The Stacks project

Exercise 111.44.5. Let $X$ be a projective surface over an algebraically closed field $k$. Prove there exists a coherent $\mathcal{O}_ X$-module $\mathcal{F}$ such that $H^2(X, \mathcal{F})$ is nonzero. Hint: Use the result of Exercise 111.44.4 and a cleverly chosen exact sequence.