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

Abstract

Let n ≥ 2 be any integer. We study the optimal lower bound vn,n−i of the canonical volume and the optimal upper bound rn,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, vn,n = 2 and rn,n = n + 2, are known to experts. In this article, we show that vn,n−1 = 6 2n+(n mod 3) and rn,n−1 = 1 3 (5n + 3 + (n mod 3)). The machinery is applicable to all canonical dimensions n − i.

Automorphisms of weighted projective hypersurfaces

“…Indeed, many of their properties are determined combinatorially by the choice of degree and weights, and they exhibit a large range of behavior. In particular, there is significant evidence that weighted projective hypersurfaces are flexible enough to solve a diverse range of optimization problems in algebraic geometry (see, for example, [12,6,11,33]). It is therefore desirable to understand basic properties of their automorphism groups.…”