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In differential geometry, it seems that a distinction is usually made between extrinsic and intrinsic definitions of real manifolds. The former is as a space defined by equations of a "nice enough" type, which lets us visualize the manifold as living in some higher-dimensional real space, while the latter is done with charts and compatibility conditions on their overlaps. Of course, for real manifolds the difference is purely psychological, but the distinction becomes quite important for complex manifolds, many of which cannot be embedded into any complex space.

My question is, what is the analogous dichotomy in algebraic geometry? I guess that the usual definition of a classical variety being defined by polynomial equations is extrinsic. Is there a similar notion of covering a variety with charts that yields an intrinsic definition? And are these definitions equivalent as with real manifolds, or is one preferred over the other?

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    $\begingroup$ Yes and no. You cover an algebraic variety with charts, but those charts are varieties defined by equations. You can't use only e.g. open subspaces of affine space. $\endgroup$
    – Zhen Lin
    Jun 1, 2021 at 22:54

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Question: "My question is, what is the analogous dichotomy in algebraic geometry? I guess that the usual definition of a classical variety being defined by polynomial equations is extrinsic. Is there a similar notion of covering a variety with charts that yields an intrinsic definition? And are these definitions equivalent as with real manifolds, or is one preferred over the other?"

Answer: An affine scheme $X=Spec(A)$ is defined using the spectrum of a commutative ring $A$. Its topological space is the set of prime ideals in $A$ (with the zariski topology) and its structur sheaf $\mathcal{O}$ is defined using localizations of the ring $A$. If $A$ is an integral domain of finite type over a field $k$ with quotient field $K(A)$, we may for every open set $U \subseteq X$ define the ring of functions $\mathcal{O}(U)$ using $K(A)$. A general scheme $(X, \mathcal{O})$ is a locally ringed space with an open cover $U_i:=Spec(A_i)$ of affine schemes.

Examples: Projective $n$-space $\mathbb{P}^n_k$ with $k$ a field. It has an open cover of affine spaces

$$\mathbb{A}^n_k:=Spec(k[y_1,..,y_n]).$$

The definitions are not "equivalent". For a manifold $M$ you cover $M$ with open subsets $U_i$ and homeomorphisms $\phi_i$ with open subsets in real $n$-space. For schemes you cover your locally ringed space with affine schemes.

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