Divisors on a compact Riemann surface.

Question

How to justify the following statement:

deg$(K-D)\le p-1 \Rightarrow $ deg(D)$\ge p-1$.

at where D is a divisors on a compact Riemann surface.

Thank You!

Answer 1

$\deg(K - D) = \deg(K) - \deg(D) = 2p - 2 - \deg(D) \leq p - 1$ by hypothesis, so it follows immediately that $\deg(D) \ \geq p - 1$.

I am assuming you know that $\deg(K) = 2p - 2$ : if not, first prove it for $\mathbb P^1$ and then you can prove it by pullback for any compact projective curve.