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.