Let $I_1,I_2\subset \mathbb{C}[x_1,\dots,x_n]=:R$ be two prime ideals (in practice we would probably work with $\mathbb{Q}$ instead). Is there an algorithm to decide whether $$R/I_1\cong R/I_2$$ as $\mathbb{C}$algebra's? i.e. whether $I_1$ and $I_2$ defined isomorphic varieties?

$\begingroup$ And probably $I_1$ and $I_2$ are input by a finite generating family. $\endgroup$ – YCor Apr 9 '19 at 21:26

$\begingroup$ @YCor Yes exactly. $\endgroup$ – user2520938 Apr 9 '19 at 21:27

$\begingroup$ Note that if you input finite sets $F_1,F_2$ of rational polynomials, the questions whether $\mathbf{Q}[x_1,\dots,x_n]/\langle F_1\rangle$ and $\mathbf{Q}[x_1,\dots,x_n]/\langle F_2\rangle$ are isomorphic, resp. whether $\mathbf{C}[x_1,\dots,x_n]/\langle F_1\rangle$ and $\mathbf{C}[x_1,\dots,x_n]/\langle F_2\rangle$ are isomorphic, are different. $\endgroup$ – YCor Apr 10 '19 at 8:42

$\begingroup$ @YCor Yes I realise that, but I'm curious about both questions. To point is that I'm actually most interested in the question over $\mathbb{C}$, but there is no hope of this being actually computable, hence the question over $\mathbb{Q}$, where more things do tend to be computable. Actually I guess we could work over $\overline{\mathbb{Q}}$. $\endgroup$ – user2520938 Apr 10 '19 at 8:47

$\begingroup$ You can work over an algebraically closed computable field (computable means that you can input its elements, compute operations, and determine whether two elements are equal). $\endgroup$ – YCor Apr 10 '19 at 8:52
Macaulay2 has an algorithm to compute minimal presentations of $I_1$ and $I_2$ ("compute a minimal presentation of the quotient ring defined by an ideal"). If those minimal presentation are equal (or otherwise observably isomorphic), then $R/I_1\cong R/I_2$.
I tried it with a few silly examples just now in Macaulay2, and it worked for those. Maybe you will get lucky with the cases you are thinking about.
Your question is a special case of the general problem of deciding whether two varieties are isomorphic. This question is discussed here:
Isomorphism problem for commutative algebras and schemes.
There is no known algorithm for this problem, and the problem is not known to be undecidable. I suspect the same is true for your problem.