Special prime Fano fourfolds of degree 10 and index 2
Special prime Fano fourfolds of degree 10 and index 2
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10 次指数为 2 的特殊素数 Fano 四倍数
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
L. Manivel
中科院分区：
文献类型：
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作者：
O. Debarre;A. Iliev;L. Manivel
Mukai证明了Grassmannian G（2，5）中包含大多数次数为10且指数为2的素Fano四重。正如罗斯在1949年所指出的，它们都是单理性的，有些是理性的。我们表明，他们的中间上同调是K3型，他们的周期映射是占主导地位的，光滑的4维纤维，到20维有界对称的IV型周期域。根据Hassett，我们说这样的四重是特殊的，如果它包含一个曲面，其上同调类不来自格拉斯曼G（2，5）。特殊的四重对应于周期域中的超曲面的可数并，用正整数d标记，判别式。我们描述了一些特殊的四倍的d值较低。我们还刻画了那些存在特殊四重的整数d。
Mukai proved that most prime Fano fourfolds of degree 10 and index 2 are contained in a Grassmannian G(2,5). They are all unirational and some are rational, as remarked by Roth in 1949. We show that their middle cohomology is of K3 type and that their period map is dominant, with smooth 4-dimensional fibers, onto a 20-dimensional bounded symmetric period domain of type IV. Following Hassett, we say that such a fourfold is special if it contains a surface whose cohomology class does not come from the Grassmannian G(2,5). Special fourfolds correspond to a countable union of hypersurfaces in the period domain, labelled by a positive integer d, the discriminant. We describe special fourfolds for some low values of d. We also characterize those integers d for which special fourfolds do exist.