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An arithmetic count of rational plane curves

Kirsten Wickelgren - Duke University

Location:  Zoom
Date & time: Wednesday, 02 February 2022 at 3:30PM - 4:30PM

Abstract: There is a unique line through 2 points in the plane, and a unique conic through 5. These counts generalize to a count of degree d rational curves in the plane passing through 3d-1 points. Surprisingly, the problem of determining these numbers is connected to string theory, and it was not until the 1990's that Kontsevich determined them with a recursive formula. Such formulas are valid when you allow your curves to be defined with complex coefficients. Over the real numbers, one can obtain a fixed number by weighting real rational curves by their Welschinger invariant, and work of Solomon provides a recursive formula. 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 fields of characteristic not 2 or 3, we give such a generalization. The resulting count is (the stable isomorphism class) of a bilinear form, or an element of the group completion GW(k) of symmetric, non-degenerate, bilinear forms over k. For example, there are 2(<1>+<-1>) + 6<1> + <2>+<2D> rational degree 3-plane curves passing through a general configuration of 6 k-points and a pair of conjugate k(sort{D}) points in the plane. No knowledge of A1-homoltopy theory or GW is assumed. This is joint work with Jesse Kass, Marc Levine, and Jake Solomon.

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