What's the definition of an isomorphism between projective planes?

Question

For some reason I can't seem to find a complete definition of what exactly it means for a map to be an isomoprism between two projective planes $f: P^n \to P^n$. I assume that f has to be bijective and satisfy some other condition. What exactly is this other condition?

Answer 1

Question: "For some reason I can't seem to find a complete definition of what exactly it means for a map to be an isomoprism between two projective planes f:Pn→Pn. I assume that f has to be bijective and satisfy some other condition. What exactly is this other condition?"

Answer: You view "projective $n$-space over a field $k$" $\mathbb{P}^n_k$ as a pair $(\mathbb{P}^n_k, \mathcal{O})$ where $\mathcal{O}$ is the "sheaf of regular functions" on (the topological space) $\mathbb{P}^n_k$. An isomorphism

$$\phi: \mathbb{P}^n_k \rightarrow \mathbb{P}^n_k$$

is by definition an isomorphism

$$(\phi, \phi^{\#}) : (\mathbb{P}^n_k, \mathcal{O}) \rightarrow (\mathbb{P}^n_k, \mathcal{O})$$

of locally ringed spaces.

Question: "What exactly is this other condition?"

Answer: An isomorphism $(\phi, \phi^{\#})$ is a pair of maps: A map $\phi$ of topological spaces and a map of sheaves

$$\phi^{\#}: \mathcal{O} \rightarrow \phi_* \mathcal{O}, $$

with an inverse map of locally ringed spaces $(\psi, \psi^{\#})$, such that the composites are the identity map. If the base field is the field of complex numbers, you may define projective $n$-space as a complex manifold, and this ringed topological space has a different structure sheaf (and topology - the strong topology). The structure sheaf $\mathcal{O}^{hol}$ is defined using an atlas for $\mathbb{P}^n_{\mathbb{C}}$ and holomorphic complex valued functions. You may similarly define real projective space (as a real smooth manifold or as an algebraic variety).

See Hartshorne, Chapter I and II in the case of algebraic varieties and schemes.

Example: If $k$ is the complex numbers and $f(x,y):=x^2-y^3$ it follows the zero set $V(f) \subseteq \mathbb{A}^2_k$ is a singular algebraic curve with a cusp singularity at the origin. There is a canonical map

$$\phi: \mathbb{A}^1_k \rightarrow C$$

defined by

$$\phi(t):=(t^3,t^2)$$

and the map $\phi$ is an "isomorphism" of the underlying topological spaces. It is not an isomorphism of algebraic varieties: The curve $C$ is singular and the affine line $\mathbb{A}^1_k$ is non-singular, and singularity is preserved under isomorphism.

Hence you must understand how we construct the structure sheaf $\mathcal{O}_C$ and the notion "locally ringed space". Note: This is not about "category theory", it is about "locally ringed spaces".

Example: Affine varieties: Let $k$ be an algebraically closed field. If $I \subseteq R:=k[x_1,..,x_n]$ is a prime ideal generated by the polynomials $f_1,..,f_l$ and $V(I) \subseteq \mathbb{A}^n_k$ is the affine $k$-variety defined by $I$ (in the sense of HH, Chapter I), the maps of affine $k$-varieties

$$\phi: V(I) \rightarrow V(J)$$

for two ideals $I,J$ are in 1-1 correspondence with maps of $k$-algebras

$$\phi': R/J \rightarrow R/I.$$

By construction $R/I \cong H^0(V(I), \mathcal{O}_{V(I)})$ equals the ring of global sections of the structure sheaf of $V(I)$.

Example: The ring of regular functions on the affine line is $k[t]$ and the ring of regular functions on $C$ is the ring $A(C):=k[x,y]/(x^2-y^3)$. If there was an isomorphism of $k$-varieties

$$\mathbb{A}^1_k \cong C,$$

this would give an isomorphism of $k$-algebras $k[t]\cong A(C)$ but $k[t]$ is a regular ring and $A(C)$ is non-regular, and regularity is preserved under isomorphism. Hence this gives another proof of the above claim.

Example: Projective varieties: The above is not true for projective varieties: There is for any $n \geq 2$ an isomorphism of $k$-algebras

$$H^0(\mathbb{P}^1_k, \mathcal{O}_{\mathbb{P}^1_k}) \cong H^0(\mathbb{P}^n_k, \mathcal{O}_{\mathbb{P}^n_k}) \cong k$$

hence the induced maps at global sections do not determine the map of algebraic varieties

$$\phi: \mathbb{P}^1_k \rightarrow \mathbb{P}^n_k.$$

There is for any $g\in Aut_k(\mathbb{P}^n_k)\cong PGL(n,k)$ an induced morphism $\phi_g:= g \circ \phi$, and $\phi_g \neq \phi$ in general. In fact: To give a map of schemes over $k$

$$\phi: \mathbb{P}^1_k \rightarrow \mathbb{P}^n_k$$

is equivalent to give an invertible sheaf $\mathcal{O}(n)$ on $\mathbb{P}^1_k$ and a surjection

$$f: \mathcal{O}^{n+1}_{\mathbb{P}^1_k} \rightarrow \mathcal{O}(n) \rightarrow 0.$$

You find this in Hartshorne, Prop.II.7.12. This property is used in HH.Ex.7.1.1 to prove the following:

$$Aut_k(\mathbb{P}^n_k) \cong PGL(n,k).$$