Weighted Hypersurfaces with Either Assigned Volume or Many Vanishing Plurigenera

Abstract

In this article, we construct, for every n, smooth varieties of general type of dimension n with the first plurigenera equal to zero. Hacon-McKernan, Takayama, and Tsuji have recently shown that there are numbers r n such that ∀ 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 . 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.

Locations

• arXiv (Cornell University) - View - PDF
• Communications in Algebra - View