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Motivic Euler numbers and an arithmetic count of the lines on a cubic surface.

Hot Topics: Galois Theory of Periods and Applications March 27, 2017 - March 31, 2017

March 30, 2017 (11:00 AM PDT - 12:00 PM PDT)
Speaker(s): Kirsten Wickelgren (Duke University)
Location: MSRI: Simons Auditorium
  • Galois theory

  • Galois orbits

  • Periods

  • motivic integration

  • Euler characteristics

  • Cayley-Salmon theorem

  • A1-homotopy theory

  • Grothendieck-Witt group

  • degree formulae

  • Poincare-Hopf theorem

  • enumerative geometry

  • generalizations of classical theorems

  • algebraic geometry

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC



A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, the number of lines depends on the surface, but Segre showed that a certain signed count is always 3. We extend this count to an arbitrary field using A1-homotopy theory: we define an Euler number in the Grothendieck-Witt group for a relatively oriented algebraic vector bundle as a sum of local degrees, and then generalize the count of lines to a cubic surface over an arbitrary field. This is joint work with Jesse Leo Kass

28375?type=thumb Wickelgren.Notes 1.74 MB application/pdf Download
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