There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne.

Here's a refresher: by $IC$ one means the intersection complex, which is just $\mathbb Q$ for a smooth scheme but more complicated for others, and by $IC_i$ one denotes the complex constructed from a subscheme $Y_i$ together  with the local system as $j_{!*}\mathcal L_i$. 

Now it turns out that
for a morphism $f: X\to Y$, you can decompose in the derived category
$$f_*IC  =  \oplus IC_i[n_i].$$
The special beauty of this decomposition theorem is in its examples. Here are some I think I know:

* For a **free action** of a group G on some X, you get the decomposition by representation of G.
* For a **resolution of singularities**, you get $f_*\mathbb Q =  IC_Y \oplus F$ (and $F$ should have support on the exceptional divisor.)
* For a **smooth algebraic bundle** $f_*\mathbb Q = \oplus \mathbb Q[-]$ (spectral sequence degenerates)

There are many known applications of the theorem, described, e.g. in the review

> [The Decomposition Theorem and the topology of algebraic maps* by de Cataldo and  Migliorini](http://arxiv.org/abs/0712.0349), 

but I wonder if there are more examples that would continue the list above, that is, "corner cases" which highlight particularly one specific aspect of the decomposition theorem?


> **Question:** What are other examples, especially the "corner" cases?