I red an article and encountered some concepts from algebraic geometry. Let $R=\mathbb{Q}[\alpha_1,\ldots,\alpha_5]$ be a polynomial ring in the variables $\alpha_i$. Define $f(x,y)\in R[x,y]$ by $$f(x,y)=y^2+\alpha_1 xy+\alpha3 y-x^3-\alpha_2 x^2-\alpha_4 x-\alpha_5.$$
We now consider the affine scheme $\mathcal{E}:f(x,y)=0$ over $R$. What does this mean? What is the definition of the n-fold fibered product of $\mathcal{E}$?
Edit:
I have a concrete question: If $\mathcal{E}=Spec\ R[X,Y]/(f(X,Y))$, how does the field of rational functions on $\mathcal{E}$ look like? I am using Geometry of schemes (Eisenbud, Harris), but I do not find any explanation about this nor the definition. And the same question for $\mathcal{E}^2=Spec\ \Big( R[X,Y]/(f(X,Y))\otimes_R R[X,Y]/(f(X,Y))\Big)$