show that $(, , )$ and $(′, ′, ′)$ define the same projective line if $(, , ) = (′, ′, ′)$ for some nonzero $ ∈ ℂ$

Question

How do I show that the triples $(, , )$ and $(′, ′, ′)$ define the same projective line if and only if $(, , ) = (′, ′, ′)$ for some nonzero $ ∈ ℂ$

I've seen that this is the definition of a projective plane, but I don't know how to actyally prove this. Any help would be great!!

Edit: A projective line in $ℙ^2(ℂ)$ is defined by $ + + = 0$ where $0 ≠ (, , ) ∈ ℂ^3$

What definition of $\mathbb{C}P^2$ are you using? Usually this is just the definition of the defining equivalence relation on $\mathbb{C}^3$. — Nate Gallup, Apr 08 '21 at 02:47
Projective lines are just equivalence classes of hyperplanes in $\mathbb{C}^3$. — mathmathmath, Apr 08 '21 at 03:00

score 0 · Accepted Answer · answered Apr 08 '21 at 03:16

This is straightforward. You show that any point $(x,y,z)$ on the projective line $(a,b,c)$ is also on the projective line $(a’,b’,c’)$; and you also show the converse i.e. that any point $(x,y,z)$ on line $(a’,b’,c’)$ is also on line $(a,b,c)$.

show that $(, , )$ and $(′, ′, ′)$ define the same projective line if $(, , ) = (′, ′, ′)$ for some nonzero $ ∈ ℂ$

1 Answers1