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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!

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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.

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

Huneke showed that

(Huneke 1980) An ideal is generated by $d$-sequence, then it is of linear type.

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.

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