Suppose that $I\subseteq\mathbb{C}[x\_0,\ldots,x\_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}\_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let $$\pi:Y=\mathrm{Bl}\_{\mathcal{I}}(\mathbb{P}^n)\longrightarrow\mathbb{P}^n$$ be the blow-up of $\mathbb{P}^n$ along $\mathcal{I}$. Since $\mathbb{P}^n$ is normal, we know that $\mathcal{J}=\pi\_\*(\mathcal{I}\cdot\mathcal{O}\_Y)$ is a coherent ideal sheaf of $\mathcal{O}\_{\mathbb{P}^n}$. In general, $\mathcal{I}$ and $\mathcal{J}$ are not the same, so my question is:


Given $I$, how can I compute the saturated homogeneous ideal $$J=\bigoplus\_{d\geq 0}\Gamma(\mathbb{P}^n,\mathcal{J}(d))\subseteq\mathbb{C}[x\_0,\ldots,x\_n]$$ (which, of course, is just the graded ideal attached to $\mathcal{J}$)? Is there a way to do it efficiently? By 'efficiently' I mean using a computer program like Macaulay2.


Note that if we took the normalised blow-up, $J$ would simply be the integral closure of $I$.



Direct image of an ideal sheaf along a blow-up
It sounds like you want to compute the pushforward of some sheaf on $Y$.


Macaulay2 can do this using the **BGG** package [HERE](http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.6.0.1-20131031-2/share/doc/Macaulay2/BGG/html/). In particular, use the **directImageComplex** function. I would apply the function **reesAlgebra** to your ideal $I$, I think that should give you a presentation of the blowup (up to Proj), right? Then use **directImageComplex** of the extended ideal. You may have to saturate again at the end I suppose.


There appears to be a bug in the **directImageComplex** function though. See [the Macaulay2 google group](https://groups.google.com/forum/#!topic/macaulay2/8N86FgJAtqs) for some discussion and a workaround suggested by Frank Schreyer.


