Invariants of vanishing Brauer classes (Q6566584)

Let \(S\) be a complex \(K3\) surface given by a double plane ramified above a smooth sextic curve \(C\). Assume that \(S\) has Picard number \(\rho (S) = 1\) or, in other words, that \(\textrm{Pic}\, S=\mathbb Z h\) with \(h^2=2\). Let \(S_2\) be a specialization of \(S\) with \(\rho (S_2)=2\). As \(S_2\) is a specialization of \(S\), there is an embedding at the level of Picard lattices \(\textrm{Pic}\, S \hookrightarrow \textrm{Pic}\, S_2\) which, in turns, induces a restriction map at the level of Brauer groups \(\textrm{Br}\, S \cong (\mathbb{Q}/\mathbb{Z})^{21}\to \textrm{Br}\, S_2 \cong (\mathbb{Q}/\mathbb{Z})^{20}\). If we only consider the \(2\)-torsion part of the Brauer group, we get the restriction map\N\[\N(\textrm{Br}\, S)_2 \cong (\mathbb{Z}/2\mathbb{Z})^{21}\to (\textrm{Br}\, S_2)_2 \cong (\mathbb{Z}/2\mathbb{Z})^{20} \; .\N\]\NGiven the difference in the rank, the kernel of the above restriction is generated by a unique element \(\alpha_{\text{van}}\in (\textrm{Br}\, S)_2\), called the \emph{vanishing Brauer class} (with respect to the specialization of \(S\) to \(S_2)\). In the Picard number \(1\) case, there is a correspondence between the elements of \((\textrm{Br}\, S)_2\) and some line bundles on \(C\).\N\NThe first result in the paper under review is to give a classification of \(\alpha_{\text{van}}\) in terms of line bundles of \(C\) using the Gram matrix of \(\mathrm{Pic}\, S_2 \cong \mathbb Z h \oplus \mathbb Z k\).\N\NTheir attention then shifts towards an application of these results to a particular geometric construction. If \(X\subset \mathbb{P}^5\) is a smooth cubic fourfold containing a plane \(P\), then it is possible to attach a double plane \(S_X\) to \(X\). \textit{C. Voisin} [Invent. Math. 86, 577--601 (1986; Zbl 0622.14009)] shows that if \(X\) does not contain a second plane intersecting \(P\), then \(S_X\) ramifies above a smooth sextic curve \(C\) and it is a \(K3\) surface. The authors consider such cubic fourfolds with the extra assumption that the group \(N^2(X)\) of algebraic codimension two cocycles has rank 3. \textit{S. Yang} and \textit{X. Yu} [Res. Math. Sci. 10, No. 1, Paper No. 2, 39 p. (2023; Zbl 1505.14090)] classify these groups, denoting them by \(M_{\tau, n}\) with \(\tau\in \{ 0,\ldots ,4\}\) and \(n\geq 2\). They also give the list of couples \((\tau, n)\) such that if \(X_{\tau, n}\) denotes a cubic fourfold with \(N^2(X)\cong M_{\tau,n}\), then the associated double cover \(S_{\tau,n}\) is a \(K3\) surface with Picard number \(2\).\N\NThe authors of the paper under review then consider the specialization of the generic cubic fourfold \(X\) with a plane \(P\) (and no other plane intersecting \(P\)) to \(X_{\tau,n}\), for the values of \((\tau, n)\) in the list of Yang and Xu. For these values, we have an induced specialization \(S\to S_{\tau, n}\) as above. The main result of the paper consists in giving explicit descriptions of \(\mathrm{Pic}\, S_{\tau,n}\) and of \(\alpha_{\text{van}}\) in these cases.