Let X be a surface. (A surface is an excellent integral normal separated 2-dimensional scheme.)

Let $\psi:Y\longrightarrow X$ be a minimal resolution of singularities and let $E$ be an irreducible component of the exceptional locus of $\psi$.

Is $E$ of genus zero? That is, do we have that $E$ is isomorphic to $\mathbf{P}^1_k$ for some field $k$?

Is the exceptional locus of $\psi$ a chain of rational curves? (This means that $(E_i,E_i) <0$, $(E_i, E_{i+1}) = (E_i, E_{i-1}) = 1$ and $(E_i,E_j) = 0$ if $j \neq i-1, i,i+1$. Here $E_i$ denotes an exceptional component.)

I know this is true if $X$ has "tame cyclic quotient singularities".

I also know that, by Lipman's theorem and what is stated on Wikipedia about it, this is true if $X$ has pseudo-rational singularities.

What about the general case?

What else can we say in general about the "shape" of the exceptional locus. If it's not a chain, does it have a loop? Can things get arbitrarily complicated? Where can I find the theory behind this?

chain, bur rather I should have saidgraphabove. $\endgroup$ – Karl Schwede Jul 21 '11 at 4:16