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From their tag descriptions, it sounds like and are synonyms.

On the other hand, I could imagine in theory how they could be differentiated (namely one doesn't necessarily have to explicitly reference the geometry of projective space at all times while studying it).

On a third hand, people might not be likely to obey such a distinction in practice, and questions concerning the geometry of projective spaces might predominate over questions concerning other aspects of them (I don't know if this is true or not).

And on a fourth hand, perhaps it might be useful to have the two tags merged so there is a larger community surrounding them. Or maybe it might not be useful for literally the exact same reason (e.g. the people studying the geometry of projective spaces could drown out those studying their other aspects).

An example of when people seem to benefit from having similar tags being separated is "probability" and "probability-theory", for example, so this might also be the situation here -- in any case, it is not clear to me what the etiquette surrounding these two projective tags is from their tag descriptions, which is why I wanted to bring it to your attention.

Anyway, I am not sure if this is something which other people have noticed before, so perhaps anyone who is an expert in either of these areas (I am not an expert in either of them) would be interested to know that both these two very similar tags exist. Again, I am not sure of anything.

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    $\begingroup$ They definitely should not be synonyms. In a sense it's like making group-theory a synonym of universal-algebra. $\endgroup$
    – Matt Samuel
    Commented Aug 30, 2016 at 0:54
  • $\begingroup$ @MattSamuel OK, so to make sure I don't use them incorrectly again in the future, which of the two is the more general tag? Keep in mind that most of my questions stem from a very introductory algebraic geometry book. $\endgroup$
    – Chill2Macht
    Commented Aug 30, 2016 at 0:56
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    $\begingroup$ That's the thing, universal algebra is not more general than group theory, but it sounds like it is. They're simply different topics. $\endgroup$
    – Matt Samuel
    Commented Aug 30, 2016 at 1:12
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    $\begingroup$ @MattSamuel Oh ok oops. So if I have a question about $\mathbb{CP}^1$ or $\mathbb{CP}^2$, should I use (projective-geometry) or (projective-spaces)? Sorry for the dumb question. $\endgroup$
    – Chill2Macht
    Commented Aug 30, 2016 at 2:09
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    $\begingroup$ It's not a dumb question. From your notation it sounds like you should use projective-space. What broader subject does it belong to? $\endgroup$
    – Matt Samuel
    Commented Aug 30, 2016 at 2:13
  • $\begingroup$ @MattSamuel I guess probably projective varieties in algebraic geometry, although I have only gotten to homogeneous cubic polynomials defined on those spaces (I also asked some questions about the topology of those spaces too) $\endgroup$
    – Chill2Macht
    Commented Aug 30, 2016 at 2:23
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    $\begingroup$ Then I would tag it algebraic-geometry first and foremost, then you can add projective-space or projective-geometry. I'm not enough of an expert to know whether or not anybody you want to see your question would even be looking for the projective-geometry tag, but it technically applies. There's a Wikipedia article on it. $\endgroup$
    – Matt Samuel
    Commented Aug 30, 2016 at 2:32

2 Answers 2

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The projective-space tag is---according to the description---about the space of lines through the origin in some vector space in geometry, so every geometric question that relates to $\mathbb{C}P^n$ or $\mathbb{R}P^n$ belongs there. No matter whether it's algebraic geometry, differential geometry, geometric topology, .... In particular, as already suggested by others, when you're asking a question about projective space in its incarnation as algebraic variety, use the algebraic-geometry tag first and then this one.

The projective-geometry tag, however, is rather concerned with this kind of geometry. (It's the projective counter part to Euclidean geometry in the classical sense.) Therefore, a question about projective space as a model for projective geometry could tag both, projective-geometry and projective-space, but a question about desargues' theorem, e.g., may only tag projective-geometry but not necessarily projective-space.

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Projective geometry could also refer to projective differential geometry, which is a type of Cartan geometry. The typical model space for projective differential geometry is the projective sphere, and a projective differential geometry can be defined on any smooth manifold.

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  • $\begingroup$ Even though projective differential geometry has 'projective geometry' in its name, the description of the Projective-Geometry tag doesn't seem to include projective differential geometry. $\endgroup$
    – Ben
    Commented Jun 21, 2017 at 11:01

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