0
$\begingroup$

I understand why nonempty open subsets of irreducible topological spaces are dense. I am having trouble comprehending how this translates to Zariski's topology. We know that a finite affine variety is irreducible iff it is a single point. How do you take an open subset of a single point? What does this look like geometrically?

$\endgroup$
10
  • $\begingroup$ The statement "an affine variety is irreducible iff it is a single point" is not true. How did you come to that? $\endgroup$
    – KReiser
    Feb 4, 2023 at 19:56
  • $\begingroup$ @KReiser Andreas Gathmann Chapter 2; example 2.11: "A finite affine variety is irreducible if and only if it is connected: namely if and only if it contains at most one point." Am I interpreting this wrong? $\endgroup$ Feb 4, 2023 at 20:00
  • $\begingroup$ @SassatelliGiulio ^ $\endgroup$ Feb 4, 2023 at 20:00
  • $\begingroup$ Lots of varieties are not finite! For instance, $\Bbb A^1$. $\endgroup$
    – KReiser
    Feb 4, 2023 at 20:01
  • $\begingroup$ @KReiser You are right. I mean my question with respect to finite affine varieties. So not $\mathbb{A}^n$ $\endgroup$ Feb 4, 2023 at 20:05

0

You must log in to answer this question.