ag.algebraic geometry - Smoothness and fullness of determinantal varieties

So, I'm in the following situation:

I have vector spaces $H,V$, and a map $A:Hlongrightarrow hom(H,V)$, 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:

1. A has no kernel;




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

Many thanks in advance!

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