An optimal boundedness on weak $\bQ$-Fano threefolds

Abstract

Let $X$ be a terminal weak $\bQ$-Fano threefold. We prove that $P_{-6}(X)>0$ and $P_{-8}(X)>1$. We also prove that the anti-canonical volume has a universal lower bound $-K_X^3\ge 1/330$. This lower bound is optimal.