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Statement. Let $X$ be a smooth complex projective variety that is a $\mathbb CP^k$ bundle over $\mathbb CP^n$. It is well known that there exists a rank $k+1$ complex vector bundle $V$ over $\mathbb CP^n$ such that $X$ is isomorphic as a projective variety to the projectivisation $\mathbb PV$.

Question. I would like to find a precise reference to this statement, is there such a reference?

(I know that one is supposed to say that the Brauer group of $\mathbb CP^n$ is trivial, and this is why the statement holds. But I can't find any place in the literature where this statement about projective bundles is stated and I need to find such a reference :( )

PS. I realised, that Section 6.2 of the beautiful paper by Arnaud Beauville https://arxiv.org/abs/1507.02476 would do as a reference to the statement. Yet this section speaks of something far more general. I still hope for something more direct, maybe along the lines of what Angelo suggested in his comment (or at least explicitly stated).

Statement. Let $X$ be a smooth complex projective variety that is a $\mathbb CP^k$ bundle over $\mathbb CP^n$ in analtytic topology. It is well known that there exists a rank $k+1$ complex vector bundle $V$ over $\mathbb CP^n$ such that $X$ is isomorphic as a projective variety to the projectivisation $\mathbb PV$.

Question. I would like to find a precise reference to this statement, is there such a reference?

(I know that one is supposed to say that the Brauer group of $\mathbb CP^n$ is trivial, and this is why the statement holds. But I can't find any place in the literature where this statement about projective bundles is stated and I need to find such a reference :( )

PS. I realised, that Section 6.2 of the beautiful paper by Arnaud Beauville https://arxiv.org/abs/1507.02476 would do as a reference to the statement. Yet this section speaks of something far more general. I still hope for something more direct, maybe along the lines of what Angelo suggested in his comment (or at least explicitly stated).

Statement. Let $X$ be a smooth complex projective variety that is a $\mathbb CP^k$ bundle over $\mathbb CP^n$. It is well known then that there exists a rank $k+1$ complex vector bundle $V$ over $\mathbb CP^n$ such that $X$ is isomorphic as a projective variety to the projectivisation $\mathbb PV$.

Question. I would like to find a precise reference to this statement, is there such a reference?

(I know that one is supposed to say that the Brauer group of $\mathbb CP^n$ is trivial, and this is why the statement holds. But I can't find any place in the literature where this statement about projective bundles is stated and I need to find such a reference :( )

Statement. Let $X$ be a smooth complex projective variety that is a $\mathbb CP^k$ bundle over $\mathbb CP^n$. It is well known that there exists a rank $k+1$ complex vector bundle $V$ over $\mathbb CP^n$ such that $X$ is isomorphic as a projective variety to the projectivisation $\mathbb PV$.

Question. I would like to find a precise reference to this statement, is there such a reference?

(I know that one is supposed to say that the Brauer group of $\mathbb CP^n$ is trivial, and this is why the statement holds. But I can't find any place in the literature where this statement about projective bundles is stated and I need to find such a reference :( )

PS. I realised, that Section 6.2 of the beautiful paper by Arnaud Beauville https://arxiv.org/abs/1507.02476 would do as a reference to the statement. Yet this section speaks of something far more general. I still hope for something more direct, maybe along the lines of what Angelo suggested in his comment (or at least explicitly stated).