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Springer fiber components in the two columns case for types \(A\) and \(D\) are normal. (English. French summary) Zbl 1268.14006
Let \(X\) he an irreducible component of the Springer fiber \({\mathcal T}_N\) in types \(A\) and \(D\) for a nilpotent matrix \(N\) with \(N^2= 0\). Then, in the paper under review, the authors give a resolution \(\pi:\widetilde X\to X\) in an arbitrary characteristic. Using this resolution, they prove that \(X\) is Frobenius split in \(\text{char\,}p> 0\). From the Frobenius splitting of \(X\), the authors deduce that \(X\) is normal in any characteristic. Moreover, they show that \(\pi\) is a rational resolution in an arbitrary characteristic. In particular, \(X\) is Cohen-Macaulay with dualizing sheaf \(\pi_*(\omega_{\widetilde X})\) in any characteristic and \(X\) has rational singularities in \(\text{char\,}0\).

MSC:
14B05 Singularities in algebraic geometry
14N20 Configurations and arrangements of linear subspaces
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