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Define a functorial scheme as in "Two functorial definitions of schemes".
Here there is one big problem.
We first need to define a projective scheme, but we also need a closed subscheme.
(cf. Projective scheme)
So The definition of a closed subscheme requires a structure sheaf.
However the definition of functorial scheme directly don't have a structure sheaf, therefore a closed subscheme is not defined.
For this reason, like Hartshorne, we also can't define projective morphism as a composition of closed immersion into projective space and natural projection because a closed immersion is not defined.

Is it possible to define a projective curve?
Is there a PDF or something that does that?

Thanks in advance.

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    $\begingroup$ A closed immersion is the same thing as a proper monomorphism, which can definitely be defined from a functor of points perspective (see the `valuative criterion for properness) $\endgroup$ May 15, 2020 at 12:54
  • $\begingroup$ Thank you very much for answering my questions. It seems that it is possible to define a projective scheme by defining closed immersion in terms of proper monomorphism and closed subscheme in terms of the scheme in which the closed immersion exists. $\endgroup$
    – undertate
    May 16, 2020 at 3:50

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Every projective curve admits a finite morphism to $\Bbb P^1_k$. Conversely, every $k$-scheme admitting a surjective finite morphism to $\Bbb P^1_k$ is one-dimensional. So if we can describe a surjective finite morphism categorically and figure out how to describe irreducibility in categorical terms, we're done.

As mentioned in the comments, proper morphisms can be defined using category-theoretic terms via the valuative criteria for properness. Surjectivity can also be checked categorically: $f:X\to Y$ is surjective iff for every field $K$ and $y\in Y(K)$, there is a field extension $L/K$ and $x\in X(L)$ whose image under $X(L)\to Y(L)$ is the image of $y$ under $Y(K)\to Y(L)$. If we can show that quasi-finiteness is a categorical property, then we'll be done as finite is proper + quasi-finite.

In order to check quasi-finiteness of a morphism $X\to Y$, it suffices to check that the fiber $X_y$ is quasi-finite for every point $y\in Y$. This is categorical: for any morphism $\operatorname{Spec} K\to Y$ where $K$ is a field, form the fiber product $X_K:= X\times_Y \operatorname{Spec} K$, and then count $X_K(K)$: if it's finite for all choices of $K$ and morphism $\operatorname{Spec} K\to Y$ then our morphism $X\to Y$ is quasi-finite.

Now to figure out irreducibility. We can certainly detect connectedness via categorical data: a $k$-scheme $X$ is disconnected iff there exists an epimorphism $X\to \Bbb A^0_k\sqcup \Bbb A^0_k$. I claim that this is actually enough to detect irreducibility, too: a scheme $X$ is irreducible iff for every closed immersion $Z\to X$ we have $X\setminus Z$ is connected. As closed immersions can be described categorically and describing complements is also categorical, this shows that irreducibility can be defined using categorical data.

In summary, the projective curves over $k$ are exactly the irreducible $k$-schemes admitting a finite morphism to $\Bbb P^1_k$, and all of these conditions are purely categorical.

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  • $\begingroup$ Thank you for your answer! I think it's beautiful to be able to define a purely categorical projective scheme. It would be a waste that there is no textbook with this perspective. $\endgroup$
    – undertate
    May 17, 2020 at 12:25
  • $\begingroup$ @undertate You're welcome. Please note that I am not claiming there's no textbook that does this - personally I don't think about the functor of points perspective that much, so it's very possible there is a book that does this but I'm unfamiliar with it. If this answers your question, please consider accepting the answer via clicking the checkmark over by the votes. $\endgroup$
    – KReiser
    May 17, 2020 at 19:08
  • $\begingroup$ I understand. I will try to find it.And, I voted. $\endgroup$
    – undertate
    May 18, 2020 at 13:47

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