How to think of algebraic geometry in characteristic p?

Question

How does a working mathematician usually think about algebraic geometry in characteristic $p$? For the sake of concreteness, and to make things more "geometric" (whatever that means), let's say we work over an algebraically closed field of characteristic $p$. It seems to me that (motivated by the étale topology, say), authors tend to use the geometry of $\mathbb{C}$ as a template, envisioning $\mathbb{A}^1$ as something like the complex plane. Then, using this as a model, one accounts for extra phenomena that occur in characteristic $p$; namely, extra automorphisms due to frobenii, as well as some extra singularities that don't occur in characteristic $0$.

Is this picture of characteristic $p$ geometry accurate? To make this question a little less soft, I'll ask the following naive, and certainly false question: Is the category of varieties over $\overline{\mathbb{F}}_p$ equivalent to the category of varieties over $\mathbb{C}$ along with extra morphisms coming from Frobenii (as well as objects obtained from glueing via these extra morphisms)?

Answer 1

The basic strategy you give is correct: To first approximation, algebraic geometry over $\overline{\mathbb F_p}$ is like algebraic geometry over $\mathbb C$ except for extra phenomena. Usually, I think of these extra phenomena as "inseparable maps" and "wild ramification".

Inseparable maps include Frobenius, but not every inseparable map is a Frobenius. They all have properties which are unexpected from a characteristic zero point of view (e.g. rank of the derivative of the map a a generic point does not equal the dimension of the image).

Wildly ramified covers also behave differently from characteristic zero maps, but in a more subtle way. Perhaps the biggest is that to specify a finite branched cover in characteristic zero it suffices up to only a finite ambiguity to specify the branch locus, while in characteristic $p$ the ambiguity can be infinite.

Of course as one gets more experience in algebraic geometry in characteristic $p$ one increasingly thinks about characteristic $p$ problems by reference to previous characteristic $p$ situations one has encountered, but I think "characteristic zero plus inseparable maps and wild ramification" is a good starting point.

In terms of visualizing, the only reasons to divert from visualizing algebraic curves as complex surfaces and so on are the same reasons that exist in characteristic $0$: sometimes, to fit more in your visual imagination, you want to visualize algebraic curves as curves, algebraic surfaces as surfaces, and so on, and sometimes you want to visualize arbitrarily-high-dimensional varieties, which means you have to imagine them as two- or maybe three-dimensional.

Let me also briefly answer your formal question in the negative: The set of maps $\mathbb P^1_{\mathbb C}\to\mathbb P^1_{\mathbb C}$ is uncountably large, and adding extra morphisms couldn't make this countable, but the set of maps $\mathbb P^1_{\overline{\mathbb F_p}}\to\mathbb P^1_{\overline{\mathbb F_p}}$ is countable.

Answer 2

Since you've mentioned $\mathbb{A}^1$, we can look at group schemes $G \subset \mathbb{G}_a$ sitting inside the additive group scheme, and contrast what is going on over characteristic 0 versus over some $\mathbb{F}_p \subset k = k^{\text{alg}}$ of positive characteristic. This should give a rather elementary exposition in how those worlds may differ.

It is easy to see that in characteristic 0, $\mathbb{G}_a$ has no nontrivial sub group schemes, for if $G(k)$ had some $k$-rational point $a \neq 0$, it must have all of its multiples $na$ for $n \in \mathbb{Z}$ corresponding to a polynomial vanishing at infinitely many points, thus $f \equiv 0$ and $G = V(f) = \mathbb{G}_a$ is the whole thing.

In contrast, the procedure for positive characteristic would give you a constant group scheme $(\mathbb{Z}/p\mathbb{Z})_k$ and furthermore at every closed point we can concentrate $\alpha_p \cong \operatorname{Spec}k[T]/(T^p)$. In fact, we can characterise these group schemes entirely by additive polynomials, which have their own arithmetic under addition and composition sometimes denoted by $k\{F,\sigma\}$.

The reason I brought this up is because even at this fundamental level there are some genuinely new objects that appear in mixed characteristic. I hope this is somewhat close to what you're looking for!