Explicit birational geometry of threefolds of general type

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Let V be a complex nonsingular projective 3-fold of general type. We prove P₁₂(V)>0 and P₂₄(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 Vol(V) ≥ 1/2660 and that the pluri-canonical map Φₘ is birational onto its image for all m≥ 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 Vol(V)≥ 1/420 among all those 3-folds V with χ(Oᵥ)≤ 1.