Bao Tran mathematics

In this post, we will discuss about a theorem which plays a crucial role in the proof of Hartshorne conjecture stated by Shigefumi Mori in [Mor79]. Theorem 1: (Collolary 7, [Mor70]): Let $X$ be an n-dimensional non singular projective variety over an algebraic closed field $k$ such that tangent bundle $T_X$ is ample. Prove that: i) $X$ contains a rational curve and any rational curve can be deformed as a cycle into a sum of rational curves $C$ such that $(C.K^{-1}_X)=n+1$. ii) If $C$ ($\subset X$) is a rational curve such that $(C.K^{-1}_X)=n+1$, the resolution $f: \mathbb{P} \longrightarrow C$ is unramified and $f^{*}(T_{X|C}) = O(2) \oplus O(1)^{n-1}$. Why is this theorem important ? In the proof of theorem 8, Mori construted a variety $Y$ which "parameterizing  the slight modification $Z$ of maximal connected family of rational curves with minimal degree" and "rational curves with minimal degree" are those curves $C$ in our theorem. How did he do that ? Fi...

In this article, i will give a detail of a proof of (8.3) and (8.4) in [Mor], 1979. (8.3) $O_S \otimes O_Z(S)\cong O_S(-1)$.  Let $L \cong \mathbb{P} ^{n-2}$ be a hyperplane, then because $S$ is a section of $Z$ over $Y$, which means $\psi (S) \cong Y \cong \mathbb{P}^{n-1}$, so in $ \mathbb{P}^{n-1}$, we can easily find a curve not contains in a hyperplane, so we can find $C \cong \mathbb{P}^{1} \subset S$ be a straight line such that $\psi(C) \nsubseteq L$. Let $D = \psi^{-1}(L)$, and then $(C . D) = 1$ because straight line intersect hyperplane at only one point. Since $\phi(S) = R$ and $S \cap D \neq { \varnothing }$, then $R \in \phi(D)$, and $V = \phi^{-1} \phi(D) \supseteq S \cup D$ but $\phi(V - S) \cong \phi(D) - {R}$ and $\phi(D \cup S - S) \cong \phi(D) - {R}$ so $\phi^{-1} \phi(D) = S \cup D$, then we can write $\phi^{-1} \phi(D) = D + aS $ as divisors for some $a > 0$. By the projection formula, $(C, \phi^{-1}\phi(D)) = (\phi(C), \phi(D)) = 0$ because $\phi(C)...