Difference between the projective space and one its part

In summary: Ssnow. In summary, the conversation discusses the differences in algebraic objects in the projective space with only positive coordinates and how classical invariants may change in this scenario. There is confusion about the specific concepts being referred to and the potential outcomes of this restriction, but it is noted that the subset ##\mathbb{R}^{+}## is not particularly interesting. The conversation also touches on the idea of a curve collapsing to a constant curve in this context.
  • #1

Ssnow

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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
 
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  • #2
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.
 
  • #3
ok, consider the projective space with an curve, something change if you limit your attention only to the positive coordinates? (I think yes...)
 
  • #4
Sorry, Ssnow, I am just as confused as before.
 
  • #5
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 ?
 
  • #6
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.
 
  • #7
micromass said:
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##
 
  • #8
Ssnow said:
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}^+##.
 
  • #9
@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 ... ,
 
  • #10
Ssnow said:
@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.
 
  • #11
yes sure I was thinking to a generic smooth curve, and I agree on the collapse to the constant curve.

Thanks
 

1. What is the projective space?

The projective space is a mathematical concept that extends the traditional Euclidean space to include points at infinity. It is often denoted as P^n, where n is the dimension of the space. In simpler terms, it is a space that allows for the representation of points, lines, and planes that do not intersect in traditional Euclidean geometry.

2. What is the difference between projective space and Euclidean space?

The main difference between projective space and Euclidean space is the inclusion of points at infinity in projective space. In Euclidean space, parallel lines never intersect, but in projective space, they intersect at a point at infinity. Additionally, projective space is a non-metric space, meaning distances and angles are not defined, while Euclidean space is a metric space with well-defined distances and angles.

3. What is the relationship between projective space and one of its parts?

Projective space is a space that is made up of smaller parts, such as projective lines, projective planes, and projective hyperplanes. These parts are defined by certain properties and equations within the projective space. For example, a projective line is a set of points that satisfy a specific equation in projective space. These parts help to define the structure and geometry of the projective space.

4. How is projective space used in science and mathematics?

Projective space has various applications in science and mathematics, particularly in fields such as computer graphics, computer vision, and algebraic geometry. It is also used in physics, specifically in the study of projective quantum mechanics. In mathematics, projective space is used to study projective geometry, which has applications in fields like robotics, computer vision, and image processing.

5. What are the advantages of using projective space over Euclidean space?

One of the main advantages of projective space is its ability to represent points, lines, and planes at infinity, which is not possible in Euclidean space. This makes it a more powerful tool for studying geometric and algebraic properties. Projective space also has simpler and more elegant equations compared to Euclidean space, making it easier to work with in certain mathematical contexts.

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