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
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
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: