TITLE: What's an example of an intersection cohomology sheaf whose stalks are pure but not pointwise pure?
QUESTION [10 upvotes]: I'll freely admit that I have a rather hard time keeping straight different notions of purity of etale sheaves, and I think part of the problem is the lack of counterexamples.
For example, it's a theorem that the stalks of intersection cohomology sheaves (with coefficients in $\overline{\mathbb{Q}}_\ell$, say) are pure.  But here, pure means "looks like the cohomology of a possibly singular projective variety" i.e., the weights are bounded above, but not below.  
This is perhaps not so surprising: if, say, one takes constant coefficients, then the IC is a summand of the pushforward of a resolution of singularities, and so the stalks are summands of the cohomology of projective varieties.  
But, if you do geometric representation theory, a funny thing happens.  In practice, one seems to always get "pointwise purity" i.e. the stalk has the weights that one expects from the cohomology of a smooth projective variety.  There are various geometric reasons for this (for example, if your resolution of singularity is symplectic, or if the fibers of your map have a paving/$\alpha$-partition by affines or other pure varieties), but it makes it hard for me to imagine anything else.  I think my understanding of algebraic geometry would be improved by knowing an example of an intersection cohomology sheaf on (say) an affine variety which isn't pointwise pure.  

What is a good example of such an intersection cohomology sheaf?

REPLY [5 votes]: Here is an example along the lines of Donu's suggestion:
We work over a (large) finite field $k$. Let $C \subset \mathbb{P}^2$ be an irreducible  plane curve of degree $d$ and geometric genus $g > 0$ having at least one node. For example, the curve given by the equation
$$y^2 = (x^2-1)(x-a_1)\cdots(x-a_{d-2})$$
 with $d>4$, all $a_i$ distinct and nonzero and $char(k) \neq 2$.
Blow up $d^2 + 1$ distinct smooth points on $C$ to get a smooth surface $X$ and let $D$ be the strict transform of $C$. We have $D\cdot D = -1$ since $C\cdot C = d^2$. By Artin's contractability criterion, we can contract $D$ to obtain a normal surface $Y$ with a singular point $p$. Now $H^1(D, \mathbb{Q}_{\ell})$ has weights $0$ and $1$, the $0$ coming from the node and the $1$ since the geometric genus is positive. By proper base change and the decomposition theorem, it follows that $IC_Y$ is not pointwise pure (at $p$).