On a particular exact sequence in cohomology

Question

The setup for my question is as follows (from page 9 of Deschamps' expository notes on the Artin-Winters proof of semi-stable reduction here). We want to prove that for $X$ the special fiber of a regular $S$-curve with generic fiber $C$, if $X = \sum_{i=1}^nr_iC_i$ as a sum of irreducible divisors $C_i$ is not reduced, and we also have $\gcd(r_i) = 1$, then, letting $X = Z+C_i$, where $Z$ is divisor containing $X_{red}$ and where $C_i\cdot Z>0$, we have$$\dim_k(H^1(X,\mathcal{O}_X)) = g> \dim_k H^1(X_{red}, \mathcal O_{X_{red}})$$Deschamps writes that from the exact sequence:$$0\to \mathcal O_{C_i}(-Z) \to \mathcal O_X \to \mathcal O_Z\to 0$$we get an exact sequence in cohomology:$$0\to H^1(\mathcal O_{C_i}(-Z))\to H^1(\mathcal O_X) \to H^1(\mathcal O_Z)\to 0$$and from this get$$h^1(X,\mathcal O_X) = g-1+h^0(X,\mathcal O_X) = g > h^1(X,\mathcal O_Z)\geq h^1(X_{red}, \mathcal O_{X_{red}})$$ My question is, why do the peripheral terms that might otherwise flank our short exact sequence in cohomology disappear? (I'm under the impression that $H^2(\mathcal O_{C_i}(-Z))$ disappears just because $C_i$ is of dimension 1, but why isn't $H^0(\mathcal O_Z)$ present?).