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I'm trying to think of a quasi-projective variety that is not isomorphic to a quasi-affine one. I image that it must be $Y \subseteq \mathbb{P}^n$ of at least $n \geq 3$, and maybe $\operatorname{dim} Y \geq 2$ as well. I am also interested in finding a low (co)dimensional example of a quasi-projective that is not homeomorphic to a quasi-affine.

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    $\begingroup$ Clearly: you obviously also mean to rule out projective varieties from your consideration of quasi-projective varieties - but just to say it. $\endgroup$ – peter a g Oct 25 '16 at 0:18
  • $\begingroup$ Just to clarify: When you say homeomorphic, in what topology are you dealing? $\endgroup$ – Ted Shifrin Oct 25 '16 at 0:39
  • $\begingroup$ Zariski topology $\endgroup$ – basket Oct 25 '16 at 0:42
  • $\begingroup$ I just realized that my use of 'Hartog's Theorem' may cause people to believe this question refers to varieties over $\mathbb{C}$ in the Hausdorff topology. I would just like to clarify that by variety I mean in the sense of Hartshorne chapter 1, $\endgroup$ – basket Oct 25 '16 at 1:50
  • $\begingroup$ In regards to your answer to @TedShifrin's question: don't the arguments below apply then? Namely, if the variety $X={\mathbb P}^2 - pt$ were homeomorphic in the Zariski topology to a quasi-affine variety, it would have non-constant global functions. Am I missing something? $\endgroup$ – peter a g Oct 25 '16 at 11:55
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I do not know of an easy way than appealing to Hartog's theorem. The example is take a high dimensional projective space and remove a point (or a line). It results in a quasi-projective variety as we are removing a Zariski closed subset. Now by Hartog's theorem (all varieties here are normal, in fact smooth), any global function on this subvariety will extend to the whole, hence a constant.

However a quasi-affine variety is rich with global regular functions. So this provides the example you are looking for.

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    $\begingroup$ I did not consider Hartog's theorem, but it provides a counter example for even $n = 2$ as David points out. I think I will refrain from accepting an answer for a while until someone addresses the homeomorphism problem though. $\endgroup$ – basket Oct 25 '16 at 0:09
  • $\begingroup$ Don't be in a hurry to accept an answer. Wait for a few hours. Also I do not understand much about topology of varieties; there are other experts who can answer with those aspects addressed. $\endgroup$ – P Vanchinathan Oct 25 '16 at 0:13
  • $\begingroup$ @PVanchinathan This provides an example of a quasi-projective non-projective variety which is not isomorphic to a quasi-affine variety in the category of varieties. But it doesn't necessarily give an example of a quasi-projective non-projective variety which is not homeomorphic to a quasi-affine variety. $\endgroup$ – Ariyan Javanpeykar Oct 25 '16 at 14:13
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I think $\mathbb{P}^2 - pt$ is an example, but I would have to think longer about why this can't be quasi-affine.

(I think you can argue that if it were quasi-affine, it would have global functions, but if that were the case, you would be able to find global functions on $\mathbb{P}^2$, which can't be. Hartog's extension theorem works over $\mathbb{C}$.)

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    $\begingroup$ Typo? "affine" should be "quasi-affine"? $\endgroup$ – peter a g Oct 25 '16 at 0:11
  • $\begingroup$ Yes! Thank you! $\endgroup$ – David Steinberg Oct 25 '16 at 0:13

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