On minimal varieties growing from quasismooth weighted hypersurfaces

Abstract

This paper concerns the construction of minimal varieties with small canonical volumes. The first part devotes to establishing an effective nefness criterion for the canonical divisor of a weighted blow-up over a weighted hypersurface, from which we construct plenty of new minimal $3$-folds including 59 families of minimal $3$-folds of general type, several infinite series of minimal $3$-folds of Kodaira dimension $2$, 2 families of minimal $3$-folds of general type on the Noether line, and 12 families of minimal $3$-folds of general type near the Noether line. In the second part, we prove effective lower bounds of canonical volumes of minimal $n$-folds of general type with canonical dimension $n-1$ or $n-2$. Examples are provided to show that the theoretical lower bounds are optimal in dimension at most $5$ and nearly optimal in higher dimensions.