Let $n,m$ be two positive integers. Let $Z$ denote the closed subvariety in $\mathbb A^n \times \mathbb A^m$ given by the equation $x_1...x_n=y_1...y_m$.
QUESTION: What is the stalk (with the action of Frobenius) of the IC sheaf of $Z$ at the point (0,....0)?
When n=m=2 you get 3-dimensional quadratic cone and the stalk is cohomology of $\mathbb P^1$ (up to the corresponding shift and Tate twist). But I don't know what happens for higher $n,m$.
Note that the stalk in this case is the same as the global intersection cohomology.