In this article, we construct, for every n, smooth varieties of general type of dimension n with the first [n-2/3] plurigenera equal to zero. Hacon-McKernan, Takayama, and Tsuji have recently shown that there are numbers rnsuch that ∀ r ≥ rn, the r-canonical map of every variety of general type of dimension n is birational. Our examples show that rngrows 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 3n+1/(n-1)n. 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.
Weighted Hypersurfaces with Either Assigned Volume or Many Vanishing Plurigenera / E. Ballico, R. Pignatelli, L. Tasin. - In: COMMUNICATIONS IN ALGEBRA. - ISSN 0092-7872. - 41:10(2013), pp. 3745-3752. [10.1080/00927872.2012.677079]
Weighted Hypersurfaces with Either Assigned Volume or Many Vanishing Plurigenera
L. Tasin
2013
Abstract
In this article, we construct, for every n, smooth varieties of general type of dimension n with the first [n-2/3] plurigenera equal to zero. Hacon-McKernan, Takayama, and Tsuji have recently shown that there are numbers rnsuch that ∀ r ≥ rn, the r-canonical map of every variety of general type of dimension n is birational. Our examples show that rngrows 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 3n+1/(n-1)n. 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.File | Dimensione | Formato | |
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