The main purpose of this paper is twofold. We first analyze in detail the meaningful geometric aspect of the method introduced in [12], concerning families of irreducible, nodal curves on a smooth, projective threefold X. This analysis gives some geometric interpretations not investigated in [12] and highlights several interesting connections with families of other singular geometric objects related to X and to other varieties. Then we use this method to study analogous problems for families of...

First we find effective bounds for the number of dominant rational maps $f:X\to Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type $\{A·K_X^n{\}}^{\{B·K_X^n{\}}^2}$, where $n=\dim X$, $K_X$ is the canonical bundle of $X$ and $A,B$ are some constants, depending only on $n$.Then we show that for any variety $X$ there exist numbers $c\left(X\right)$ and $C\left(X\right)$ with the following properties:For any threefold $Y$ of general type the number of dominant rational maps $f:X\to Y$ is bounded above by $c\left(X\right)$.The number of threefolds $Y$, modulo birational...

Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}\left(V\right):=dim{H}^0(V,12K_V)\>0$ and $P_{m_0}\left(V\right)\>1$ for some positive integer $m_0\le 24$. A direct consequence is the birationality of the pluricanonical map ${\varphi }_m$ for all $m\ge 126$. Besides, the canonical volume $\text{Vol}\left(V\right)$ has a universal lower bound $\nu \left(3\right)\ge \frac{1}{63·{126}^2}$.