2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/140663Let $V$ 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 $\text{Vol}(V) \geq 1/2660$ and that the pluri-canonical map $Φ_m$ is birational onto its image for all $m\geq 77$. As an application of our method, we prove Fletcher's conjecture on weighted hyper-surface 3-folds with terminal quotient singularities. Another featured result is the optimal lower bound $\text{Vol}(V)\geq {1/420}$ among all those 3-folds $V$ with $χ({\mathcal O}_V)\leq 1$.(updated version on October 15, 2007) 55 pages, a couple of missing P_2 terms in Section 5 added and slight rearrangements to the contextAlgebraic Geometrytext