Could you recommend any **references** to (some of) the following very basic assertions in algebraic geometry? (It seems unreasonable to reprove them in a research paper.)

(1) Let a surface $X$ in $\mathbb{P}^n$ be the solution set of a system of polynomial equations with real coefficients. Assume that $X$ has a smooth linear normalization $\bar X$ in $\mathbb{P}^N$. Then the complex conjugation on $\mathbb{P}^n$ lifts to an antiholomorphic involution on the linear normalization $\bar X$ (i.e., **a** normalization map $\bar X\to X$ commutes with the involutions on $\bar X$ and $X$).

**EDIT3.** The following examples show that this is not completely trivial:

Ex1. *Even if $\bar X$ is a solution set of a system of polynomial equations with real coefficients as well, a complex normalization map $\bar X \to X$ may not be given by real polynomials, and complex conjugation in $\mathbb{P}^N$ may not be the required antiholomorphic involution on $\bar X$.* E.g., the cone $\bar X=\{x^2-y^2-z^2=0\}$ in $\mathbb{P}^3$ is

**a**normalization of the cone $X=\{x^2+y^2-z^2=0\}$ in $\mathbb{P}^3$ but the map $\bar X\to X$, $(x,y,z)\mapsto (x,iy,z)$ is not given by polynomials with real coefficients. And the map is indeed

**a**complex linear normalization map because the latter is only defined up to composition with a

*complex*isomorphism.

Ex2. *The lift of a fixed point may not be a fixed point.* E.g., the curve $y^2=x^2(x-1)$ in $\mathbb{P}^2$ has linear normalization $(1:t:t^2:t^3)$ in $\mathbb{P}^3$. A normalization map is $x(t)=t^2+1, y(t)=t^3+t$ (or more accurately $(x_0:x_1:x_2:x_3)\mapsto (x_0:x_0+x_2:x_1+x_3)$). Then the real point $(x,y)=(0,0)$ of the curve is covered by two distinct complex conjugate points $(1:\pm i:-1:\mp i)$ of the linear normalization.

Assertion (1) might seem a tautology, but it has highly nontrivial consequences: e.g., (1)-(3) together with [1, Theorems 5-7 and Proposition 1] imply that a (nonruled) surface in $\mathbb{P}^n$ with a 2-dimensional set of real points parametrized by *complex* polynomials of degree $2$ has a parametrization by *real* polynomials of degree $2$ as well. And analogous assertion for higher degree polynomials does *not* remain true [1, Remark 4] although there is still a nice map $\mathbb{C}P^2\to X$, just not a linear mormalization anymore.

(2) (The closure of) the Veronese surface $(1:u:v:u^2:uv:v^2)$ in $\mathbb{P}^5$ is biregular to $\mathbb{P}^2$.

(3) The only antiholomorphic involution of $\mathbb{P}^2$ up to projective automorphism is the complex conjugation.

And also a question:

(4) Is (the closure of) the ruled surface $(1:u:v:u^2:uv)$ in $\mathbb{P}^4$ smooth? Which 'standard' surface is it isomorphic to? What are the antiholomorphic involutions on (a desingularization of) the surface?

Ideally, a reference to a particular published theorem, which can be just applied `as is' by a nonspecialist, is requested. Notice that (2)-(3) are mentioned in wikipedia and mathoverflow, but without a proof or a reference.

One should remark that there are some references where a version of (1) is stated even without the assumption that $\bar X$ is smooth but these versions cannot be correct just because an `antiholomorphic involution' is undefined for a nonsmooth surface (https://en.wikipedia.org/wiki/Complex_manifold). **EDIT:** In fact a generalization to locally ringed spaces is used there without mentioning that; thanks to Angelo for clarification.

[1] J. Schicho, The multiple conical surfaces // Contrib. Algeb. Geom. 42:1 (2001), 71-87.

Algebraic Geometry and Arithmetic Curves, in particular proposition 4.1.27 there). $\endgroup$someembedding in $\mathbb{P}^n$ given by real coefficients such that the projection $\tilde{X}\to X$ is given by real polynomials (this is because the map $\tilde{X}\to X$ is finite and so projective, moreover it is a morphism of varieties over $\mathbb{R}$ by definition!). I suspect some miscommunication is present. $\endgroup$5more comments