# Class of the universal conic

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 $$\mathcal L$$?

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