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cohomology Cohomology of a stratified projective bundle

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).)

cohomology of a stratified projective bundle

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).

Cohomology of a stratified projective bundle

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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cohomology of a stratified projective bundle

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).