first cohomology of the n-th power of the sheaf of relative differentials

Question

Imagine you have a curve $X$ (integral scheme of dimension 1, proper over $k$ (algebraically closed) whose local rings are regular) of genus $g$. What can be said about $H^1(X, ( \Omega^{1}_{X / k})^n)$?

Greetings

Marc

Answer 1

By Serre duality, your $H^1$ is isomorphic to the dual (as vector space) of $H^0(X, (\Omega_X)^{\otimes (1-n)})$. Let $e_n$ be its dimension. Then:

1. If $n=1$, $e_n=1$.
2. If $n\ge 2$ and $g\ge 2$, then $e_n=0$ because $(\Omega_X)^{\otimes (1-n)}$ has degree $(1-n)(2g-2)<0$.
3. If $n\ge 2$ and $g=1$, then $e_n=1$ because $\Omega_X=O_X$.
4. If $n\ge 2$ and $g=0$, then $e_n=2n-2+1=2n-1$ again by Riemann-Roch for the projective line.