A support theorem for nested Hilbert schemes of planar curves (Q2226557)

For a complex, complete algebraic curve \(C\), one can form the Hilbert schemes \(C^{[m]}\) parametrizing subschemes \(Z \subset C\) of length \(m\) and nested Hilbert schemes \(C^{[m,m+1]}\) parametrizing pairs \(\{(Y,Z): Y \in C^{[m]}, Z \in C^{[m+1]}, Y \subset Z\}\). If \(\pi: \mathcal C \to B\) is a proper flat family of curves, there are relative versions \(\pi^{[m]}: {\mathcal C}^{[m]} \to B\) and \(\pi^{[m,m+1]}: {\mathcal C}^{[m,m+1]} \to B\). For families of curves that are reduced and locally planar over a smooth base \textit{V. Shende} showed that if \(\mathcal C \to B\) is a locally versal family of reduced, locally planar curves over a smooth base \(B\), then the total space \({\mathcal C}^{[m]}\) is smooth over sufficiently small analytic open sets in \(B\) provided \(m \leq \dim B\) [Compos. Math. 148, 531--547 (2012; Zbl 1312.14015)]. The author uses similar methods to prove an analogous smoothness result for \({\mathcal C}^{[m,m+1]}\). For a smooth family \(\pi^{[m]}: {\mathcal C}^{[m]} \to B\), the decomposition theorem of \textit{Beilinson, Bernstein and Deligne} [Astérisque 100, (1982; Zbl 0536.14011)] says that the complex \(R \pi_*^{[m]} \mathbb C\) decomposes as a direct sum of shifted intersection complexes. In this setting, \textit{L. Migliorini} and \textit{V. Shende} determined the decomposition of \(R \pi^{[m]}_* \mathbb Q [m + \dim B]\) [J. Eur. Math. Soc. (JEMS) 15, 2353--2367 (2013; Zbl 1303.14019)], showing that none of the summands have proper support in \(B\). The author determines the analogous decomposition of \(R \pi_*^{[m,m+1]} \mathbb Q [m+1+\dim B]\) assuming that \({\mathcal C}^{[m,m+1]}\) is smooth, again showing that none of the summands has proper support in \(B\). The proof uses the theory of higher discriminants introduced by \textit{L. Migliorini} and \textit{V. Shende} [Algebr. Geom. 5, No. 1, 114--130 (2018; Zbl 1406.14005)].