GAAG seminar - Quadratic Segre indices: counting lines on hypersurfaces over any field

Every smooth complex cubic surface contains exactly 27 lines, but a real one may contain 3, 7, 15 or 27 real lines. Segre divided real lines into hyperbolic and elliptic ones, depending on whether a natural involution on the line has real fixed points. Assigning +1 to the elliptic lines and -1 to the hyperbolic ones, the count is always 3. Kass and Wickelgren showed that this generalizes to a single statement valid over any field: instead of signs, each line contributes with a quadratic form over the field, read off from its Segre involution. The total count is then the quadratic form 15〈1〉 + 12〈−1〉 in the Grothendieck–Witt ring.

In this talk I will explain how this picture extends to lines on general hypersurfaces of degree 2n−1 in P^{n+1} for every n, from the 27 lines on a cubic surface through the 2875 on a quintic threefold and beyond. Over the real numbers, Finashin and Kharlamov attached to each line a Segre index, built from involutions determined by the secants of its Gauss curve. We show that a quadratic version of this index computes the line’s local contribution to the A^1-Euler number over any perfect field of characteristic not 2. This generalizes the cases of cubic surfaces (Kass–Wickelgren) and quintic threefolds (Pauli). The real proof uses wall-crossing, but ours is algebraic: a “conic model” replaces the Gauss curve by a plane conic and a configuration of points, and the local index then factors as a product of resultants.

This is joint work with Stephen McKean and Sabrina Pauli.