Suppose that $X$ is a quasi-projective $k$-scheme with a right $B$-action (where $B$ is a linear algebraic group or Lie group) and that the quotient $X/B$ exists. Let the canonical projection $X \to X/B$ be a principal $B$-bundle (with fibre $B$).

Now consider the associated bundle $f: Y \to X/B$ with fibre $\mathbb{P}^1$. Suppose that $X/B$ is a projective scheme. Can someone please help me prove the following:

(1) $f$ is a projective morphism.

(2) If $X/B$ is a projective scheme, then so is $Y$.

  • $\begingroup$ The term "associated bundle" is generally used for a bundle arising out of a representation of the group $B$. For the fibres to be projective lines it should be a 1-dimensional representation. $\endgroup$ – P Vanchinathan Apr 5 '14 at 12:00

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