Bao Tran mathematics

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 ...

In [Arau06], Araujo proved an interesting theorem which after that, used for proving a generalization of Hartshrone's conjecture. That theorem is: Theorem: Suppose $C_x$ is a $d$- dimensional linear subspace of $\mathbb{P}(T_{x}X)$ for a general point $x \in X$. There is a dense open subset  $X^0$ of $X$ and a $\mathbb{P}^{d+1}$-bundle $\varphi: X^0 \rightarrow T^0$ such that any curve from a minimal covering of rational curves, denote $H$, meeting $X^0$ is a line on a fibre of $\varphi$. For the application of this theorem, I will write later. Now, I will give a part of the construction of $X^0$ and $\varphi$, for the second statement, you can see theorem 3.1 in [Arau06]. First, there is some notion and properties that reader should know. I will copy the open part of [Arau06] right here. Let $X$ be a smooth complex projective variety and, assume that X is uniruled. Let  $H$ be an irreducible component of $RatCurves^{n}(X)$ . We say that $H$ is a covering family i...

This is my preparation for a small talk in commutative algebra conference in free discussing time, but it didn't happen because the organizers didn't have time for that (although my proof said we would have time to talk about our sections). Introduction:    When I was reading about COhen-Macaulay local ring, I found an example of a non-CM local ring, that is the ring $k[x,y,z,w]/(x,y) \cap (z,w)$. Because examples are important in mathematics, so I extend that example to a more general ring in light of a theorem in Eisenbud's book.    Main talk: My Non-CM rings that I've proved is: $ k[x_1, x_2,..,x_n]/(x_1,...,x_k) \cap (x_k,x_{k+1},...,x_{2k})$ with $n \geq 4, k \geq 2, 2k \leq 2$. To prove this, we need a theorem in Eisenbud's book "commutative algebra: a view toward to algebraic geometry". Theorem:   Let $I, J$ are two ideals of a local ring $R$ whose radical ideals are incomparable. If $I \cap J = 0$, then grade($I+J)\leq 1$ proof:   Assume ...

In 1966, Kleimann's paper ([Klei66]) was published on Annals of math, and it mostly determined a new kind of technique which would be used widely after that in studying birational geometry, especially by Mori. He (Kleimann) constructed a space of 1-cycle under the numerical relation generated by curves which isomorphics to $\mathbb{R}^n$ with usual topology and that gave us chances to study algebraic geometry in the light of euclidian space. Both Kleimann and Mori used this to study about the set of positive multiplicity 1-cycle $NE(X) = \sum_{C \subset X} \mathbb{R}_{\geq 0} [C]$ and its closure in  $\mathbb{R}^n$ in usual topology. The $\overline{NE(X)}$ is also called Kleiman-Mori cone [Mum-Oda]. Kleimann provided a criterion of ampleness by using $\overline{NE(X)}$. Mori even went futher. He pointed out the relation between rational curves, ample invertible sheaf and $\overline{NE(X)}$ by a theorem which is called "cone theorem" in [Mor82].In this post, I will give so...

In this section, I will give a half of a proof of Theorem 4 in [Mor79]. We usually call this"bend and break" lemma which widely used in birational geometry. Theorem : Let $f: \mathbb{P}^1 \rightarrow C$ be a morphism such that $f(0) \neq f(\infty)$. Suppose there exists a smooth connected curve $D$, a closed point $t_0 \in D$ and a morphism $\varphi: \mathbb{P}^1 \times D \rightarrow C$ such that:                                                                                      $\varphi |_{\mathbb{P}^1 \times t_0}=f$          ...

In this post, I will rewrite a prove for a proposition of extremal nbd from [Mor88]. First, there is some lemmas we need to know first. Lemma 1: For an extremal nbd $X \subset C \cong \mathbb{P}^{1}$: $H^{1}(O_Z) = H^1(\omega_X \otimes O_Z) = 0$ for all $Z \subset X$ with $Z_{red}=C$. Lemma 2: Over a smooth projective curve, for any locally free module $ \xi$ of rank $r$, we have equation: $\chi (C, \xi) =$ deg (det $\xi$)+($1-g$)rk $\xi$. Now we will come to the main theorem: Theorem:  Let $X \subset C \cong \mathbb{P}^{1}$ is smooth extremal nbd. Then $O_C(K_X) \cong O_C(-1)$, $I_C/I_C^{2} \cong O_C \oplus O_C(1)$ and $|-K_X|$ has smooth member, where $I_C$ is an ideal sheaf of $C$. Proof of theorem. By the definition of extremal nbd, $K_X$ is not ample, hence $O_X(K_X) \otimes O_C$ must not be ample and deg  $O_X(K_X) \otimes O_C$ = deg $(K_X.C)$  $< 0$ (it cannot equal to $0$ because $K_X$ is not equivalence to $0$). By Remainn-Roch theorem, $h^0(O_X...