Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces

Let k be a perfect 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 χ(X/k)∈GW(k) for X a smooth hypersurface in a projective space and in 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 Pn+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 for similar types of degenerations, and we interpret the quadratic conductor formulas in terms of Ayoub's motivic nearby cycles functor.