Given a complex vector space $V$ of dimension $n>2$, the universal conic $\mathcal C$ of $\mathbb P(V^\*)$ is a divisor in $\mathbb P(t^\*\mathcal E\_3)\overset{\pi}{\rightarrow}\mathbb P({\rm Sym}^2\mathcal E\_3^\*)\overset{t}{\rightarrow} Gr(3,V)$ where $\mathcal E\_3$ is the natural rank $3$ quotient bundle on $Gr(3,V)$.  

The Hilbert scheme of conics in $\mathbb P(V^\*)$ can be identified with $\mathbb P({\rm Sym}^2\mathcal E\_3^\*)$.  

The class in ${\rm Pic}(\mathbb P(t^\*\mathcal E\_3))$ of $\mathcal C$ is of the form $\pi^\*\mathcal L\otimes \mathcal O\_{\mathbb P(t^\*\mathcal E\_3)}(2)$ for some line bundle $\mathcal L$ on $\mathbb P({\rm Sym}^2\mathcal E\_3^\*)$.  

Is there a simple way to identify $\mathcal L$?



Class of the universal conic
The projective bundle $t \colon \mathbb{P}(\mathrm{Sym}^2\mathcal{E}\_3^\*) \to \mathrm{Gr}(3,V)$ comes with the tautological subbundle
$$
\mathcal{O}\_t(-1) \hookrightarrow t^\*\mathrm{Sym}^2\mathcal{E}\_3^\*.
$$
This embedding gives a global section in
$$
H^0(\mathbb{P}(\mathrm{Sym}^2\mathcal{E}\_3^\*), 
\mathcal{O}\_t(1) \otimes p^\*\mathrm{Sym}^2\mathcal{E}\_3^\*) 
\cong
H^0(\mathbb{P}(t^\*\mathcal{E}\_3), 
\pi^\*\mathcal{O}\_t(1) \otimes \mathcal{O}\_\pi(2)),
$$
which is precisely the equation of the universal conic. So, $\mathcal{L} = \mathcal{O}\_t(1)$.


