Freeness of Pluricanonical Linear System on Smooth Minimal n-folds of General Type

Source abstract

Let $X$ be a smooth minimal projective variety of general type of dimension $n$ over $\mathbb C$. We prove that $|\frac{n^2+n+2}{2}K_X|$ is base point free. This gives a bound for the pluricanonical case of Fujita Freeness Conjecture with merely nef and big divisor. Let $X$ be a smooth projective variety of dimension $n$ and $B$ be a nef and big Cartier divisor. We also prove that $\operatorname{Bs}|K_X+mB|\subseteq \mathbf{B}_+(B)$ for every integer $m\geq m_n\sim n^2/\log n$, where $\mathbf{B}_+(B)$ is the augmented base locus.