Kirsten Wickelgren (I): There are 1 + -1 lines meeting 4 lines in space: on A1-enumerative geometry
Channel: Schubert Seminar
Date: 18/11/2021

It is a feature of A1-homotopy theory that analogous real and complex results can indicate the presence of a common generalization, valid over a general field. Counts in enumerative geometry often fit into this framework, and this talk will present a paradigm. Given four general lines in 3-dimensional space, it is a classical result that the number of lines intersecting all four is two, provided you allow the coefficients of the lines to be complex numbers. Over a general field k, say with characteristic not 2, the two lines may be a conjugate pair over a quadratic extension of the field. Weighting these lines by a quadratic form produced from their field of definition and arithmetic-geometric information about the configuration produces the count 1 + -1 in the Grothendieck--Witt group GW(k) of quadratic forms over k. Similarly, there are 160,839 x 1 + 160,650 x -1  3-planes in a 7-dimensional cubic hypersurface. Another example is the count -3 +4( 1 + -1) + 1 + 2  + 2D of rational curves on the Fermat Cubic surface in divisor class O_{P^3}(1) passing through 2 general points with field of definition k(sqrt{D}). No knowledge of A1-homotopy theory will be required. This talk contains joint work with Tom Bachmann, Jesse Kass, Marc Levine, Jake Solomon and Padma Srinivasan.