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
Cayley-Salmon theorem
A1-homotopy theory
Grothendieck-Witt group
degree formulae
Poincare-Hopf 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 higher-dimensional 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 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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