Let $i:\mathbb F\hookrightarrow\mathbb P^4$ be a cubic scroll i.e. $\mathbb F\simeq \mathbb P(\mathcal O_{\mathbb P^1}(1)\oplus\mathcal O_{\mathbb P^1}(2))\overset{n}{\rightarrow} \mathbb P^1$ with $\mathcal O_{\mathbb P(\mathcal O_{\mathbb P^1}(1)\oplus\mathcal O_{\mathbb P^1}(2))}(1)\simeq \mathcal O_{\mathbb P^4}(1)_{|\mathbb F}$. The main component of the space of conics on it should be $|i^*\mathcal O_{\mathbb P^4}(1)\otimes n^*\mathcal O_{\mathbb P^1}(-1)|\simeq \mathbb P^2$.

But how does it sit inside the Hilbert scheme $\mathbb P({\rm Sym}^2\mathcal E_3^*)\overset{t}{\rightarrow}Gr(3,5)$ of conics in $\mathbb P^4$ (namely what are the degrees on $\mathbb P^2$ of the restriction of the main classes on the Hilbert scheme)?

A natural idea that seems not working: as $\mathbb F$ in $\mathbb P^4$ is given by the $2\times 2$-minors of a $2\times 3$-matrix with linear entries, say $M_{\mathbb F}\in |M_{2\times 3}\otimes\mathcal O_{\mathbb P^4}(1)|$, one can use the universal conic on the Hilbert scheme (and the fact that it is a divisor in $\mathbb P(t^*\mathcal E_3)$ ) to get a section $M_{C(\mathbb F)}\in H^0(M_{2\times 3}\otimes t^*\mathcal E_3)$ whose degeneracy locus, when seen as an injective morphism $\varphi_{M_{C(\mathbb F)}}:\mathcal O_{\mathbb P({\rm Sym}^2\mathcal E_3^*)}^{\oplus 2}\rightarrow t^*\mathcal E_3^{\oplus 3}$, $\{rk(\varphi_{M_{C(\mathbb F)}})\leq 1\}$ should be the space of conics contained in $\mathbb F$. But unfortunately it has dimension $3$ (codimension $(2-1)(9-1)$).