Counting Lines on Hypersurfaces over general fields

One of the most famous results in enumerative geometry is the fact that, over an algebraic 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 A^1 homotopy theory from Morel and Voevodsky, we can assign a local index in the set of square classes of the field to each one of the lines, such the sum of them is invariant. In our work, we consider general hypersurfaces and give a geometrical interpretation for the local indices of lines, following ideas from Finishing and Khalarmov who worked on the real case. This is joint work with Stephen McKean and Sabrina Pauli.