Suppose $Y$ is an algebraic variety and $\mathcal{E}$ a coherent sheaf on $Y$. Suppose $f:X=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E})) \to Y$ is a morphism of algebraic varieties with all fibres scheme theoretically projective spaces.

If the fibres all had the same dimension, I would have $\mathrm{R}f_* \mathbb{C}_X = \mathrm{H}^*(\mathbb{P}^n) \otimes \mathbb{C}_Y$.

In the case that the fibre dimension varies, let $Y_k$ be the locus where the fibre dimension is at least $k$. Then is it true that $\mathrm{R}f_* \mathbb{C}_X = \bigoplus \mathbb{C} _{Y_k}\[-2k\](k)$?

(and if not in general, are there any reasonable assumptions which make it true?)