Related papers: A Note on Anti-Pluricanonical Maps for 5-Folds

We investigate birationality of the anti-pluricanonical map $\phi_{-m}$, the rational map defined by the anti-pluricanonical system $|-mK|$, on $\mathbb{Q}$-Fano threefolds.

For a terminal weak $\mathbb{Q}$-Fano $3$-fold $X$, we show that the $m$-th anti-canonical map defined by $|-mK_X|$ is birational for all $m\geq59$.

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 $m\geq 5$.

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…

We prove that for a smooth projective irregular $3$-fold $X$ with $K_X\equiv 0$ and a nef and big divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\geq 3$ and all $P\in \text{Pic}^0(X)$. We also use the same method to deal with…

In this paper we will prove that for a threefold of general type and large volume the second plurigenera is positive and the fifth canonical map is birational.

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…

We show that $|mK_X|$ defines a birational map and has no fixed part for some bounded positive integer $m$ for any $\frac{1}{2}$-lc surface $X$ such that $K_X$ is big and nef. For every positive integer $n\geq 3$, we construct a sequence of…

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…

In this paper we prove that for any smooth projective variety of Albanese fiber dimension two and of general type, the 6-canonical map is birational. And we also show that the 5-canonical map is birational for any such variety with some…

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.

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…

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…

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…

In continuation of our paper in Math. Ann. 333 we classify smooth complex projective threefolds X with -K_X big and nef but not ample and Picard number 2, whose anticanonical map is small. We assume also that the Mori contraction of X and…

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…

We start the classification of smooth projective threefolds X whose anticanonical bundles -K_X are big and nef but not ample. In this paper we treat the case b_2(X) = 2 and the morphism associated with the base point free linear system…

We show that if X is a nonsingular projective variety of general type over an algebraically closed field k of positive characteristic and X has maximal Albanese dimension and the Albanese map is separable, then |4K_X| induces a birational…

Let $X$ be a smooth complex projective algebraic variety of maximal Albanese dimension. We give a characterization of $\kappa (X)$ in terms of the set $V^0(X,\omega_{X})$ $:=\{P\in {\text{\rm Pic}}^0(X)|h^0(X, \omega_X \otimes P) \ne 0\}$.…

We study canonical and pluricanonical maps of varieties isogenous to a product of curves, i.e., quotients of the form $X = (C_1 \times \dots \times C_n)/G$ with $g(C_i)\ge 2$ and $G$ acting freely. For this purpose, we provide a technical…