Bao Tran mathematics

 I. Introduction: In this article, I would like to generate the work in bend and break lemma of professor Andreas Horing and professor Thomas Peternell in paper [HP16]. In the orginal paper, authors excellently resolved the problem of finding minimal model program for $\mathbb{Q}$- factorial Kahler threefold with terminal singularities and $K_X$ is not pseudo-effective. From [HP16] in general and especially in part five, I saw many proofs where the "threefold" assumption may not take a crucial role, and I try to generalize them in dimension four, at least for the case non-nef locus of $N(K_X)$ ("negative" part of Zariski decompostion of $K_X$) is not contained in any prime divisors which appear in Zariski decomposition of $K_X$. Since the closure of the union of curves' deformation could be surfaces, the restriction of $N(K_X)$ might not be pseudo-effective. So to reserve the pseudo-effectiveness, the assumption "non-nef locus of $N(K_X)$ is not containe...

In this section, I would like to generate a lemma in [HOR2016] which has improtant meaning in constructing "bend and break" for Kahler three-fold. The statement below is for the surface: Theorem (Lemma 5.5):  Let $S'S be a smooth projective surface that is uniruled, and let $C' \subset S'$ be an irreducible curve.: a) Suppose that $K_{S'}C < 0$ . Then there exists an effective 1-cycle $\sum_{i} \alpha_{i} C_i$ with coefficients in $\mathbb{Q^+}$ such that $[\sum_{i} \alpha_{i} C_i] = [C'] $ and $C_1$ is a rational curve such that $K_{S'}.C_1 < 0$ . b) Suppose that $K_{S'}.C \leq -4$ . Then there exists an effective 1-cycle $\sum_{i} \alpha_{i}C_{i}$ with $i \geq 2$ such that $[\sum_{i} \alpha_{i} C_i]=[C']$ such that $K_{S'}C_1 < 0$ and $K_{S'}.C_2 < 0$. Now the statement for higher-dimensional will be: Let $T$ be a smooth projective n-fold that is uniruled, and let $C \subset T$ be an irreducible c...