Theorem 1.2 of Deligne-Mumford's 1969 IHÉS paper, "The irreducibility of the space of curves of given genus," is as follows.

If $g \ge 2$ and $C$ is a stable curve of genus $g$ over an algebraically closed field $k$, then $$H^1(C, \omega_{C/k}^{\otimes n}) = (0)$$if $n \ge 2$, and $\omega_{C/k}^{\otimes n}$ is very ample if $n \ge 3$.

I have two questions, as follows.

- What is the easiest way to see that the theorem is true?
- Do there exist any alternate presentations of the proof in the literature?

Many thanks in advance.