Felipe Espreafico Guelerman (Université Paris-Sorbonne)

One of the most famous results in enumerative geometry is the fact that, over an algebraically closed field, there are exactly 27 lines on a smooth cubic surface. One may ask, however, what happens if the field is not algebraically closed. Is there a way to get an "invariant count," i.e., a count that does not depend on the cubic? Over the reals, if one counts lines with signs, there are exactly 3 real lines. In general, using tools from A1 homotopy theory by Morel and Voevodsky, we can assign a local index in the set of square classes of the field to each of the lines, such that the sum of them is invariant. In our work, we consider general hypersurfaces and provide a geometric interpretation for the local indices of lines, following ideas from Finashin and Kharlamov, who worked on the real case. This is joint work with Stephen McKean and Sabrina Pauli.