On explicit birational geometry for minimal n-folds of canonical dimension n-1

TL;DR

This paper determines the precise bounds for the canonical volume and stability index of minimal projective n-folds of general type with canonical dimension n-1, extending known results to this specific case.

Contribution

It explicitly computes the optimal bounds for canonical volume and stability index for minimal n-folds with canonical dimension n-1, generalizing previous results.

Findings

The canonical volume bound is v_{n,n-1} = 6 / (2n + (n mod 3))

The stability index bound is r_{n,n-1} = (5n + 3 + (n mod 3)) / 3

The methods apply to all canonical dimensions n-i.

Abstract

Let $n\geq 2$ be any integer. We study the optimal lower bound $v_{n, n-i}$ of the canonical volume and the optimal upper bound $r_{n,n-i}$ of the canonical stability index for minimal projective $n$-folds of general type, which are canonically fibered by $i$-folds ($i=0,1$). The results for $i = 0$, $v_{n,n}=2$ and $r_{n, n}=n+2$, are known to experts. In this article, we show that $v_{n,n-1}=\frac{6}{2n+(n \bmod 3)}$ and $r_{n,n-1}=\frac{1}{3}(5n+ 3 + (n \bmod 3))$. The machinery is applicable to all canonical dimensions $n-i$.