Weighted hypersurfaces with either assigned volume or many vanishing plurigenera

Details Weighted hypersurfaces with either assigned volume or many vanishing plurigenera Weighted hypersurfaces with either assigned volume or many vanishing plurigenera

2011-02-15

arxiv.org/abs/1102.3105v2

Mathematics

Algebraic Geometry

7 pages; v2: minor corrections, final version to be published in Communications in Algebra

Scientific paper

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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