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.



Example of wall-crossing formulae?
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


$$ e(C^+)-e(C^-)= 2. $$


