Example of wall-crossing formulae? In a nutshell my question is "Are there any easy, educational wall-crossing formulae?".
Recently I often hear the word "(Kontsevich-Soibelman's etc) wall-crossing formula" in algebraic geometry talks. I wonder what they are like but am not really ready to read those relevant papers with high technology. I understand that this kind of formula expresses how your invariants of moduli spaces varies when you change the stability condition to construct the moduli spaces. Could someone provide me with a toy example? I am looking for a "wall-crossing formula" which can be understood by those with basic AG and maybe some GIT. 
Thank you very much.
 A: I can tell you  the gist of a  "wall crossing formula".    Typically you have a  space of parameters $\newcommand{\eS}{\mathscr{S}}$ $\Lambda$,  a configuration space   $\newcommand{\eC}{\mathscr{C}}$ $\eC$, and a parametrized moduli space $\newcommand{\eM}{\mathscr{M}}$ $\eM$ which is a  subvariety $\eM\subset \eC\times \Lambda$.
We have a natural projection
$$\pi:\eM\to \Lambda. $$
The fiber of this map over $\lambda\in\Lambda$, denoted by $\eM_\lambda$,  is called the moduli space corresponding to the  parameter $\lambda$.
One could associate to $\eM_\lambda$ various invariants. For example, there might exists a sheaf $\eS\to\eM$ which restricts to a  sheaf $\eS_\lambda\to\eM_\lambda$. We denote   by  $e(\lambda)$ the Euler characteristic  of $H^\bullet(\eM_\lambda,\eS_\lambda)$.
It could happend that  $e(\lambda)$ depends on $\lambda$, but it could depend in a rather  specal way. Namely, there could exists  real codimension one  subvarieties  $W_i\subset \Lambda$, $i\in I$, called walls,  so that, if
$$ W:=\bigcup_{i\in I} W_i, $$
then $\lambda\to e(\lambda)$ is continuous on $\Lambda^\ast:=\Lambda\setminus W$.  In particular, the  function $\lambda\to e(\lambda)$ is constant on the connected components   of $\Lambda^*$, which are called chambers.
If    two chambers $C_0, C_1$ are adjacent, i.e.,  they sit on opposite sides of a wall  $W_i$ ,   then  $e(\lambda)$  has constant values $e(C_0)$ and $e(C_1)$ in these two chambers and a wall crossing formula  will tell you what the difference  $e(C_1)-e(C_0)$ is.
Here is a  trivial example.    Let $\Lambda=\mathbb{R}^2$.  We let $(b,c)$ denote the coordinates  of a point in $\Lambda$.   The configuration space $\eC$ is $\mathbb{R}$ and the parametrized moduli space is
$$\eM= \bigl\lbrace (t,b, c)\in\eC \times \Lambda;\;\;t^2+bt+c=0\,\bigr\rbrace. $$
The moduli space $\eM_{b,c}$ can be identified with the set of real roots of the quadratic polynomial $t^2+bt+c$. We denote by $e(b,c)$ the number of such roots. In other words, $e(b,c)$ is the (topological) Euler characteristic of the space $\eM_{b,c}$.
In this case we have a single  wall
$$ W=\bigl\lbrace (b,c)\in \Lambda;\;\; b^2- 4c=0\bigr\rbrace, $$
with two chambers,
$$ C^\pm=\bigl\lbrace \;\pm(b^2-4c)>0\;\bigr\rbrace. $$
In this case the Wall crossing formula is
$$ e(C^+)-e(C^-)= 2. $$
