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

Abstract

Let $n\ge 2$ 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 $v_{n,n-1}=\frac{6}{2n+(n\mathrm{mod}3)}$ and $r_{n,n-1}=\frac{1}{3}(5n+3+(n\mathrm{mod}3))$. The machinery is applicable to all canonical dimensions $n-i$.

arXiv comment16 pages, comments are welcome

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