I have 2 questions.
Let $X$ be a smooth projective curve. Let $L$ be a line bundle.
Why is $L$ slope-stable?
What I need to show is that every non trivial subbundle (in the sense that it is a subsheaf which is a vector bundle in its own right) has strictly smaller slope. So a subbundle can have rank $= 0$ or $1$. The 0 case gives the trivial subbundle. What contradiction does the 1 case give?
Does the 1 case at all occur? That is, can a line bundle have a proper subbundle which has rank 1?