The Stacks project

Exercise 111.59.4. Let $k$ be an algebraically closed field. Let $X$ be a reduced, projective scheme over $k$ all of whose irreducible components have the same dimension $1$. Let $\omega _{X/k}$ be the relative dualizing module. Show that if $\dim _ k H^1(X, \omega _{X/k}) > 1$, then $X$ is disconnected.