Suppose that I blow up an ideal sheaf $J$ in $\mathbb A^2$ via a map $\pi : X \to \mathbb A^2$. I'd like to compute, from the ideal, how many exceptional divisors there are for $\pi$, and be able to get my hands on them. For example, it would be nice to know the intersection form (I am willing assume $X$ is smooth.)
What do these divisors correspond to in the ideal? How can I read this off from $J$? By "compute", and "read off", I really mean I would like a Macaulay2 or Singular command to do it all for me!
