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: MSRI: Simons Auditorium
Galois theory
Galois orbits
Periods
motivic integration
Euler characteristics
CayleySalmon theorem
A1homotopy theory
GrothendieckWitt group
degree formulae
PoincareHopf theorem
enumerative geometry
generalizations of classical theorems
algebraic geometry
11Rxx  Algebraic number theory: global fields {For complex multiplication, see 11G15}
14C30  Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture
55Mxx  Classical topics {For the topology of Euclidean spaces and manifolds, see 57Nxx}
57R22  Topology of vector bundles and fiber bundles [See also 55Rxx]
14Jxx  Surfaces and higherdimensional varieties {For analytic theory, see 32Jxx}
14J25  Special surfaces {For Hilbert modular surfaces, see 14G35}
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 A1homotopy theory: we define an Euler number in the GrothendieckWitt 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
Wickelgren.Notes

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