My question: Prove that if $X$ is an irreducible variety in $\mathbb{C}^n$, then $\mathbb{C}^n \setminus X$ is path-connected in the standard metric topology.

For every two distinct points $p,q$ in $\mathbb{C}^n \setminus X$,let $L$ denote the line through $p$ and $q$. I can prove the line is a continuous path but I don't know how to show $L \subseteq \mathbb{C}^n \setminus X$ i.e. how to use the condition that $X$ is irreducible?

I'm stuck on this question for several hours. Can anyone give me a hint? Thank you !


Hint: Think about the complex line $\{tp+(1-t)q\in\mathbb{C}^n\mid t\in\mathbb{C}\}\cong\mathbb{C}$.

  • $\begingroup$ I still don't get it.....I can see the isomorphism but how should I use it? Can you help a little bit more? $\endgroup$ – bbw Jun 10 at 1:59
  • $\begingroup$ Bigger hint: What is the intersection of $X$ with this line? $\endgroup$ – user10354138 Jun 10 at 2:06
  • $\begingroup$ I'm confused....$X$ is just an arbitrary irreducible variety and $L$ is the line through $p$ and $q$. We don't have specific "data" about them, how do we even know $X \cap L$? $\endgroup$ – bbw Jun 10 at 2:12
  • $\begingroup$ Put it another way, what can you say about the intersection? $\endgroup$ – user10354138 Jun 10 at 2:15
  • 1
    $\begingroup$ $X$ need not be finite, but $L\cap X$ is finite since it is an algebraic set and can't be $L$. Irreducibility doesn't come in here, the result is true for all algebraic sets $X$. $\endgroup$ – user10354138 Jun 10 at 4:54

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