Question

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:

1. A has no kernel;

2. 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!

Answer 1

I would suggest to consider the scheme $Z \subset P(H) \times P(H^*)$ defined by $$ Z = \lbrace(x,f)\ |\ A_x(Ker f) = 0\rbrace. $$ It is clear that $Z$ is the zero locus of a global section (given by $A$) of a vector bundle $V\otimes O_{P(H)}(1)\otimes T_{P(H^*)}(-1)$. This $Z$ is useful because it is a zero locus, so you can check smoothness by looking at the differential. On the other hand it is clear that $\Sigma_1(A) = p(Z)$, where $p:P(H)\times P(H^*) \to P(H)$ is the projection. Moreover, if $Ker \hat{A} = 0$ then $p$ is an isomorphism, so $\Sigma_1(A)$ is smooth iff $Z$ is.