# Conics on a cubic scroll

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 $$(2-1)(9-1)$$).

• What are "the main classes on the Hilbert scheme" for you? Sep 17 '21 at 18:26
• Thank you for your answer. By "main classes" I mean the chern classes $t^*c_i(\mathcal E_3)$ coming from $Gr(3,5)$ and $c_1(\mathcal O_t(1))$.
– pi_1
Sep 17 '21 at 18:44

Let me first describe the pullback of the tautological bundle of $$\mathrm{Gr}(3,5)$$. Let $$V$$ be a 3-dimensional vector space; then we can take $$\mathbb{F} = \mathrm{Bl}_{[f]}(\mathbb{P}(V^\vee)) \subset \mathbb{P}(S^2V^\vee/ \langle f^2 \rangle) =: \mathbb{P}(W),$$ where $$0 \ne f \in V^\vee$$, and then the main component of the Hilbert scheme of conics on $$\mathbb{F}$$ is $$\mathbb{P}(V)$$ (because lines on $$\mathbb{P}(V)$$ go to conics on $$\mathbb{F}$$). Consider the Euler sequence $$0 \to \mathcal{O}(-1) \to V \otimes \mathcal{O} \to T(-1) \to 0$$ on $$\mathbb{P}(V)$$ and its symmetric square $$0 \to V \otimes \mathcal{O}(-1) \to S^2V \otimes \mathcal{O} \to S^2T(-2) \to 0.$$ The morphism $$W^\vee \otimes \mathcal{O} \hookrightarrow S^2V \otimes \mathcal{O} \to S^2T(-2)$$ is not surjective; its cokernel is isomorphic to the cokernel of $$V \otimes \mathcal{O}(-1) \to S^2V \otimes \mathcal{O} \to \mathcal{O}$$ (where the second arrow is given by $$f^2$$), which allows one to check that the cokernel si the structure sheaf of the line $$L \subset \mathbb{P}(V)$$ corresponding to $$f$$. Consider the kernel of the corresponding map $$\mathcal{E} := \mathrm{Ker}(S^2T(-2) \to \mathcal{O}_L).$$ The construction implies that there is an epimorphism $$W^\vee \otimes \mathcal{O} \to \mathcal{E}$$ which induces a morphism $$\mathbb{P}(V) \to \mathrm{Gr}(3,W)$$. It is easy to see that this is the morphism from the above description, hence the pullbacks of the tautological classes of the Grassmannian are the Chern classes of $$\mathcal{E}$$.