Let $S$ be a smooth algebraic variety, and suppose $X\to S$ is a smooth morphism of schemes such that the geometric fibers are all projective spaces. Let us suppose that the dimension of the fibers is constant on an open subset of $S$, and changes on some strata in positive codimension (I have in mind the projectivization of the image of a map of vector bundles, and its degeneracy loci). Is there a formula to compute the cohomology of $X$? (I guess in terms of the classes of the strata.)
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2$\begingroup$ There would be a spectral sequence abutting to the cohomology of $X$, with $E_1$ term the relative cohomology of strata. And unless you know that the stratification is "perfect", that's the best you can hope for. $\endgroup$– Donu ArapuraCommented Jun 18, 2023 at 14:40
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1$\begingroup$ The spectral sequence, or a more elementary argument, will give you the Euler characteristic of $X$. $\endgroup$– Donu ArapuraCommented Jun 18, 2023 at 14:43
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1$\begingroup$ @DonuArapura, what does "perfect" mean for the stratification? and how does the situation get better in that case? and BTW what is the spectral sequence that abuts to that? (too many questions, I know) $\endgroup$– IMeasyCommented Jun 18, 2023 at 15:00
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1$\begingroup$ Perfect means that the restriction maps $H^i (X_j)\to H^i(X_{j-1})$ are surjective, where $\ldots X_{j-1}\subset X_j\ldots$ are the strata. I learned this terminology from a paper of Atiyah and Bott. $\endgroup$– Donu ArapuraCommented Jun 18, 2023 at 15:47
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1$\begingroup$ I should probably make this an answer, and I will when I have more time. $\endgroup$– Donu ArapuraCommented Jun 18, 2023 at 15:50
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