends classical enumerative geometry results about conics meeting lines from complex numbers to perfect fields using motivic homotopy theory, providing a weighted count of such conics.

Contribution

It introduces an enriched counting method for conics meeting lines over perfect fields, utilizing motivic homotopy theory and Euler classes.

Findings

Weighted count of conics over perfect fields

Real conics form two equally sized families

Extension of classical results to broader fields

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.