Weighted Hypersurfaces with Either Assigned Volume or Many Vanishing Plurigenera

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.