Quadric surface bundles over surfaces (Q273850)

Let \(S\) be a noetherian integral regular scheme on which \(2\) is invertible. Consider a line-bundle valued quadratic form \(q:\mathcal{E}\to\mathcal{L}\) over \(S\). Here, \(\mathcal{E}\) is a locally free \(\mathcal{O}_S\)-sheaf of rank \(n\) and \(\mathcal{L}\) is a line bundle over \(S\). The quadratic form \(q\) is said to have \textit{simple degeneration of multiplicity \(1\)} along a reduced effective Weil divisor \(D\) if for every \(x\in S\) of codimension \(1\), the specialization of \(q\) to the local ring \(\mathcal{O}_{S,x}\) is regular (i.e.\ unimodular) when \(x\) is not the generic point of a component of \(D\), and otherwise, it can be diagonalized to the form \(\langle\alpha_1,\dots,\alpha_{n-1},\alpha_n\rangle\) with \(\alpha_1,\dots,\alpha_{n-1}\in\mathcal{O}_{S,x}^\times\) and \(\alpha_n\) a generator of the maximal ideal of \(\mathcal{O}_{S,x}\).NEWLINENEWLINENEWLINEVery loosely speaking, the ``zeros'' of \(q\) in \(\mathbb{P}(\mathcal{E})\) form a quadric bundle \(Q\) over \(S\). When \(n\) is even and \(q\) is simply degenerate of multiplicity \(1\) along a divisor \(D\), one can associate with \(Q\to S\) a discriminant cover \(T\to S\), which is the finite morphism associated with center of the even Clifford algebra of \((\mathcal{E},q,\mathcal{L})\). The branch divisor of \(T\to S\) is \(D\).NEWLINENEWLINENEWLINESuppose \(\dim S=2\) and let \(T\to S\) be a double cover with regular branch divisor \(D\). The main result of the paper establishes a bijection between isomorphism classes of: {\parindent=0.7cm\begin{itemize}\item[(1)] quadric bundles of rank \(4\) (equivalently, of relative dimension \(2\)) with simple degeneration of multiplicity \(1\) along \(D\) and discriminant \(T\to S\), \item[(2)] Azumaya quaternion algebras over \(T\) whose norm to \(S\) is split. NEWLINENEWLINE\end{itemize}} The bijection from (1) to (2) is given by taking the even Clifford algebra. If \(L\), \(K\) denote the function fields of \(T\), \(S\) respectively, the norm of a quaternion Azumaya algebra \(\mathcal{A}\) over \(T\) is given as the unique Azumaya algebra \(\mathcal{B}\) over \(S\) for which \(\mathrm{cor}_{L/K}(\mathcal{A}\otimes_{\mathcal{O}_T} L)\cong \mathcal{B}\otimes_{\mathcal{O}_S}K\). The existence of \(\mathcal{B}\) is nontrivial, since \(T\to S\) is ramified, and is shown by the authors under the previous assumptions.NEWLINENEWLINEThe authors give two applications of their result. The first application uses the equivalence to demonstrate the failure of a local-global principle for isotropy of quadratic forms over surfaces. Specifically, it is shown that if \(K\) is field which is finitely generated and of transcendence degree \(2\) over an algebraically closed field of characteristic \(0\), then there exists an anisotropic quadratic form \(q\) of rank \(4\) over \(K\) such that \(q\) is isotropic over the completion of \(K\) relative to any discrete valuation.NEWLINENEWLINENEWLINEThe second application is an algebraic proof of the global Torelli Theorem for general cubic fourfolds containing a plane. The global Torelli theorem for cubic fourfolds, proved by \textit{C.\ Voisin} [Math. 86, 577--601 (1986; Zbl 0622.14009)], states that a cubic hypersurface \(Y\) in \(\mathbb{P}^5\) is determined up isomorphism by \(\mathrm{H}^4(Y,\mathbb{Z})\) and its polarized Hodge structure.NEWLINENEWLINEIt is well-known that when the cubic \(Y\) is sufficiently general and contains a plain \(P\), one can view the blow up \(\tilde{Y}\) of \(Y\) along \(P\) as a quadric bundle \(\tilde{Y}\to \mathbb{P}^2\) of rank \(4\) with simple degeneration of multiplicity \(1\). The discriminant cover \(f:T\to \mathbb{P}^2\) is a (smooth) \(K3\) surface, and the pullback \(\mathcal{F}:=f^*\mathcal{O}_{\mathbb{P}^2}(1)\) is a polarization of \(T\). The authors use their main theorem to show that the data of \(T\), \(\mathcal{F}\), and a certain relaxation of the datum of the even Clifford algebra of \(\tilde{Y}\to \mathbb{P}^2\), determine \(Y\) up to isomorphism; this was shown by Voisin (loc.\ cit.) using Hodge theory. Together with the global Torelli theorem for \(K3\) surfaces [\textit{I. I.\ Piatetski-Shapiro} and \textit{I. R.\ Shafarevich}, Izv.\ Akad.\ Nauk SSSR, Ser.\ Mat.\ 35, 530--572 (1971; Zbl 0219.14021)], this implies the global Torelli theorem for general cubic fourfolds containing a plane.NEWLINENEWLINEThe paper also includes useful results about isometry group schemes of quadratic forms with simple degeneration, and gluing of tensors.