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Given an ample line bundle on a curve is there any chance for its pullback to the projective bundle of some vector bundle to be big? It is known that first cohomology of inverse of nef and big line bundles can be killed by Frobenius if the dimension of variety $\geq 2$. I was wondering whether such a vanishing can hold for pullbacks of ample line bundles to projective bundles.

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No. Just use the projection formula to compute the cohomology of powers of the pullback.

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