Hodge theory and derived categories of cubic fourfolds
Hodge theory and derived categories of cubic fourfolds
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DOI：
10.1215/00127094-2738639
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发表时间：
2012-11
影响因子：
2.5
通讯作者：
N. Addington;Richard P. Thomas
中科院分区：
文献类型：
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作者：
N. Addington;Richard P. Thomas
三次四维体在很多方面表现得像K3曲面。某些三次曲面——据推测是有理的那些——在几何上有特定的K3曲面与之相关联。哈塞特在霍奇理论层面研究了与K3曲面相关联的三次曲面，库兹涅佐夫在导出范畴层面研究了与K3曲面相关联的三次曲面。这两种具有相关联K3曲面的概念应该是一致的。我们证明它们在一般情况下是一致的：哈塞特的三次曲面在模空间中形成不可约诺特 - 莱夫谢茨除子的可数并集，并且我们表明库兹涅佐夫的三次曲面是这些除子的一个稠密子集，在每个除子中形成一个非空的、扎里斯基开子集。
Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics with K3s associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3s associated to them at the level of derived categories. These two notions of having an associated K3 should coincide. We prove that they coincide generically: Hassett's cubics form a countable union of irreducible Noether-Lefschetz divisors in moduli space, and we show that Kuznetsov's cubics are a dense subset of these, forming a non-empty, Zariski open subset in each divisor.