Today in my introductory algebraic geometry class we defined the so-called Rees algebra associated with an ideal $I$ of a ring $R$ (with strong conditions on $R$, if you like: I don't mind restricting to finitely generated reduced algebras $R$ over an algebraically closed field $k$). If we want to think of (maximal) Proj applied to the Rees algebra as Spec $R$ blown up along the vanishing set of $I$, what happens when $I$ is not radical? 


In particular, if $R = k[X\_1,\cdots,X\_n]$ and $I = (X\_1,\cdots,X\_n)$ we get $\mathbb{A}^n$ blown up at the origin. Have people tried to interpret geometrically the Rees algebra of $I^m$ for $m > 1$, or even $I^2$? I asked my professor, and (in an unusual turn of events) he did not have an answer available off the top of his head.



Rees algebra for non-radical ideals
The blowup of $Spec R$ along $I$ and $I^m$ give isomorphic results. This is Hartshorne exercise II.7.11.a. In general, any birational projective morphism is realized as the blowing up of the target along some ideal sheaf, which in general will be quite complicated (e.g. not radical, but not a power of a radical ideal either).


