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 whether 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!