Algebraic Geometry – Class of the Universal Conic

Question

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 $\mathcal L$?

Answer 1

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, $\mathcal{L} = \mathcal{O}_t(1)$.