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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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