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
Location: SLMath: Eisenbud 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
14D24 - Geometric Langlands program (algebro-geometric aspects) [See also 22E57]
53Zxx - Applications of differential geometry to sciences and engineering
01-06 - Proceedings, conferences, collections, etc. pertaining to history and biography
14J50 - Automorphisms of surfaces and higher-dimensional varieties
Wickelgren
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
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