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Let $X\subseteq \mathbf{P}^3$ be a smooth cubic. We know that $X$ contains a line $\ell$, which is a $-1$-curve. Therefore there exists a map $f:X\to Y$ to a smooth surface $Y$ which contracts $\ell$ to a point $p$ and is an isomorphism away from $p$. Let $U\subseteq Y$ be an affine neighborhood of $p$ in which $p$ is defined by $f=g=0$, and let $D\subseteq X$ be the union of $\ell$ and $f^{-1}(Y\setminus U)$. I claim that $H^1(X\setminus D, \mathcal{O}_{X\setminus D})\neq 0$. The map $f$ identifies $X\setminus D$ with $U\setminus \{p\}$. It then suffices to prove that if $U$ is a smooth affine surface with a point $p$ defined by $f=g=0$, then $H^1(U\setminus\{p\},\mathcal{O}_{U\setminus \{p\}})\neq 0$. This can be seen using Cech cohomology for the affine covering by $V=\{f\neq 0\}$ and $W=\{g\neq 0\}$: the functions $f^a g^b$ with $a,b<0$ are defined on $V\cap W$, but are not in the image of $\Gamma(V, \mathcal{O}_V)\times \Gamma(W, \mathcal{O}_W)\to \Gamma(V\cap W, \mathcal{O}_{V\cap W})$, $(x,y)\mapsto x-y$, and hence yield a nonzero element in the cokernel, which is $H^1(U\setminus \{p\}, \mathcal{O}_{U\setminus \{p\}})$.