What is the difference between an algebraic and a complex analytic variety?

Question

I have some simple questions about scheme theory which are not clear in my mind :

$1)$ What is the difference between a scheme $X$ or, a $ \mathrm{Spec} \mathbb{Z} $ - scheme : $ X $, and a $ Y $ - scheme $ X $ or a scheme $ X $ over a scheme $ Y $ ? Who's big and general ?.

$ 2) $ What is the difference between a scheme $ X $ and a scheme which takes the form : $ X(L) = \mathrm{Hom}_k ( k[T_1 , \dots , T_n ] / ( P_1 , \dots , P_r ) , L ) $ $ \simeq \{ (x_1 , \dots , x_n ) \in L^n \ \mathrm{t.q.} \ P_1 (x_1 , \dots , x_n ) = ... = P_r ( x_1 , \dots , x_n ) = 0 \} $ ?

Who's big and general ? $ L $ is an extension of the base field $ k $.

$ 3) $ If $ X $ is an integral scheme of finite type, over $\mathbb{Q}$, then $ X(\mathbb{Q} ) $ is a $ \mathbb{Q} $ - scheme, and $ X( \mathbb{C} ) = X $, right ? why are $ X(\mathbb{Q}) $ an algberaic variety and $ X(\mathbb{C} ) $ a complex analytic variety ? When $ X $ is more general than an integral scheme of finite type over $\mathbb{Q}$, what is the difference between $ X $ and $ X( \mathbb{C} ) $ ? In other words, How can we construct, or what is the nature of a point $ x \in X \backslash X( \mathbb{C} ) $ ?

Thanks in advance for your answers and for your patience. :-)

Answer 1

1) Let $Y$ be any scheme. A $Y$-scheme $X$ is, technically, a morphism of schemes $X\to Y$. It might be helpful to consider two different ideas here: firstly, if $Y = \text{Spec}(k)$ for some field $k$, then topologically $Y$ is just a point. We can then imagine the $Y$-scheme $X\to Y$ as some geometric object projecting down onto the point. If $Y$ is more general than a point (say, e.g. $Y = \mathbb{A}_k^1$ is the affine line over $k$), then we can imagine $X\to Y$ as a family of fibers varying over the points of $Y$. Since $\text{Spec}(\mathbb{Z})$ is the terminal object in the category of schemes, there is a unique morphism $Y\to\text{Spec}(\mathbb{Z})$ for every $Y$. Therefore every $Y$-scheme $X\to Y$ is naturally a scheme over $\mathbb{Z}$ (an "absolute scheme") via $$X\to Y\to\text{Spec}(\mathbb{Z})$$ However, I would say that it is more general to be able to consider constructions working over any base scheme $Y$, because these can always be "base-changed" by another morphism of schemes.

2) If you mean this in a functorial sense then they are two aspects of the same thing. Every scheme $X$ over a ring $R$ determines a functor of points which, due to glueing, can be defined as a covariant functor from the category of $R$-algebras to the category of sets. When $k$ is a field (and also an $R$-algebra), the set $X(k)$ corresponds to the points of the topological space $X$ whose residue field is contained in $k$. But it's important to note that there are other $R$-algebras $A$ that aren't fields where $X(A)$ doesn't correspond to "typical" points.

3) This isn't quite right. $X(\mathbb{Q})$ and $X(\mathbb{C})$ aren't schemes - they're just identifiable with subsets of its points (more traditionally, you can think of them as points having coordinates in those fields). But $X$ also has non-closed points that [don't live in $X(k)$ for any field $k$] edit: live in much bigger transcendental extensions. When $X$ is an integral scheme of finite type over $\mathbb{Q}$, there is also a generic point $\xi$ that is not closed, and corresponds to the irreducible closed subscheme $X$ itself. You can think of $\xi$ as a point "spread out over the shape of $X$". This point is not in $X(\mathbb{C})$.

Edit: Along with the great references given in the other answer by @paf, a fun and informal introduction to the functor of points view is given in this blog post by Lieven le Bruyn. I would say that getting to grips with viewing schemes functorially has vastly improved my understanding of them, especially when you work in "arithmetic" situations.

Answer 2

1) A $Y$-scheme is given by a scheme $X$ with a particular morphism $X\to Y$. Every scheme is a $\mathrm{Spec} \Bbb Z$-scheme.

2) $X(L)$ is not a scheme. It's only a set (the set of $L$-rational points of $X$). But there are two possibilities for defining a scheme $X$: either as a locally ringed space or as a functor of points from the category of schemes (not only field extensions of the base field) to the category of sets defined as $\mathrm{Hom}(\dot,X)$. This functor is not given by only one $X(L)$ but by all of them!

3) See above. If $X$ is a scheme over a field $\Bbb K$, then its closed points are $\bar{\Bbb K}$-rational (the set of closed points is in bijection with $X(\bar{\Bbb K})/\mathrm{Gal}(\bar{\Bbb K}/\Bbb K)$). The other points of $X$ are the generic points of its irreducible closed subschemes. For example, if $X=\Bbb A^3_{\Bbb K}$, we have one generic point for each irreducible curve and one generic point for each irreducible surface in $X$. Moreover, since $X$ is irreducible, it has one generic point for itself, namely the zero ideal of $\Bbb K[x,y,z]$. This is a general fact: the generic point of an irreducible affine scheme $X=\text{Spec} A$ is the zero ideal of $A$.

Finally, you should note that the generic point $\eta$ of an irreducible scheme $X$ is dense in it : $\overline{\{\eta\}}=X$ (that's why they are so useful and one of the reasons of the power of scheme theory). This is different from closed points, whose closure are themselves.

Good references for all of this (and more!) are Vakil's notes on the Web and the book The Geometry of Schemes by Eisenbud and Harris.