Let $\mathbb{P}^n$ be the projective $n$-space and consider $\mathbb{P}(H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(d))) \cong \mathbb{P}^{\binom{n+d}{n} -1}$ be the complete linear system of divisors of degree $d$ in $\mathbb{P}^n$. This linear system parametrizes divisors linearly equivalent to $dH$ where $H$ is a hyperplane in $\mathbb{P}^n$.
Consider the subset
$$ V = \{ D \in Div(\mathbb{P}^n) \mid deg(D)= d \text{ and } Supp(D) \text{ is irreducible} \} \subset \mathbb{P}^{\binom{n+d}{n} -1}. $$
Is it known if $V$ is and open or closed (or maybe locally closed) subset of $\mathbb{P}^{\binom{n+d}{n} -1}$? In other words is it a quasi-projective variety? What about reducibility of $V$?