algebraic-geometry
This is from Qing Liu's Algebraic Geometry and Arithmetic Curves, page 131:
Why can we assume that $X$ is integral? Can we prove that an irreducible component of a geometrically reduced algebraic variety given with the reduced closed subscheme structure is geometrically reduced?
Yes, your scheme $X$ has exactly one irreducible component - i.e., it is an irreducible scheme. Note that an affine scheme $\operatorname{Spec} R$ is irreducible if and only if the nilradical of $R$ is prime. Here, the nilradical of $k[X,Y]/(X^2,XY)$ is $(\overline{X})$ (where $\overline{X}$ denotes the image of $X$ in $k[X,Y]/(X^2,XY)$) and this is a prime ideal since $(k[X,Y]/(X^2,XY))/(\overline{X}) \cong k[Y]$. However, you should be careful to understand your claim that "the subscheme $V(X)$" is the irreducible component of $X$ correctly.
Let me explain. There is a natural way to view $X$ as a closed subscheme of $\mathbb{A}_k^2 = \operatorname{Spec} k[X,Y]$. The underlying set of this subscheme contains precisely those prime ideals $\mathfrak{p}$ of $k[X,Y]$ such that $\mathfrak{p} \supset (X^2,XY)$. As A.P.'s comment to your question shows, we have $$ \mathfrak{p} \supset (X^2,XY) \Leftrightarrow \mathfrak{p} \supset (X) .$$ It is customary to express this by writing $$ V(X) = V(X^2,XY), $$ which is perfectly fine as long as you understand this as an equality of sets only (of course, the subspace topology induced by the Zariski topology on $\mathbb{A}_k^2$ then agrees as well, which is why you could also view the above equality as an equality of topological spaces).
However, sometimes $V(X)$ is not used to denote a subset, but to denote a subscheme (and writing "the subscheme $V(X)$" suggests just that) - here, that would be the scheme $\operatorname{Spec} k[X,Y]/(X)$. But $\operatorname{Spec} k[X,Y]/(X^2,XY)$ and $\operatorname{Spec} k[X,Y]/(X)$ are not isomorphic as schemes. In fact, there is a bijective correspondence between the closed subschemes of any affine scheme $\operatorname{Spec} R$ and the ideals of $R$ - but clearly, $(X^2, XY) \neq (X)$.
(Well, $\operatorname{Spec} k[X,Y]/(X)$ is [naturally isomorphic to] a closed subscheme of $\operatorname{Spec} k[X,Y]/(X^2,XY)$ and, strictly speaking, it is true that $\operatorname{Spec} k[X,Y]/(X)$ is the irreducible component of $\operatorname{Spec} k[X,Y]/(X^2,XY)$ because the underlying topological spaces of those two schemes agree. But, at least to me, this seems to be a rather strange way of asserting that $\operatorname{Spec} k[X,Y]/(X^2,XY)$ is irreducible.)
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I now feel a bit silly for writing this pedantic and probably superfluous remark so let me add another comment which might be of more interest to you: While the scheme $X$ is irreducible, it has an embedded prime. For this reason, it might help you to understand the "geometric meaning" of associated primes and primary decompositions - in particular, in what sense they yield more refined information than the decomposition into irreducible components.
Each one of these definitions is morally just a restricted version of each of the more general definitions. To be precise, there are fully faithful functors from the less general definitions to the more general definitions which in some cases are equivalences of categories. Let's rewrite the definitions here so we have a quick reference. We'll cover the affine case first and then explain how to patch everything together to the global case afterwards.
"Classical" definition (affine case): A $k$-variety is an irreducible Zariski-closed subset of $k^n$ for an algebraically closed field $k$ and some integer $n$.
Milne's definition (affine $k$-variety): An affine $k$-variety is a locally ringed space isomorphic to $(V,\mathcal{O}_V)$ where $V\subset k^n$ is a "classical" $k$-variety and $\mathcal{O}_V$ is the sheaf of regular functions on $V$.
Liu's definition: An affine $k$-variety is the affine scheme $\operatorname{Spec} A$ associated to a finitely generated reduced $k$-algebra $A$.
General definition: An affine $k$-variety is $\operatorname{Spec} A$ for a finitely generated $k$-algebra $A$.
Basically what's going on here is that each of these definitions is slowly, grudgingly accepting greater generality and more extensible structure on the road to the general definition.
Milne's definition adds the structure sheaf, but is not yet all the way to a scheme - it's missing generic points. This in particular shows that generally $(V,\mathcal{O}_V)$ is not the spectrum of a ring. (Milne's definition is set up in such a way that there's only one way to get the structure sheaf, so there's an equivalence of categories between the "classical" category and Milne's category.)
From here, Liu's definition adds the generic points - there is a fully faithful functor between Milne's definition and Liu's definition, which has image exactly the irreducible varieties in Liu's definition.
The road from Liu's definition to the general definition is easy: we stop requiring reducedness, which is a technical advantage for some more advanced properties one may wish to consider later on (eg those involving cohomology).
The proof that there are fully faithful functors between all these definitions can be found (among other places) in Hartshorne II.2.6:
Proposition (Hartshorne II.2.6): Let $k$ be an algebraically closed field. There is a natural fully faithful functor $t:\mathfrak{Var}(k)\to \mathfrak{Sch}(k)$ from the category of varieties over $k$ to schemes over $k$. For any variety $V$, it's topological space is homeomorphic to the closed points of the underlying topological space of $t(V)$, and it's sheaf of regular functions is obtained by restricting the structure sheaf of $t(V)$ via this homeomorphism.
The idea of the proof is that one can add the generic points of each irreducible positive-dimensional closed subset and then construct the structure sheaf on this new space in a canonical way, which produces for you a scheme verifying the properties claimed. (In case you're wondering about Hartshorne's definition, Hartshorne defines his category of varieties as quasiprojective integral varieties, of which the affine varieties of the "classical" and Milne's definitions are full subcategories. This same idea of the proof works in all cases.)
This provides us the answer to the first part of your main question: there are fully faithful functors which lets you consider each category as a part of the next more general category. This means that you can generalize without fear.
Now we can talk about gluing and non-affine varieties. In full generality, just like a manifold is some space locally modeled on $\Bbb R^n$, we should have that varieties are locally modeled on affine varieties (and schemes are locally modeled on affine schemes). This is what Milne's getting at with his definition of a prevariety, and what Liu is getting at with the finite cover condition.
There are some pathologies one may wish to avoid, like the line with two origins, which one can get by gluing to copies of $\Bbb A^1$ along the open sets which are the complements of the origin in each copy. Such varieties are non-separated, and that's what the separated condition in Milne's "algebraic $k$-varieties" excludes.
The most general definition one normally sees of a variety over a field is the following:
Most general definition: A $k$-variety is a scheme of finite type over the field $k$.
This allows non-reduced, non-irreducible, non-separated schemes, but keeps the essential finiteness condition of "finite type", which implies that any $k$-variety has a finite cover by affine open $k$-varieties, which is exactly the finiteness condition that Liu and Milne both require. Be warned that many modern authors of papers will take this general definition plus some adjectives, and are not always clear on which adjectives they take. (If you're writing papers in algebraic geometry, please include a sentence in your conventions section which makes it clear what adjectives you take when you write "variety"!)
In this most general situation, affineness and projectiveness are easy to describe. Each is exactly the condition that our variety admits a closed embedding in to $\Bbb A^n_k$ or $\Bbb P^n_k$, respectively, for some $n$. (To connect this with the affine definition as $\operatorname{Spec} A$ of a finitely-generated $k$-algebra, note that we can choose a surjection $k[x_1,\cdots,x_n]\to A$, which gives us $A\cong k[x_1,\cdots,x_n]/I$ for some ideal $I$, and this exactly shows us that $\operatorname{Spec} A \to \operatorname{Spec} k[x_1,\cdots,x_n]=\Bbb A^n_k$ is a closed immersion.)
Best Answer
Liu's definition of an algebraic variety over a field $k$ is a scheme of finite type over $k$. In particular, such a scheme $X$ is noetherian and has finitely many irreducible components $X_1\cup\cdots\cup X_n$. Then $X_1\setminus (X_2\cup\cdots\cup X_n)$ is an open irreducible subscheme, and so we may pick an affine open irreducible subscheme $U\subset X_1\setminus (X_2\cup\cdots\cup X_n)\subset X$.
As geometrically reduced implies reduced and any open subscheme of a reduced scheme is reduced, $U$ is reduced. As open immersions are preserved under base change, we have that $U_{\overline{k}}$ is an open subscheme of $X_{\overline{k}}$, which implies $U_{\overline{k}}$ is reduced by the same logic as in the previous sentence. So $U$ is an affine, irreducible, geometrically reduced subscheme. In particular, $U$ is an integral affine scheme. Further, any regular closed point of $U$ is a regular closed point of $X$, so if $U$ has a regular closed point, then $X$ must have a regular closed point.
It should be pointed out that it's much easier to work with irreducible opens for this reduction rather than (closed) irreducible components. It's automatic that any open subscheme of a reduced scheme is reduced, but one needs to add more conditions when speaking of closed subschemes. So one might as well make one's life easier by just jumping straight to an irreducible open affine.