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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 $\mathcal{E}^2=Spec\ \Big( R[X,Y]/(f(X,Y))\otimes_R R[X,Y]/(f(X,Y))\Big)$

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    You have not asked a question, really. Do you want to know what *what* means?2011-12-12
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    Please ask the question in the question body, not I'm the comments! :)2011-12-13
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    Dear Nadori, Mariano is absolutely right. However while adding the new question, you have completely deleted the old one, so that shaye's and my answer now look strangely irrelevant. Please roll back, reestablish the old question,and follow it by the new one. That said, I have addressed your new question in an Edit to my answer.2011-12-13
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    Dear Georges, you're right. I'm sorry. I have changed it again.2011-12-13

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