Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces

Let k be a field and let GW (k) be the Grothendieck-Witt ring of virtual non-degenerate symmetric bilinear forms over k. We develop methods for computing the quadratic Euler characteristic (X/k) GW (k) for X a smooth hypersurface in a projective space or a weighted projective space. We raise the question of a quadratic refinement of classical conductor formulas and find such a formula for the degeneration of a smooth hypersurface X in P^n+1 to the cone over a smooth hyperplane section of X; we also find a similar formula in the weighted homogeneous case. We formulate a conjecture that generalizes these computations to similar types of degenerations. Finally, we give an interpretation of the quadratic conductor formulas in terms of Ayoub's nearby cycles functor.