Irreducibility of the $k$-secant variety of the Veronese variety. I have read that the $k$-secant variety $\sigma_{k}(V_{d}^{n})$ of the Veronese variety $V_{d}^{n}$ is irreducible, but I do not know how to prove it.
If $k=2$, it is clear, because the variety of secant lines of any irreducible variety is irreducible (see for example Harris' Algebraic Geometry: A first Course). Now, I do not know if it is true that the $k$-secant variety of any irreducible variety is irreducible for $k>2$.
I have tried to prove that at least $\sigma_{k}(V_{d}^{n})$ is irreducible considering some incidence correspondences and morphisms, but I have not been able to get anything. Any help would be appreciated.
 A: From pp. 87-88 of "On the superadditivity of secant defects" by B. Fantechi Bull. Soc. Math. France 118 (1990), 85—100.

We shall denote by ${\mathbb P}^N$ the $N$-dimensional projective space over a fixed algebraically closed field $k$. If $X_1, \ldots , X_n$ are subsets of ${\mathbb P}^N$, we shall denote by $\langle X_1, \ldots , X_n\rangle$ their linear span, i.e. the smallest projective subspace of ${\mathbb P}^N$ containing all the $X_j$’s.
Definition 1.1. — Let $X_1, \ldots , X_n$ be irreducible subvarieties of ${\mathbb P}^N$, and let
$h = \operatorname{max} \{\operatorname{dim}\langle x_1,\ldots,x_n\rangle | x_j \in X_j\}$· 
We denote by $S(X_1,\ldots,X_n)$ the closure in $X_1 \times\cdots \times X_n \times {\mathbb P}^N$ of the set
  
($\bigstar$) $\{(x_1,\ldots,x_n,z) | z \in \langle x_1,\ldots,x_n\rangle, \operatorname{dim}\langle x_1,\ldots,x_n\rangle = h\}$.
The projection of $S(X_1, \ldots , X_n)$ in ${\mathbb P}^N$ will be called the join of $X_1,...,X_n$ and will be denoted by $J(X_1,...,X_n)$.
Remark 1.2. — Let $X_1, . . . , X_n$ be irreducible subvarieties of ${\mathbb P}^N$ . Then the following hold :
(i) $S(X_1, \ldots , X_n)$ is irreducible ; 
(ii) $J(X_1, \ldots , X_n)$ is irreducible ;
(iii) $\operatorname{dim} S(X_1,\ldots,X_n)=h+\sum \operatorname{dim}X_j$;
(iv) If $X_1 \cup X_2$ contains at least two distinct points
$\operatorname{dim} S(X_1,X_2) = \operatorname{dim}X_1 + \operatorname{dim}X_2 + 1$.
  
Proof. — Let $\pi$ denote the natural map from $S(X_1, \ldots , X_n)$ to $X_1\times\ldots\times X_n$.
  (i) and (iii) both follow by observing that the open set defined in ($\bigstar$) is a projective bundle of rank $h$ over the set $\{(x_1,\ldots ,x_n) | \operatorname{dim}\langle x_1,\ldots,x_n\rangle = h\}$, which is open in $X \times\cdots\times X$.
  (ii) is a consequence of (i), and (iv) is a consequence of (iii).

[...]

Definition 1.5. — Let $X$ in ${\mathbb P}^N$ be an irreducible variety. We denote $S(X,...,X)$ ($k$ copies) by $S^k(X)$; in the same way we denote $J(X,...,X)$ by $J^k(X)$. $J^k(X)$ will be called $k$-th secant variety of $X$.

