arXiv subjects
Publications and source records attributed to Pinxian Bie.
We prove an explicit lower bound of order $\varepsilon^3/(1+\log(1/\varepsilon))$ for the anticanonical volume of every complex $\varepsilon$-log canonical weak del Pezzo surface. This general estimate is close to being sharp, as it differs from the conjectured optimal quadratic order only by one power of $\varepsilon$ and a logarithmic factor.
We investigate the DCC property of the set of Iitaka volumes of a given set of pairs of varieties. We both generalize previous results of Birkar and Li about usual pairs to the real coefficient case, and also establish similar results on generalised pairs, where some natural boundedness assumptions are required for technical reasons.
We give some explicit upper bounds on the effective birationality of the canonical or anti-canonical system for a singular surface. In particular, we show that for any surface $X$ with $\epsilon$-lc singularity and the canonical divisor $K_X$ or the anti-canonical divisor $-K_X$ is big and nef, then there exists explicit natural numbers $m$ and $l$ depending only on $\epsilon$ such that $|mK_X|$ or $|-lK_X|$ defines a birational map. Although these explicit values are expected to be far from optimal, they are the first explicit upper bounds of this type for surfaces.
This paper is devoted to the generalization of the construction of minimal varieties from the previous work of Meng Chen, Chen Jiang and Binru Li. We first establish several effective nefness criterions for the canonical divisor of weighted blow-ups over a weighted complete intersection, we both consider the high codimensional case and the blowing up several points case, from which we construct plenty of new minimal $3$-folds including $79$ families of minimal $3$-folds of general type, several infinite series of minimal $3$-folds of Kodaira dimension $2$, $16$ families of minimal $3$-folds of general type on or near the Noether lines.