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so I just started studying projective varieties (over algebraically closed fields) and I simply want to understand why $$V_{P_n}(T_i) \simeq P_{n-1}$$ whereas $V_{P_N}(T_i):= \{[z_0,..., z_n]\in P_n| z_i=0\}$ $i=0,...,n$.

So clearly I can get a morphism $P_{n-1}\mapsto V_{P_n}(T_i)$, $[z_o,...,z_{n-1}]\mapsto [z_0, .., 0, ..,z_{n-1} ]$ whereas the $0$ is at the i-th position, using that for every open set $U\subset P_n$, every prevariety Z and every morphism $\phi:\pi^{-1}(U)\mapsto Z$ such that $\phi(tz)= \phi(z)$ $\forall t\in K^{*}$ and $z\in\pi^{-1}(U)$, there exists a morphism $\psi:U\mapsto Z$ such that $\psi\circ\pi =\phi$. (whereas $\pi:\mathbb{K}^{n+1}\setminus\{0\}\mapsto P_n$).

Now I would like to construct the inverse morphism, which is given straightforwardly but I can't really get why it's also a morphism.

Does the same statement as above also hold for closed $A\subset P_n$ (i.e. the construction with the morphism $\psi:A\mapsto Z$)? Because thus I would immediately get what I want. More generally I don't really know how to construct isomorphism of closed sub varieties of $P_n$ (like, that $V_{P_2}(T_0T_1-T_2)\simeq P_1$).

I feel like at my stage, I really want to check all the details, but I don't really get it.

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  • $\begingroup$ Wlog you can consider $i = 0$. Now think about the case of real projective plane, as the space classifying linear subspaces of the real vector space of dimension $3$. Consider for example the plane $\pi = \{z = 1\}$, then each line passing through $\pi$ is uniquely determined by a point on this affine plane. All the lines not intersecting $\pi$ lie in the plane $\{z = 0\}$, and to classify them you use a projective line. This is the case where $n = 2$. Maybe try to think about this case to get an idea for the inverse map. $\endgroup$
    – user716004
    Mar 18, 2020 at 14:13
  • $\begingroup$ The statement above does not hold for every closed subset, ie. there are closed subsets which are not $\mathbb{P}^{n-1}$. Think for example one point or the union of two hypersurfaces. $\endgroup$
    – user716004
    Mar 18, 2020 at 14:14
  • $\begingroup$ Sorry I was not clear on that: by "the same statement as above" I meant the construction of the morphism $\psi:U\mapsto Z$ for open subsets $A\subset P_n$. Does this also work for closed Subsets $U\ subsetP_n$? $\endgroup$
    – Vasco1008
    Mar 18, 2020 at 14:19
  • $\begingroup$ Correct me if I'm wrong, $\pi$ is the canonical projection $\pi \colon \mathbb{A}^{n+1} \setminus \{0\} \twoheadrightarrow \mathbb{P}^n$? And your claim is that the map is well defined on the quotient space as long as it is invariant by the quotient group $K^{\ast}$ (considering an eventual restriction of the canonical projection to an open subset)? And so the question is that if this works in the case where the restriction is to a closed subset? You can uses charts, as this map is given by polynomials on affine charts, but I'm a bit rusty on geometry to answer the question about the quotient. $\endgroup$
    – user716004
    Mar 18, 2020 at 14:37
  • $\begingroup$ Hm, any other ideas? I really don't know how to construct Morphisms from closed sub-varieties of projective varieties :/ $\endgroup$
    – Vasco1008
    Mar 20, 2020 at 9:57

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