Weighted Hypersurfaces with Either Assigned Volume or Many Vanishing Plurigenera
Weighted Hypersurfaces with Either Assigned Volume or Many Vanishing Plurigenera
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DOI：
10.1080/00927872.2012.677079
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发表时间：
2011-02
影响因子：
0.7
通讯作者：
E. Ballico;R. Pignatelli;L. Tasin
中科院分区：
文献类型：
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作者：
E. Ballico;R. Pignatelli;L. Tasin
本文对任意n构造了维数为n且第一多属为零的一般型光滑簇。Hacon-McKernan、Takayama和Tsuji最近证明了存在数r n使得n ≥ r n，每个维数n的一般类型的簇的r-正则映射是双有理的。我们的例子表明，r n增长至少二次作为n的函数。并且证明了各种一般类型的维数n的最小体积都小于。此外，我们还证明了对任意正有理数q，都存在体积q和维数任意大的一般型光滑簇。
In this article, we construct, for every n, smooth varieties of general type of dimension n with the first plurigenera equal to zero. Hacon-McKernan, Takayama, and Tsuji have recently shown that there are numbers r n such that ∀ r ≥ r n , the r − canonical map of every variety of general type of dimension n is birational. Our examples show that r n grows at least quadratically as a function of n. Moreover, they show that the minimal volume of a variety of general type of dimension n is smaller than . In addition, we prove that for every positive rational number q there are smooth varieties of general type with volume q and dimension arbitrarily big.