Assume that we work over an algebraically closed field $k$, with $char(k)\neq 2$. Suppose that one has a quadric bundle $\pi:Q\rightarrow S$ over a smooth scheme $S$, and that $Q$ degenerates simply over a divisor $D\subset S$. Can one find a stack $\mathcal{S}\rightarrow S$ over S, such that the pull-back of Q to $\mathcal{S}$ is a nondegenerate quadric bundle over $\mathcal{S}$?
The question is interesting also just over a geometric point, without considering $D$. Say we have a simply degenerate quadric $Q$ over $Spec(k)$, is there a stack $\mathcal{S} \rightarrow Spec(k)$ s.t the pullback of $Q$ to $\mathcal{S}$ is nondegenerate?
Take ${\mathcal S} = S \setminus D$! – Sasha Apr 7 2011 at 20:29