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Through October 6, 2019
A meeting of the Midwest Topology Seminar (Oct. 4-6) celebrating the
80th birthday of Peter May.
Friday, October 4th Colloquium:
Kirsten Wickelgren (Duke) to give special colloquium (Kent 120, Oct.
4, 4pm), titled "An arithmetic count of rational plane curves"
Abstract: There are finitely many degree d rational plane
curves passing through points, and over the complex numbers, this
number is independent of (generically) chosen points. For example,
there are 12 degree 3 rational curves through 8 points, one conic
passing through 5, and one line passing through 2. Over the real
numbers, one can obtain a fixed number by weighting real rational
curves by their Welschinger invariant, and work of Jake Solomon
identifies this invariant with a local degree. It is a feature of
\(A^1\)-homotopy theory that analogous real and complex results can
indicate the presence of a common generalization, valid over a general
field. We develop and compute an \(A^1\)-degree, following Morel, of the
evaluation map on Kontsevich moduli space to obtain an arithmetic
count of rational plane curves, which is valid for any field of
characteristic not 2 or 3. This shows independence of the count on the
choice of generically chosen points with fixed residue fields,
strengthening a count of Marc Levine. This is joint work with Jesse
Kass, Marc Levine, and Jake Solomon.
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