I have recently been learning about algebraic geometry and came across this question in "Algebraic Geometry A Problem Solving Approach" by Thomas Garrity et al. His problem 1.12.18 asks about the dual curve to a projective line given by $L=\{(x:y:z)\in \mathbb{P}^2:ax+by+cz\}$ and why it might be strange.
When I try and answer the question on my own following the recipe on https://en.wikipedia.org/wiki/Dual_curve I get $X=Y=Z=\lambda$, which doesn't make sense to me.
I also looked at the Plücker formula, which I think is saying that a degree 1 curve should have a degree zero dual.
Does anyone have any thoughts? Thanks!
The dual of $x+y+z=0$ is the point $(1:1:1).$
Your link says: “The dual of a line (a curve of degree 1) is an exception to this and is taken to be a point in the dual space (namely the original line). The dual of a single point is taken to be the collection of lines though the point; this forms a line in the dual space which corresponds to the original point.”
But as you can interpret $ax+by+cz=0$ as a line in the plane, changing the roles of coefficients and variables it’s also the equation of a line in the dual plane. This is the universal line.