Let $K$ be a field and let $F\in K[x,y]$ be a polynomial such that
$$ X_1:=\mathrm{spec}(K[x,y]/(F)) $$
is a reduced irreducible affine variety over $K$. You can also add smooth if you want to.
Now as we don't know the exact equation $F$, I have no information about an equation $F_2$ at infinity and I can define
$$ X:=X_1\cup_\text{glue}\mathrm{spec}(K[z,v]/(F_2)) $$
but I do not know if $X$ is reduced.
I have two questions:
- Does the equation at infinity always exist?
- Is there a statement like: If the variety is reduced everywhere except one geometric point, then it is reduced?
I read Proposition 2.4.2 (page 60) in Qing Liu's AG, but I do not know how to apply it here.
Kind regards
I don't know what you mean by an "equation at infinity." However your other question, which I'll interpret to mean "if $X$ is a $K$-scheme locally of finite type which is regular except at one closed point, is it regular everywhere?," has a negative answer in general. When you say "geometric point" of $X$, the natural interpretation is an element of $X(\overline{K})$, but this is a $K$-morphism $\mathrm{Spec}(\overline{K})\to X$, not just a physical point of $X$ (it's really the data of a pair consisting of a point $x\in X$ and a $K$-map $K(x)\hookrightarrow\overline{K}$). But if you forget about the second part of the pair, and just look at all the points you have, you're just talking about the closed points of $X$ (because $X$ is locally of finite type). Anyway, if you take something like $R=K\times K[t]/(t^2)$, then $\mathrm{Spec}(R)=\mathrm{Spec}(K)\coprod\mathrm{Spec}(K[t]/(t^2))$ consists of two closed points, one regular (even $K$-smooth), the other not reduced.
If you want your scheme to be integral, this still fails. Take the closed subscheme $X$ of $\mathbf{P}^2_K$ cut out by a Weierstrass equation with discriminant zero. There will be a single non-smooth closed point (if we assume $K$ is perfect, this is the same as having a single non-regular closed point).
In the title of your question, you ask about reducedness, but in the body, you ask about regularity. For reducedness, the answer is still "no," as the first example above indicates.