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Severi varieties and their varieties of reductions

Atanas Iliev·Laurent Manivel
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Abstract

We study the varieties of reductions associated to the four Severi varieties, the first example of which is the Fano threefold of index 2 and degree 5 studied by Mukai and others. We prove that they are smooth but very special linear sections of Grassmann varieties, and rational Fano manifolds of dimension $3a$ and index $a+1$, for $a=1, 2, 4, 8$. We study their maximal linear spaces and prove that through the general point pass exactly three of them, a result we relate to Cartan's triality principle. We also prove that they are compactifications of affine spaces.

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Atanas Iliev, Laurent Manivel. 2003-06-23. Severi varieties and their varieties of reductions. https://arxiv.org/abs/math/0306328

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