I've had this question for a while and haven't seen anyone else answer it to my knowledge.
Here goes: Construct all degree $d$ algebraic varieties in $n$ independent variables: $x\in \mathbb{R}^n$, $[$let the algebraic variety be defined by $X_i=\{x\in \mathbb{R}^2|f_i(x)=0\}$. $f$ is in the set of polynomials. We can view $X_i$ as an algebraic variety immersed in $\mathbb{R}^n$$]$ such that one or more of their homology groups is given.
Not the most precise, I know, but here is a more concrete example that will hopefully help elucidate what I am going for.
Let $X_i$ be the algebraic variety generated by the equation $f_i(x,y)=0$. I am using algebraic varieties so that spaces such as $xy=0$ are considered. Now, I want to construct all $X_i$ explicitly by writing out $f_i(x,y)$, such that $H_1(X_i)=\mathbb{Z}$. In other words, that the algebraic variety has a single hole.
I can make at least 1 very simple solution to this question off the top of my head. Say $X_1=f_1(x,y)=x^2+y^2-1=0$. Quickly, I can see that this defines a circle with unit radius which satisfies all the conditions. I can also make a second one defined by $X_2=f_2(x,y)=a(x-x_0)^2+b(y-y_0)^2-r^2=0 \ni a,b,r>0, $ and $ a,b,r,x_0,y_0\in \mathbb{R}$; this one defines a family of varieties satisfying the conditions. But I really already knew that a priori.
I'd like to be able to run some algorithm to generate these polynomials, the $X_i$'s but I have no idea where to start looking, or what I should start doing. I would really like to answer this question in 2-5 variables so that I can make 1D-4D spaces, and ask "Make all the varieties with $H_1(X_i)=\mathbb{Z},H_2(X_i)=\mathbb{Z},H_3(X_i)=\mathbb{Z},H_4(X_i)=\mathbb{Z}$, etc.
I would love to hear any thoughts/ideas/references that you all have