I've been thinking recently about moduli spaces defined over $\mathbb{Z}$, and this led me to the following question:

### Question

Riemann existence says that if we have a variety over $\mathbb{C}$, $X_{\mathbb{C}}$, then $\widehat{\pi_1^{top}(X_{\mathbb{C}}(\mathbb{C}))}\cong\pi_1^{et}(X_{\mathbb{C}})$. For which (the first, the second, or both) of the following interpretations of $X_{\mathbb{C}}(\mathbb{C})$ does this theorem work: a. It is the set of all morphisms $Spec(\mathbb{C})\rightarrow X_{\mathbb{C}}$.

or

b. It is the set of all sections of the structure morphism $X_{\mathbb{C}}\rightarrow Spec(\mathbb{C})$.

### Motivation

The motivation to the question, as I said before, is because I've been thinking about moduli spaces over $\mathbb{Z}$. Let's say $X$ is a priori a scheme of finite type over $\mathbb{Z}$. Here $X(\mathbb{C})$ has only the first interpretation, since the structure morphism is not going to $\mathbb{C}$, but to $\mathbb{Z}$. I want to say something about the geometry of $X(\mathbb{C})$, but how could I if this is not what Riemann Existence is talking about?

### Remark

Let me give you a sense of why the two interpretations are so different. If $X_{\mathbb{C}}$ is a variety over $\mathbb{C}$, the second interpretation would mean that $X_{\mathbb{C}}(\mathbb{C})$ is in correspondence with the maximal points of $X_{\mathbb{C}}$. However, in the first interpretation we would also have phenomena like the following: If $Y_{\mathbb{C}}$ is a subvariety of $X_{\mathbb{C}}$, then $\kappa(Y_{\mathbb{C}})$ (the function field of $Y_{\mathbb{C}}$) is going to be a field of cardinality $2^{\aleph_0}$ and so its algebraic closure is isomorphic to $\mathbb{C}$. This means that we have a $Spec(\mathbb{C})\rightarrow X_{\mathbb{C}}$ such that its image is at the generic point of $Y$. Obviously, the first interpretation makes more geometric sense, but as I mentioned in the motivation, I am very interested in how to deal with the second. And so, in particular, I'm asking if we have a Riemann Existence theorem for the second interpretation.

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# In Riemann Existence, what is the interpretation of the space of complex-geometric points?

I've been thinking recently about moduli spaces defined over $\mathbb{Z}$, and this led me to the following question:

### Question

Riemann existence says that if we have a variety over $\mathbb{C}$, $X_{\mathbb{C}}$, then $\widehat{\pi_1^{top}(X_{\mathbb{C}}(\mathbb{C}))}\cong\pi_1^{et}(X_{\mathbb{C}})$. For which (the first, the second, or both) of the following interpretations of $X_{\mathbb{C}}(\mathbb{C})$ does this theorem work: a. It is the set of all morphisms $Spec(\mathbb{C})\rightarrow X_{\mathbb{C}}$.

or

b. It is the set of all sections of the structure morphism $X_{\mathbb{C}}\rightarrow Spec(\mathbb{C})$.

### Motivation

The motivation to the question, as I said before, is because I've been thinking about moduli spaces over $\mathbb{Z}$. Let's say $X$ is a priori a scheme of finite type over $\mathbb{Z}$. Here $X(\mathbb{C})$ has only the first interpretation, since the structure morphism is not going to $\mathbb{C}$, but to $\mathbb{Z}$. I want to say something about the geometry of $X(\mathbb{C})$, but how could I if this is not what Riemann Existence is talking about?