13
$\begingroup$

I am starting to learn algebraic geometry and in the notes I am reading there is the following remark:

" Over the complex numbers and with the strong topology we see that $A^n$ and affine varieties (except for points) are not compact. " (The strong topology seems to be the standard topology in this text.)

Why is that true ?

  • 5
    Dear readingframe, this is an excellent question and you are quite right to ask about this non trivial assertion: +1.2012-03-12

2 Answers 2

9

They are never bounded. ${}{}{}{}{}{}{}$

  • 0
    That is nice. Thank you a lot.2012-03-12
19

Noether's normalization theorem (Mumford, Red book, page 42 ) says that if $X$ is a variety of dimension $n$ , there exists a finite surjective morphism $X\to \mathbb A^n$.
Since, in the transcendental topology over $\mathbb C$, affine space $\mathbb A^n$ is not compact for $n\geq 1$ , $X$ is not compact either.

A comment
Exciting as it definitely is, algebraic geometry has the drawback that many very intuitive facts, like the above, are difficult to justify without some fairly technical tools.
I think it is the duty of a teacher to acknowledge this explicitly in an introductory course (and maybe give a reference for the student to come back to later), rather than throw offhand remarks which might discourage a student and make him feel it is his fault that he can't find the (actually quite hard) rigorous proof.

  • 0
    yes, my comment was on the remark called "comment":)2012-03-15