Is the join of two irreducible varieties is irreducible? (reference + real fied)

Question

The following definition and theorem are taken from J.M. Landsberg, Tensors: Geometry and Applications, Graduate Studies in Mathematics, v. 128 (p. 118)

Definition 5.1.1.1: The $join$ of two varieties $X$, $Y$ is $$ J(X,Y)=\overline{\bigcup\limits_{x\in X,\ y\in Y,\ x\ne y} \mathbb P^1_{xy}}. $$ Theorem 5.1.1.4: Joins and secant varieties of irreducible varieties are irreducible.

For the proof the author refers to p.144 of Joe Harris, Algebraic Geometry: A First Course. But this book contains the proof only for secant varieties.

I have not found any other books or papers where the statement

`` Join of two irreducible varieties is irreducible''

is formulated explicitly. And I have not found the proof.

Question 1: Where I can find the proof for joins of intersecting varieties?

Question 2: Does the statement hold over $\mathbb R$? I will appreciate any references here.

Answer 1

I am going to prove that the join of two irreducible varieties is irreducible, but first, I am going to give a definition of join that is more easy to drive.

Let $X,Y\subseteq\mathbb{P}^{n}$ be two irreducible varieties. Let us consider $$ S^{0}_{X,Y}:=\{(x,y,z)\in X\times Y\times \mathbb{P}^{N}:x\neq y, z\in\langle x,y\rangle\}, $$ and $$ S_{X,Y}:=\overline{S^{0}_{X,Y}}\subseteq X\times Y\times\mathbb{P}^{N} $$ (where the line means Zariski closure). Let us consider the projections $$ \pi_{1}:S_{X,Y}\rightarrow X\times Y, $$
$$ \pi_{2}:S_{X,Y}\rightarrow \mathbb{P}^{N}. $$ We define the join of $X,Y$ to be $$ S(X,Y):=\pi_{2}(S_{X,Y}). $$ The fibers of $\pi_{1}|_{S^{0}_{X,Y}}$ are isomorphic to lines, so they are irreducible of the same dimension. In consequence, $S^{0}_{X,Y}$ is irreducible. Accordingly, its Zariski closure $S_{X,Y}$ is irreducible, and the image of $S_{X,Y}$ under a morphism is irreducible too. In particular, $S(X,Y)=\pi_{2}(S_{X,Y})$ is irreducible, as we wanted to prove.

Answer 2

Lemma. Let $X$ be an irreducible variety inside a grassmanian, $X\subset G(k,\mathbb{P}^n)$. Then the union of k-planes in $\mathbb{P}^n$ is irreducible.

proof. Consider the incidence correspondence $\{(p,V), p\in V\}\subset\mathbb{P}^n\times X$. It is easy now to see that the union of k-planes in $X$ is irreducible in $\mathbb{P}^n$. \qed

As a corollary, the Join is irreducible. The image of $X\times Y ---> Gr(1,n)$ is irreducible. Hence, the union of lines joining $X$ and $Y$ is also irreducibe.