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I'm a beginner of algebraic geometry, sorry to ask basic question.

I heard 'curve' over algebraically closed field $k$ is defined as 'integral separated scheme of finite type scheme over $k$' in terms of scheme theoretic description.

But I cannot understand why this is generalization of the 'curve' in terms of algebraic variety.

For example, elliptic 'curve' in terms of scheme is, like pair of $Spec \Bbb{C}[x,y]/(y^2-x^3-x)$ and scheaf as a set, but elliptic 'curve' in terms of algebra variety is

{$(x,y) \in \Bbb{C}× \Bbb{C}｜y^2=x^3-x$}∪{∞}$⊂ \Bbb{P}^2$. They are not the same even as a set, but why can we say scheme theoretic 'curve' is generalization of 'curve' as algebraic variety?

Pont
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    Potential duplicate of What is an algebraic variety? The point is that you can generalize because they're both in some sense determined by the functions on them and those are the same. – KReiser Jan 16 '23 at 04:15
  • Do you have objections to the claim that, say, real numbers are generalizations of rational numbers, even though $1$ as a rational number is a different set from $1$ as a real number? – Zhen Lin Jan 16 '23 at 05:13
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    In the elliptic curve example, if you are worried that the point $\infty$ has no counterpart in your scheme, that's because your Spec scheme is an affine scheme. To get the point at infinity, you must use a projective scheme. – Ted Jan 16 '23 at 05:43
  • @Zhen Lin But rational number $x$ can be naturally embed into $ \Bbb{R}$ by $x→(x,x,・・・)$. But I don't know how to embed algebraic variety into scheme. – Pont Jan 16 '23 at 16:59
  • @buoyant I explain how to do that in the linked answer - check the proposition there. – KReiser Jan 16 '23 at 21:49

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You could ask the same question about many situations in math where a concept has been generalized: why replace permutation groups with abstract groups, why replace Riemann integrals with Lebesgue integrals, why replace metric spaces with topological spaces, and why replace submanifolds of $\mathbf R^n$ with abstract smooth manifolds that need not be embedded in some Euclidean space?

First of all, nothing is lost: in each case there is a standard way to regard the first thing as a special case of the second thing (e.g., each metric space is a topological space in a standard way, even if the axioms of a topological space may initially look more abstract). Second, the reason for interest in the second thing (the generalization) is that it has better properties or is more flexible and it lets us prove stronger theorems. For classical varieties as schemes, see Varieties as schemes and Why study schemes?.

Although algebraic geometry was initially developed over $\mathbf C$ and then over any algebraically closed field, there were strong pressures to relax the setting, such as allowing varieties to be defined over a field that is not algebraically closed. The motivation for making constructions beyond classical algebraic geometry came from both within algebraic geometry itself and also number theory (e.g., the Weil conjectures). See https://hsm.stackexchange.com/questions/1813/why-did-algebraic-geometry-need-alexander-grothendieck

KCd
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