Algebraic Groups I. Properties of orthogonal groups

Let V be a vector bundle of constant rank n ≥ 1 over a scheme S, and let q : V → L be a quadratic form valued in a line bundle L, so we get a symmetric bilinear form Bq : V × V → L defined by Bq(x, y) = q(x+ y)− q(x)− q(y). Assume q is fiberwise non-zero over S, so (q = 0) ⊂ P(V ∗) is an S-flat hypersurface with fibers of dimension n − 2 (understood to be empty when n = 1). By HW2, Exercise 4 (and trivial considerations when n = 1), this is smooth precisely when for each s ∈ S one of the following holds: (i) Bqs is non-degenerate and either char(k(s)) 6= 2 or char(k(s)) = 2 with n even, (ii) the defect δqs is 1 and char(k(s)) = 2 with n odd and qs|V ⊥ s 6= 0. (Likewise, δqs ≡ dimVs when char(k(s)) = 2.) In such cases we say (V, q) is non-degenerate; (ii) is the “defect-1” case at s. (In [SGA7, XII, §1], such (V, q) are called ordinary.) Clearly the functor S′ {g ∈ GL(VS′) | qS′(gx) = qS′(x) for all x ∈ VS′}