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In Mathematics we are always classifying things up to some transformations, for example, we classify Vector Spaces up to isomorphisms (in the finite-dimensional case it's easy, there is a invariant that works every time, namely, dimension), we classify Differentiable Manifolds up to diffeomorphism in Differential (Riemannian) Geometry or up to homeomorphism in Topology. We are always trying to classify things in every branch of Mathematics, we are used to this.

Yesterday I attended a talk on the history of Algebraic Geometry, the speaker mentioned the introduction of Analytic Geometry (by Fermat and Descartes) and said that at that time people were interested in the classification of (plane) curves of degree 2 and 3 (he didn't mentioned up to which transformations), things were messy at that time, but later with the introduction of Projective Geometry we were able to prove some nice classification theorems of low degree curves in this context, for example, there is only one 2-degree and three 3-degree curves in the projective context (if I remember it right).

Thinking about this talk on my way home I noticed something strange, this type of classification is not the same as those mentioned on the first paragraph. I explain myself: in the "usual" (first paragraph) king of classification we are classifying things in a unique context, but as the speaker mentioned, we change context to get a "nicer" classification of curves, we go from the real plane to projective plane over the complex numbers, we not only change the "scalars" but the whole concept of "space".

My questions:

1) What transformations were people classifying curves up to in the Fermat/Descartes time?

2) How classifying things up to projective transformation (complex plane) answer this question on the real domain? If we change the transformations we are classifying things up to we start a new game, right?

Hope I was clear, thanks.

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  1. Up to rotation, translation, and scaling in the real affine plane.
  2. Well, given a real affine algebraic plane curve (up to isomorphism), there's a corresponding complex projective algebraic plane curve (up to isomorphism). So, if you want to classify real affine curves up to isomorphism, start with an isomorphism class of complex projective plane curves and then determine all the isomorphism classes of real affine algebraic plane curves corresponding to it.

Anyway, it's misleading to think that this was the only reason for moving to complex projective geometry. The complex projective setting is just more natural for algebraic geometry, just like complex numbers are a more natural setting for talking about roots of polynomials.

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