Exercise 111.44.4. Let $X$ be a projective surface over an algebraically closed field $k$. Show that for every $n \geq 0$ there exists a proper closed subscheme $Z \subset X$ such that $\dim _ k H^1(Z, \mathcal{O}_ Z) > n$. Only explain how to do this by modifying the arguments in Exercise 111.44.3 and 111.44.2; don't give all the details.
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