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!



Equations for blow-ups non-regular centers
I believe that you are looking for a description of the defining equations of Rees algebras in commutative algebra. In general, this is a hard question. The complete intersection case can be generalized to ideals of linear type. 



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> Definition: Let $R$ be a ring. An ideal $I$ is call of linear type if $\operatorname{Sym} (I) \cong R[It].$
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Huneke showed that



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> (Huneke 1980) An ideal is generated by $d$-sequence, then it is of linear type. 
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I believe the book Arithematic of blowup algebras by Vasconcelos has good information and references of ideals of linear type. Also, there are other cases one can say a bit more about the defining equations of Rees algebras under additional assumptions. 


