We are going to prove the inequality $$ \dim_k H^1(D, \mathcal{O}_D) \geq g_{top}(D) + g_{geom}(D/k) $$ for all connected reduced effective Cartier divisors $D \subset (X_k)_{red}$ containing $x$ by induction on the number of irreducible components of $D$. Here $g_{top}(D) = 1 - m + e$ where $m$ is the number of irreducible components of $D$ and $e$ is the number of unordered pairs of components of $D$ which meet. \medskip\noindent Base case: $D$ has one irreducible component. Then $D = C_i$ is the unique irreducible component containing $x$. In this case $\dim_k H^1(D, \mathcal{O}_D) = g_i$ and $g_{top}(D) = 0$. Since $C_i$ has a $k$-rational smooth point it is geometrically integral (Varieties, Lemma \ref{varieties-lemma-variety-with-smooth-rational-point}). It follows that $g_i$ is the genus of $C_{i, \overline{k}}$ (Algebraic Curves, Lemma \ref{curves-lemma-genus-base-change}). It also follows that $g_{geom}(D/k)$ is the genus of the normalization $C_{i, \overline{k}}^\nu$ of $C_{i, \overline{k}}$. Applying Algebraic Curves, Lemma \ref{curves-lemma-genus-normalization} to the normalization morphism $C_{i, \overline{k}}^\nu \to C_{i, \overline{k}}$ we get \begin{equation} \label{equation-genus-change-special-component} \text{genus of }C_{i, \overline{k}} \geq \text{genus of }C_{i, \overline{k}}^\nu \end{equation} Combining the above we conclude that $\dim_k H^1(D, \mathcal{O}_D) \geq g_{top}(D) + g_{geom}(D/k)$ in this case. \medskip\noindent Induction step. Suppose we have $D$ with more than $1$ irreducible component. Then we can write $D = C_i + D'$ where $x \in D'$ and $D'$ is still connected. This is an exercise in graph theory we leave to the reader (hint: let $C_i$ be the component of $D$ which is farthest from $x$). We compute how the invariants change. As $x \in D'$ we have $H^0(D, \mathcal{O}_D) = H^0(D', \mathcal{O}_{D'}) = k$. Looking at the short exact sequence of sheaves $$ 0 \to \mathcal{O}_D \to \mathcal{O}_{C_i} \oplus \mathcal{O}_{D'} \to \mathcal{O}_{C_i \cap D'} \to 0 $$ (Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-union}) and using additivity of euler characteristics we find \begin{align*} \dim_k H^1(D, \mathcal{O}_D) - \dim_k H^1(D', \mathcal{O}_{D'}) & = -\chi(\mathcal{O}_{C_i}) + \chi(\mathcal{O}_{C_i \cap D'}) \\ & = w_i(g_i - 1) + \sum\nolimits_{C_j \subset D'} a_{ij} \end{align*} Here as in Lemma \ref{lemma-numerical-type-of-model} we set $w_i = [\kappa_i : k]$, $\kappa_i = H^0(C_i, \mathcal{O}_{C_i})$, $g_i$ is the genus of $C_i$, and $a_{ij} = (C_i \cdot C_j)$. We have $$ g_{top}(D) - g_{top}(D') = -1 + \sum\nolimits_{C_j \subset D'\text{ meeting }C_i} 1 $$ We have $$ g_{geom}(D/k) - g_{geom}(D'/k) = g_{geom}(C_i/k) $$ by Algebraic Curves, Lemma \ref{curves-lemma-bound-geometric-genus}. Combining these with our induction hypothesis, we conclude that it suffices to show that $$ w_i g_i - g_{geom}(C_i/k) + \sum\nolimits_{C_j \subset D'\text{ meets } C_i} (a_{ij} - 1) - (w_i - 1) $$ is nonnegative. In fact, we have \begin{equation} \label{equation-genus-change} w_i g_i \geq [\kappa_i : k]_s g_i \geq g_{geom}(C_i/k) \end{equation} The second inequality by Algebraic Curves, Lemma \ref{curves-lemma-bound-geometric-genus-curve}. On the other hand, since $w_i$ divides $a_{ij}$ (Varieties, Lemma \ref{varieties-lemma-divisible}) it is clear that \begin{equation} \label{equation-change-intersections} \sum\nolimits_{C_j \subset D'\text{ meets } C_i} (a_{ij} - 1) - (w_i - 1) \geq 0 \end{equation} because there is at least one $C_j \subset D'$ which meets $C_i$.