Weighted hypersurfaces with either assigned volume or many vanishing plurigenera

TL;DR

This paper constructs smooth varieties of general type with specific vanishing plurigenera and analyzes their volume and canonical maps, showing growth bounds and volume diversity across dimensions.

Contribution

It provides explicit examples of varieties with zero initial plurigenera, establishes lower bounds on the r-canonical map, and demonstrates the existence of varieties with arbitrary volume.

Findings

r_n grows at least quadratically with dimension n

Minimal volume of varieties of general type is bounded above by a specific exponential function

Existence of varieties with any positive rational volume in arbitrarily high dimensions

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.