Let $U_{n}$ be a vector space of dimension $n$.
From plethysm we obtain an isomorphism $$\mathrm{Ker}(S^2(U_4\otimes U_3)^{\vee}\stackrel{p}{\rightarrow} S^2U_4^{\vee}\otimes S^2 U_3^{\vee})\simeq \Lambda^2U_4^{\vee}\otimes \Lambda^2U_3^{\vee}.$$
Projectivisation $\mathbb{P}(\mathrm{Ker}(p))$ has the following geometric interpretation: it parametrizes quadrics in $\mathbb{P}(U_4\otimes U_3)$ containing Segre variety $\mathbb{P}(U_4)\times \mathbb{P}(U_3)$, so from the above isomorphism we obtain that quadrics containig Segre variety are parametrized by $\mathbb{P}(\Lambda^2U_4^{\vee}\otimes \Lambda^2U_3^{\vee})$. For an element $f\in \Lambda^2U_4^{\vee}\otimes \Lambda^2U_3^{\vee}$ we will denote by $Q_f$ the corresponding quadric.
Note that $f$ determines a map $f\colon \Lambda^2U_3\to \Lambda^2U_4^{\vee}$. Suppose that this map is an embedding, then $f$ determines a conic $$C_f:= \mathbb{P}(\Lambda^2 U_3)\cap \mathrm{Gr}(2,U_4^{\vee}).$$
I would like to find a reference where the connection between ranks of $C_f$ and $Q_f$ is described. More precisely I'm interested in two specific cases: when $C_f$ is a smooth conic and when $C_f$ has rank 2.
I have computed in coordinates that for a smooth conic $C_f$ the corresponding quadric $Q_f$ is smooth, and for $C_f$ of rank 2 the corank of $Q_f$ is 2. But this problem looks "classical" so I'm pretty sure that it should be written somewhere. Maybe someone knows where it is written?
There is a similar construction with quadrics in $\mathbb{P}(S^2U_3)$ containing Veronese surface $\mathbb{P}(U_3)$ and it is written in 13.4 in Fulton and Harris.
