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CATEGORIES:Differential Geometry and Topology Seminar
SUMMARY:An arithmetic count of rational plane curves - Kir
sten Wickelgren\, Duke
DTSTART;TZID=Europe/London:20200219T160000
DTEND;TZID=Europe/London:20200219T170000
UID:TALK134392AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/134392
DESCRIPTION: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 o
f (generically) chosen points over the complex num
bers. The problem of determining these numbers was
solved by Kontsevich with a recursive formula wit
h connections to string theory. Over the real numb
ers\, one can obtain a fixed number by weighting r
eal rational curves by their Welschinger invariant
\, and work of Solomon identifies this invariant w
ith a local degree. It is a feature of A1-homotopy
theory that analogous real and complex results ca
n indicate the presence of a common generalization
\, valid over a general field. For generically cho
sen points with coordinates in chosen fields\, we
give such a generalization\, providing an arithmet
ic count of rational plane curves over fields of c
haracteristic not 2 or 3. This is joint work with
Jesse Kass\, Marc Levine\, and Jake Solomon.
LOCATION:MR13
CONTACT:Ivan Smith
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