Undergrad Difference between the projective space and one its part

[Ssnow]

Hi everybody, I want to ask if there are big differences consider algebraic objects in the projective or in the projective space with only positive coordinates? I know that the question is generic (so permit the discussion ... ). I am interested to know what happen to classical invariants if someone has informations ...
Thank you in advance.
Ssnow

 

[zinq]

I don't understand what "algebraic invariants" you mean. Or what is meant by "the projective or in the projective". Or by "the question is generic". Or what "classical invariants" you might be referring to.

So I'd need a lot of clarification before understanding your post.

 

[Ssnow]

ok, consider the projective space with an curve, something change if you limit your attention only to the positive coordinates? (I think yes...)

 

[zinq]

Sorry, Ssnow, I am just as confused as before.

 

[Ssnow]

Mmmm, think $\mathbb{P}^{1}$ as a sphere I want to know if a curve in $\mathbb{P}^{1}$, when you restrict the attention only to the positive coordinates , can change totally of something remain invariant prom the previous representation ?

 

[micromass]

Do you mean to take only those points in $\mathbb{P}^n$ with nonzero homogeneous coordinates (with respect to some base)? I don't understand why you say positive, since positivity is not a projective concept.

 

[Ssnow]

Do you mean to take only those points in Pn\mathbb{P}^n with nonzero homogeneous coordinates (with respect to some base)? I don't understand why you say positive, since positivity is not a projective concept.

points $[p_{0}:p_{1}]$ only with $p_{0}>0,p_{1}>0$

 

[micromass]

points $[p_{0}:p_{1}]$ only with $p_{0}>0,p_{1}>0$

So that would be a subset of the affine space, namely $[p_0/p_1: 1] = [a:1]$ with $a>0$. So in this case, we will obtain $\mathbb{R}^+$.

 

[Ssnow]

@micromass yes exactly,
ok $\mathbb{R}^{+}$ is not so interesting but I was thinking what happen to a curve in $\mathbb{P}^{1}$ when you restrict the attention only to this subset, I cannot visualizing if a part of it collapse or nothing happens ... ,

 

[micromass]

@micromass yes exactly,
ok $\mathbb{R}^{+}$ is not so interesting but I was thinking what happen to a curve in $\mathbb{P}^{1}$ when you restrict the attention only to this subset, I cannot visualizing if a part of it collapse or nothing happens ... ,

What do you mean with a curve? Any smooth curve? Or an algebraic curve determined by a polynomial?
You know $\mathbb{P}^1$ is a circle, so there are curves which do not collapse to a point. On the other hand, $\mathbb{R}^+$ is contractible. Any curve can collapse to a constant curve.

 

[Ssnow]

yes sure I was thinking to a generic smooth curve, and I agree on the collapse to the constant curve.

Thanks