Let $X$ the projective cone of $C$, here $C$ a curve of genus $g$.
We can compute its intersection cohomology : $IH^0(X) = \mathbb Q, IH^1(X) = \mathbb{Q}^{2g}, IH^2(X) = \mathbb 0, IH^3(X) = \mathbb{Q}^{2g} , IH^4(X) = \mathbb Q$.
We have a resolution of singularity : $W:= \mathbb P^1 \times C \to X$ (blow-up at the vertex). We obtain that $H^i(W) = IH^i(X)$ for $i \neq 2$ and $H^2(W) = \mathbb{Q}^2$.
Is there an geometric interpretation of this $\mathbb Q^2$ ? The decomposition theorem for perverse sheaves gives us that $IH^k(X)$ are direct summand of $H^k$ but I don't know how if there is an geometric interpretation of the other summand.
Thanks in advance !