Let $X$ be a 1-dimensional scheme and it contains a cycle of curves $C_1,...,C_n$ that intersect at $x_1,...,x_n$. At this point I want to study $H^1(X,O_X)$ to find a closed subscheme of an algebraic surface with $h^1>n$, so I want to prove that $H^1(X,O_X)$ is nonzero and its dimension is at least n.
My approach: In order to do this I was thinking of constructing some short exact sequence (probably there is a canonical one in this case that I am missing) in order to deal with the n curves and their intersection points starting with $O_X\rightarrow \oplus O_{C_i}$.