Mathematics Department Colloquium: An arithmetic count of rational plane curves

Colloquium | November 21 | 4:10-5 p.m. | 60 Evans Hall

 Kirsten Wickelgren, Duke University

 Department of Mathematics

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 this number is independent of (generically) chosen points over the complex numbers. The problem of determining these numbers turns out to be deep and connected to string theory. It was not until the 1990's that Kontsevich determined them with a recursive formula. Over the real numbers, one can obtain a fixed number by weighting real rational curves by their Welschinger invariant, and work of Solomon gives the analogous recursion. 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 an arithmetic count of rational plane curves in characteristic not 2 or 3. This is joint work with Jesse Kass, Marc Levine, and Jake Solomon.