Weighted hypersurfaces with either assigned volume or many vanishing plurigenera

E. Ballico; R. Pignatelli; L. Tasin

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arXiv · 1102.3105

Weighted hypersurfaces with either assigned volume or many vanishing plurigenera

E. Ballico·R. Pignatelli·L. Tasin

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Abstract

In this paper we construct, for every n, smooth varieties of general type of dimension n with the first $\lfloor \frac{n-2}{3} \rfloor$ plurigenera equal to zero. Hacon-McKernan, Takayama and Tsuji have recently shown that there are numbers $r_n$ such that, for all 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 $\frac{3^{n+1}}{(n-1)^{n}}$. 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.

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BibTeX RIS

E. Ballico, R. Pignatelli, L. Tasin. 2012-03-13. Weighted hypersurfaces with either assigned volume or many vanishing plurigenera. https://arxiv.org/abs/1102.3105

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