# Equations for blow-ups non-regular centers

Let $X = \textrm{Spec} A$ be a reasonable scheme and $I\subset A$ an ideal generated by a regular sequence. Then we have a full set of generators/relations for the blow-up of $X$ along $V(I)$.

Are there other situations where one can compute a presentation for the Rees algebra by hand? The schemes I'm interested are things like arithmetic surfaces.

References will be greatly appreciated!

Definition: Let $R$ be a ring. An ideal $I$ is call of linear type if $\operatorname{Sym} (I) \cong R[It].$
(Huneke 1980) An ideal is generated by $d$-sequence, then it is of linear type.