Abstract

A rational plane curve of degree d is a polynomial map from the line to the plane of degree d. There are finitely many such curves passing through 3d-1 points, and the number of them is independent of (generically) chosen points over the complex numbers. The problem of determining these numbers was solved by Kontsevich with a recursive formula with connections to string theory. Over the real numbers, one can obtain a fixed number by weighting real rational curves by their Welschinger invariant, and work of Solomon identifies this invariant with a local degree. It is a feature of A1-homotopy theory that analogous real and complex results can indicate the presence of a common generalization, valid over a general field. For generically chosen points with coordinates in chosen fields, we give such a generalization, providing an arithmetic count of rational plane curves over fields of characteristic not 2 or 3. This is joint work with Jesse Kass, Marc Levine, and Jake Solomon.