Bao Tran mathematics

In this post, I will try to give an Introduction and some basic properties of canonical hyperplane section of a projective surface. The definitions, theorems or properties related to this kind of surface are taken from the thesis of Epema. This is also a part of my master 2 thesis studies. I. Introduction:  First we come to the definition of surface with canonical hyperplane sections: Definition 1.1 : A curve $C$ with genus $g \geq 2$ is called canonically embedded if there exists an embedding $i_{|K_C|}: C \hookrightarrow \mathbb{P}^{g-1}$. Here this curve must be smooth since we assume that $deg K_C = 2g - 2$. Definition 1.2 : $X$ is called a surface with canonical hyperplane sections if there exists an embedding $i : X \hookrightarrow \mathbb{P}^g$, $g \geq 3$, such that a general hyperplane sections $C$ of $i(X)$, i.e., the intersection of $X$ with a hyperplane of degree $1$ is a canonically embedded curve with genus $g$. Remark 1 : Here, $C$ is the general curve of th...

 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...

In my previous post, I mentionned about a generalization of Hartshorne's conjecture in [Jie17] : Theorem 1: Let $X$ be a complex projective manifold of dimension $ n$ . Suppose that $T_X$ contains an ample subsheaf $\mathcal{F}$  of positive rank $r$ , then $(X, \mathcal{F})$ is isomorphic to $(\mathbb{P}^n, T_{\mathbb{P}^n})$ or $(\mathbb{P}^n, O_{\mathbb{P}^1}^{\oplus r} )$.   This theorem only needs $T_X$ containing an ample subsheaf, which is far stronger than theorem 8 in [Mor79]. To prove Theorem 1, Jie represented the concept of foliation and then combined with Theorem 1.1 in [Arau06] to derive to two key theorems, those are 2.2.9 and 2.3.8 in [Jie17]. For detail of proofs and definitions, please c.f. [Jie17].  Now we will come to key theorems for our proofs of Theorem 1: Theorem 2 (theorem 2.2.9): Let $X$ be a projective manifold. Assume that $T_X$ contains an ample subsheaf $\mathcal{F}$ of positive rank $r < dim(X)$ . Then its saturation ...