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In one the the answers to this thread " Can one embedd the projectivezed tangent space of CP^2 in a projective space? " it was mentioned that " $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the variety of complete flags in the vector space $\mathbb{C^3}$ ".

I'm having a hard time understanding why this is true, and can't seem to find any references.

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  • $\begingroup$ Someone answered this and I was reading it, but then my page refreshed and it was gone... does that just mean they deleted their answer? $\endgroup$
    – Lauren
    May 22, 2014 at 20:58
  • $\begingroup$ One way to look at it is that if $z\in \mathbb{P}^2$, corresponding to a line $L_z\subset \mathbb{C}^3$, then a line in $T_z$ corresponds uniquely to a direction orthogonal to $L_z$, which corresponds uniquely to a $2$-dimensional subspace of $\mathbb{C}^3$ containing $L_z$. $\endgroup$
    – Kevin Ventullo
    May 22, 2014 at 21:01

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Let $\pi:\mathbb{P}(T_{\mathbb{P}^2})\rightarrow\mathbb{P}^2$ be the projectivized tangent bundle. The point $x:=(p,[L])\in\mathbb{P}(T_{\mathbb{P}^2})$ corresponds to the point $p = \pi((p,[L]))\in\mathbb{P}^2$ and to the class of the line $L\subset\mathbb{P}^2$ passing through $p$. Now, the point $p$ is a line through the origin $V_p\subset\mathbb{C}^3$, the line $L_p$ corresponds to a plane $\Pi_p\subset\mathbb{C}^3$. Clearly, we have $V_p\subset\Pi_p\subset\mathbb{C}^3$. Now, the morphism $$\phi:\mathbb{P}(T_{\mathbb{P}^2})\rightarrow F(1,2,3),\: x:=(p,[L])\mapsto (V_p,\Pi_p),$$ is an isomorphism because $\phi$ is injective, and $\mathbb{P}(T_{\mathbb{P}^2})$, $F(1,2,3)$ are both smooth.

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  • $\begingroup$ Thanks! I think the main problem I'm having is that I'm not understanding $T\mathbb{P}^2$ correctly. Given a point $z$ in $\mathbb{P}^2$, how to you describe the fiber over the point $z$; i.e. what is $T_z\mathbb{P}^2$? And then my understanding is that you projectivize each point in $T_z\mathbb{P}^2$ to obtain the fiber over $z$ in $\mathbb{P}(T\mathbb{P}^2)$. $\endgroup$
    – Lauren
    May 22, 2014 at 23:21
  • $\begingroup$ Exactly, a point in the fiber of $\mathbb{P}(T_\mathbb{P}^2)$ over $z\in\mathbb{P}^2$ corresponds to a line in $\mathbb{P}^2$ through $z$. $\endgroup$
    – Puzzled
    May 23, 2014 at 9:01
  • $\begingroup$ And a point in the fiber of $T\mathbb{P}^2$ over $z\in\mathbb{P}^2$, would that be a 2-dimensional subspace of $\mathbb{C}^3$ containing the line <z>? $\endgroup$
    – Lauren
    May 24, 2014 at 15:00
  • $\begingroup$ Exactly. That's it. $\endgroup$
    – Puzzled
    May 24, 2014 at 15:22
  • $\begingroup$ @Lauren Another way of thinking about it is as follows: Given a point $\ell\in \Bbb P^2$, the space $T_\ell \Bbb P^2$ is simply $\operatorname{Hom}(\ell,\ell^\perp)$, where $\ell^\perp$ denotes the orthogonal complement with respect to the standard inner product on $\Bbb C^3$. Now pick an element $[v]$ of $\Bbb P(T_\ell \Bbb P^2)$. Then any representative of $[v]$ corresponds to a linear map $\ell\to \ell^\perp$. Thus, picking any $0\neq z\in \ell$, it is sent to $w\in \ell^\perp$. [continued in the next comment] $\endgroup$
    – Danu
    Dec 5, 2018 at 10:59
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I will take $P^2$ to mean the space of lines in $C^3$. The tangent space at a particular point in $P^2$ (say represented by a line $L$) is a linear map from $L$ to $C^3/L$. Let $\mathcal{L}$ be the line bundle whose total space is $\{(L,x) \in P^2 \times C^3 \mid x \in L\}$ (this is $O(-1)$ but it doesn't matter). Then from the first fact, the tangent bundle is $Hom(\mathcal{L}, C^3/\mathcal{L}) \cong \mathcal{L}^* \otimes (C^3/\mathcal{L})$ where I use $C^3$ to be the trivial bundle $C^3 \times P^2$.

The projectivization of a vector bundle ignores tensoring with line bundles, so $P(TP^2)$ is the same as the space of projectivization of $C^3/\mathcal{L})$. So its points correspond to a choice of line $L$ and a choice of line in $C^3/L$. The latter is equivalent to a 2-dimensional subspace in $C^3$ containing $L$.

In general, $P(TP^n)$ is just the space of partial flags of type $(1,2)$ in $C^{n+1}$.

EDIT. A derivation of the above fact for tangent spaces of projective spaces can be found here: http://concretenonsense.wordpress.com/2009/08/17/tangent-bundle-of-the-grassmannian/

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