Related papers: Explicit birational geometry of 3-folds of general…

Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V)>0$ and $P_{24}(V)>1$ (which answers an open problem of J. Kollar and S. Mori). We also prove that the canonical volume has an universal lower bound…

Let $V$ be a complex nonsingular projective 3-fold of general type. We shall give a detailed classification up to baskets of singularities on a minimal model of $V$. We show that the $m$-canonical map of $V$ is birational for all $m\geq 73$…

For nonsingular projective 4-folds V of general type with plurigenus $P_{m_0}(V) \geq 2$ for some positive integer $m_0$, we show that $\varphi_{m}$ is birational onto its image for all integers $m \geq 76m_0+77$ and the canonical volume…

Nonsingular projective 3-folds $V$ of general type can be naturally classified into 18 families according to the {\it pluricanonical section index} $\delta(V):=\text{min}\{m|P_m\geq 2\}$ since $1\leq \delta(V)\leq 18$ due to our previous…

Let $V$ be a complex nonsingular projective 3-fold of general type with $\chi(\omega_V)\geq 0$ (resp. $>0$). We prove that the m-canonical map $\Phi_{|mK_V|}$ is birational onto its image for all $m\ge 14$ (resp. $\geq 8$). Known examples…

We prove that for all nonsingular projective 3-folds of general type with third plurigenus $P_3 \geq 2$, the pluricanonical map $\varphi_m$ is birational onto its image for all $m \geq 14$, which is optimal.

There are many examples of 3-folds of general type with $\chi(\mathcal {O})=1$ found by Iano-Fletcher and Reid about twenty years ago. Iano-Fletcher has ever proved $P_{12}(X)\ge 1$ and $P_{24}(X)\ge 2$ for all minimal 3-folds $X$ of…

Let $V$ be a nonsingular projective 3-fold of general type. When the pluricanonical section index $\delta(V)>12$, Chen-Chen \cite{Chen3} has a complete list of the possibility for the weighted basket ${\mathbb B}(V)$. However the…

In this paper we study pluricanonical maps of minimal projective 3-folds of general type with geometric genus $1$, $2$ and $3$. We go in the direction pioneered by Enriques and Bombieri, and other authors, pinning down, for low projective…

We show that, for nonsingular projective 4-folds V of general type with geometric genus $p_g\geq 2$, the 33-canonical map is birational onto the image and the canonical volume has the lower bound $1/520$, which improves a previous theorem…

We explicitly find lower bounds on the volume of threefolds and fourfolds of general type in order to have nonvanishing of pluricanonical systems and birationality of pluricanonical maps. In the case of threefolds of large volume, we also…

We prove that for any smooth projective $3$-fold of general type with canonical volume greater than $12^6$, the image of its bicanonical map has dimension at least $2$. We also study pluricanonical maps of $3$-folds of general type with…

We prove that the canonical volume $K^3\geq {1/30}$ for all projective 3-folds of general type with $\chi(\mathcal{O})\leq 0$. This bound is sharp.

Let $V$ be a smooth projective 3-fold of general type. Denote by $K^{3}$, a rational number, the self-intersection of the canonical sheaf of any minimal model of $V$. One defines $K^{3}$ as a canonical volume of $V$. The paper is devoted to…

Let $X$ be a complex smooth projective threefold of general type. Assume $q(X)>0$. We show that the $m$-canonical map of $X$ is birational for all $m\geq 5$.

This paper aims to improve a theorem of Janos Kollar. For a given Complex projective threefold X of general type, suppose the plurigenus p_k(X)\ge 2, Kollar proved that the (11k+5)-canonical map is birational. Here we show that either the…

Let V be a smooth projective 3-fold of general type. Denote by $K^3$, a rational number, the self-intersection of the canonical sheaf of any minimal model of V. One defines $K^3$ as the canonical volume of $V$. Assume $p_g\ge 2$. We show…

By a canonical (resp. terminal) weak $\mathbb{Q}$-Fano $3$-fold we mean a normal projective one with at worst canonical (resp. terminal) singularities on which the anti-canonical divisor is $\mathbb{Q}$-Cartier, nef and big. For a canonical…

Let $n\geq 2$ be any integer. We study the optimal lower bound $v_{n, n-i}$ of the canonical volume and the optimal upper bound $r_{n,n-i}$ of the canonical stability index for minimal projective $n$-folds of general type, which are…

For a $\mathbb{Q}$-Fano 3-fold $X$ on which $K_X$ is a canonical divisor, we investigate the geometry induced from the linear system $|-mK_X|$ in this paper and prove that the anti-$m$-canonical map $\varphi_{-m}$ is birational onto its…