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

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 $n-i$.

Theorems & Definitions (20)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1
• proof
• Definition 2.2
• Definition 2.3
• Proposition 2.4: Birationality principle
• Corollary 2.5
• proof
• Example 3.1