A tale of two maps into a Grassmannian

Question

I suspect that the answer to this question is well-known to the experts. However, I was not able to find it in the literature, so let me ask here.

Setup. In the sequel, $X$ is a compact complex surface and $E$ is a rank $2$ vector bundle on $X$. We call $\pi \colon \mathbb{P}(E) \to X$ the projective bundle in the sense of Hartshorne. In my applications, $X$ is of general type and $E=\Omega_X^1$, but let me ask the question in generality.

We assume that $S^nE$ is globally generated, namely, that the evaluation of sections $$H^0(X, \, S^nE) \otimes \mathcal{O}_X \to S^nE$$ is surjective.

If $V$ is a vector space, we denote by $G(r, \, V)$ the Grassmannian of $r$-dimensional subspaces of $X$, and by $G(V, r)$ the Grassmannian of the $r$-dimensional quotients of $V$, so that $G(r, \, V) \simeq G(V^*, \, r)$. Moreover, we denote by $\mathbb{G}(r-1, \, \mathbb{P}(V))$ and $\mathbb{G}(\mathbb{P}(V), \, r-1)$ the corresponding projective Grassmannians.

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We are now going to construct two morphisms $X \to \mathbb{G}(n, \, \mathbb{P}H^0(X, \, S^nE)^*)$.

The first morphism. Take a point $x \in X$. The fibre $\pi^{-1}(x)$ is the projective line $\mathbb{P}(E(x))$, and $|\mathcal{O}_{\mathbb{P}(E)}(n)|$ cuts on it the complete linear system $\mathcal{O}(n)$. Identyfying the global sections of $\mathcal{O}_{\mathbb{P}(E)}(n)$ with those of $S^nE$, we see that $\pi^{-1}(x)$ is embedded in $\mathbb{P}H^0(X, \, S^nE)^*$ as a rational normal curve $C_x$ of degree $n$. There is exactly one $n$-plane $L_x$ containing $C_x$, and we get a morphism $$\mathscr{G} \colon X \to \mathbb{G}(n, \, \mathbb{P}H^0(X, \, S^nE)^*), \quad x \mapsto L_x.$$

The second morphism. Take a point $x \in X$. There is a surjection $$H^0(X, \, S^nE) \to S^nE(x) \to 0,$$ where $S^nE(x)$ is the fibre of $S^nE$ over $x$, namely, a $n+1$-dimensional vector space. This provides by definition an element $$s_x \in G(H^0(X, \, S^nE), \, n+1) \simeq G(n+1, \, H^0(X, \, S^nE)^*) \simeq \mathbb{G}(n, \, \mathbb{P}H^0(X, \, S^nE)^*),$$ and we can define a morphism $$\mathscr{H} \colon X \to \mathbb{G}(n, \, \mathbb{P}H^0(X, \, S^nE)^*), \quad x \mapsto s_x.$$

Question. How are the morphisms $\mathscr{G}$ and $\mathscr{H}$ related? Are they the same?

Every answer and/or reference to the relevant literature will be greatly appreciated.

Answer 1

I guess, when you say "in the sense of Hartshorne" you mean the projective spectrum of $\oplus S^kE$.

Yes, the morphisms are the same, and to see this just note that there is a natural (relative Veronese) embedding $$ \mathbb{P}_X(E) \to \mathbb{P}_X(S^nE), $$ such that every fiber of the (former) $\mathbb{P}^1$-bundle becomes a rational normal curve in the (latter) $\mathbb{P}^n$-bundle. Moroever, the composition $$ \mathbb{P}_X(E) \to \mathbb{P}_X(S^nE) \to \mathbb{P}(H^0(X,S^nE)^*) $$ (where the second arrow is induced by the dual of the evaluation morphism $H^0(X,S^nE) \otimes \mathcal{O}_X \to S^nE$) coincides with the morphism induced by the line bundle $\mathcal{O}_{\mathbb{P}_X(E)}(n)$. This shows that the two maps coincide.