Universal bundle of grassmannian of planes and projective bundle over grassmannian of lines

Question

Let $p:Y=\mathbb P(\mathcal E_3^{\vee})\rightarrow G(3,n+1)$ be the universal family of hyperplanes (i.e. lines) of the planes of $\mathbb P^{n}$. The following isomorphism seems natural $$\mathcal O_{\mathbb P(\mathcal E_3^{\vee})}(1)\simeq q^*\mathcal O_{G(2,n+1)}(1)$$ where the last line bundle the Plucker polarization and $q:Y\rightarrow G(2,n+1)$ is the projection. But by $q$, $Y$ has also a structure of projective bundle over $G(2,n+1)$ and we should have $Y=\mathbb P(\mathcal Q)$, where $\mathcal Q$ is the kernel of $\mathcal O^{n+1}\rightarrow \mathcal E_3$, and an integer $l$ such that $$\mathcal O_{\mathbb P(\mathcal Q)}(1)\simeq p^*\mathcal O_{G(3,n+1)}(1)\otimes q^*\mathcal O_{G(2,n+1)}(l).$$
But using $p$, we have $$K_Y=-3c_1(q^*\mathcal O_{G(2,n+1)}(1)) + p^*(-(n+1)c_1(\mathcal O_{G(3,n+1)}(1)) - c_1(\mathcal O_{G(3,n+1)}(1)))$$ and using $q$, we have $$K_Y = -(n-1)p^*c_1(\mathcal O_{G(3,n+1)}(1)) + (n(l+1) + (l-1)- 1)q^*c_1(\mathcal O_{G(2,n+1)}(1))$$ so that one cannot expect these formulas to match. So (it seems to be the only possibility) why the first isomorphism is not true?

Answer 1

Let $V$ be a vector space of dimension $n+1$ and $F(2,3;V)$ the flag variety parameterizing $U_2 \subset U_3 \subset V$ with $\dim U_i = i$. Then $F(2,3;V)$ comes with projections to $Gr(2,V)$ and $Gr(3,V)$ respectively.

Let me denote by $U_2$ and $U_3$ the tautological vector bundles on $Fl(2,3;V)$, so that $0 \subset U_2 \subset U_3 \subset V \otimes O$ is the tautological flag. Then we have natural embeddings $$ \Lambda^2U_2 \subset \Lambda^2U_3 \qquad\text{and}\qquad U_3/U_2 \subset (V \otimes O)/U_2. $$ The give identifications $$ Fl(2,3;V) \cong P_{Gr(3,V)}(\Lambda^2U_3) \qquad\text{and}\qquad Fl(2,3;V) \cong P_{Gr(2,V)}((V \otimes O)/U_2) $$ with $\Lambda^2U_2$ and $U_3/U_2$ being the tautological line bundles on the projectivizations.

Note also that $$ \Lambda^2U_2 \cong q^*O_{Gr(2,V)}(-1) $$ and $$ U_3/U_2 \cong \det(U_3) \otimes \det(U_2)^{-1} \cong p^*O_{Gr(3,V)}(-1) \otimes q^*O_{Gr(2,V)}(1). $$ Altogether, this gives a compatible system of identifications.