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.
Best Answer
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.