Exercise 111.44.2. Let $k$ be an algebraically closed field. Let $X$ be a projective $1$-dimensional scheme. Suppose that $X$ contains a cycle of curves, i.e., suppose there exist an $n \geq 2$ and pairwise distinct $1$-dimensional integral closed subschemes $C_1, \ldots , C_ n$ and pairwise distinct closed points $x_1, \ldots , x_ n \in X$ such that $x_ n \in C_ n \cap C_1$ and $x_ i \in C_ i \cap C_{i + 1}$ for $i = 1, \ldots , n - 1$. Prove that $H^1(X, \mathcal{O}_ X)$ is nonzero. Hint: Let $\mathcal{F}$ be the image of the map $\mathcal{O}_ X \to \bigoplus \mathcal{O}_{C_ i}$, and show $H^1(X, \mathcal{F})$ is nonzero using that $\kappa (x_ i) = k$ and $H^0(C_ i, \mathcal{O}_{C_ i}) = k$. Also use that $H^2(X, -) = 0$ by Grothendieck's theorem.

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