# Morphism from projective varieties

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.

• 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.
– user716004
Mar 18, 2020 at 14:13
• 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.
– user716004
Mar 18, 2020 at 14:14
• 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 $U\ subsetP_n$? Mar 18, 2020 at 14:19
• 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.
– user716004
Mar 18, 2020 at 14:37
• Hm, any other ideas? I really don't know how to construct Morphisms from closed sub-varieties of projective varieties :/ Mar 20, 2020 at 9:57