I'm confused about the use of expression "rational curve" by Griffiths Harris:I'm reading the chapter on algebraic complex surfaces of the book "Principles of algebraic geometry". Throughout the chapter, they refer to irreducible curve as just an effective divisor which is notdecomposable as the sum of two other effective divisors.
I'm reading in particular the paragraph where they show Noether lemma about rational surfaces.
Here they talk about rational irreducible curves and they say they are smooth: I do not understand the use of word rational here:are there subvariety which admit a map to $\mathbb{P}^1$ which is birational? So , why if they are irreducible are smooth?
In particular at the end of the first part of the proof,they get a pencil of irreducible disjoint rational curves and they say they are smooth.
After , it seems to me that they really use these curves as projective lines, but so I do not get the use of the word irreducible (as the projective line is irreducible).