Weighted hypersurfaces with either assigned volume or many vanishing plurigenera

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.