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Register a new userExplicit birational geometry of 3-folds of general type, I
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Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V):=text{dim} H^0(V, 12K_V)>0$ and $P_{m_0}(V)>1$ for some positive integer $m_0leq 24$. A direct consequence is the birationality of the pluricanonical map $varphi_m$ for all $mgeq 126$. Besides, the canonical volume $text{Vol}(V)$ has a universal lower bound $ u(3)geq frac{1}{63cdot 126^2}$.
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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 $mgeq 73$ and that the canonical volume $text{Vol}(V)geq {1/2660}$. When $chi(mathcal{O}_V)leq 1$, our result is $text{Vol}(V)geq {1/420}$, which is optimal. Other effective results are also included in the paper.
Nonsingular projective 3-folds $V$ of general type can be naturally classified into 18 families according to the {it pluricanonical section index} $delta(V):=text{min}{m|P_mgeq 2}$ since $1leq delta(V)leq 18$ due to our previous series (I, II). Based on our further classification to 3-folds with $delta(V)geq 13$ and an intensive geometrical investigation to those with $delta(V)leq 12$, we prove that $text{Vol}(V) geq frac{1}{1680}$ and that the pluricanonical map $Phi_{m}$ is birational for all $m geq 61$, which greatly improves known results. An optimal birationality of $Phi_m$ for the case $delta(V)=2$ is obtained. As an effective application, we study projective 4-folds of general type with $p_ggeq 2$ in the last section.
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 $text{Vol}(V) geq 1/2660$ and that the pluri-canonical map $Phi_m$ is birational onto its image for all $mgeq 77$. As an application of our method, we prove Fletchers 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 $chi({mathcal O}_V)leq 1$.
We prove that rationally connected Calabi--Yau 3-folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected $3$-folds of $epsilon$-CY type form a birationally bounded family for $epsilon>0$. Moreover, we show that the set of $epsilon$-lc log Calabi--Yau pairs $(X, B)$ with coefficients of $B$ bounded away from zero is log bounded modulo flops. As a consequence, we deduce that rationally connected klt Calabi--Yau $3$-folds with mld bounded away from $1$ are bounded modulo flops.
This version corrects a wrong proof of Proposition 6.3.2 and simplifies the exposition in Section 6.
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