So, I'm in the following situation:

I have vector spaces $H,V$, and a map $A:H\longrightarrow \hom(H,V)$, sending $x$ to $A_x$ (notation); i need to consider the variety

$\Sigma_1(A)=\{[x]\in \mathbb{P}(H)|\\; rank(A_x)\leq 1\}$.

Mostly, I'm interested in wether this variety is smooth, reducible, and/or full (by full, i mean not contained in any proper projective subspace).

Are there any conditions on $A$ that allow me to know any of these properties?

I don't know what $A$ is, but here is a couple of things i know:

A has no kernel;

If I define $\hat{A}:V\longrightarrow \hom(H,H)$ by $\hat{A}_v(x):=A_x(v)$, then $\hat{A}_v$ is skew-symmetric for all $v\in V$.

Many thanks in advance!