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Types of lines and Euler Numbers enriched in GW(k)

Duke Math
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Monday, March 09, 2020
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3:15 pm - 4:15 pm
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Sabrina Pauli (University of Oslo)
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Geometry/topology Seminar

Motivated by Morel's degree in A1-homotopy theory which takes values in the Grothendieck-Witt ring of a field k, Kass and Wickelgren define the Euler number of an oriented vector bundle valued in GW(k) to be the sum of local A1-degrees of the zeros of a generic section. Using this definition they get an enriched count of lines on a smooth cubic surface in GW(k). In my talk I will compute several Euler numbers valued in GW(k). In particular, I will count lines on quintic threefolds. In addition, I will give a geometric interpretation of the local contribution of a line on a quintic threefold to the enriched Euler number. When k = R this geometric interpretation agrees with the Segre type defined by Finashin and Kharlamov.