a Gorenstein stable log surface which is non-normal. $D$ is the non-normal locus of $X$.

Let $C\subset D$ be a subcurve of the non-normal locus of a Gorenstein stable log surface $(X,\Lambda)$ and $\nu\colon:\tilde{X}\to X$ be the partial normalization of $X$ along $C$. Denote $\tilde{C}$ and $\tilde{\Lambda}$ as the proper transformation of $C$ and $\Lambda$.

Then $(\tilde{X},\tilde{\Lambda}+\tilde{C})$ is a Gorenstein stable log surface.

We first notice that $\nu^*(K_X+\Lambda)=K_{\tilde{X}}+\tilde{\Lambda}+\tilde{C}$ is ample and Cartier. Second it is easy to verify that $(\tilde{X},\tilde{\Lambda}+\tilde{C})$ still has Gorenstein slc singularities only by the classification of Gorenstein slc singularities. Therefore $(\tilde{X},\tilde{\Lambda}+\tilde{C})$ is a Gorenstein stable log surface as well.

The map $\nu|_{\tilde C}\colon \tilde{C} \to C$ is generically a double cover and thus induces a rational involution $\tau$ on $\tilde{C}$.

\[prop: descend section\] If $(X, \Lambda)$ is a non-normal Gorenstein stable log surface and $\nu$, $C$, $\tilde{C}$, $\tau$ are defined as above, then $\nu^*H^0(X, K_X+\Lambda)\subset H^0(\tilde{X}, \nu^*(K_X+\Lambda)))$ is the subspace of those sections $s$ whose restriction to $\tilde{C}$ is $\tau$-anti-invariant.

\[separateNonnormalCurve\] It is easy to see that $\nu^*H^0(X, K_X+\Lambda)$ can not separate $\tilde{C}$. Actually, $\Phi_{\nu^*H^0(X, K_X+\Lambda)}|_{\tilde{C}}\circ \tau=\Phi_{\nu^*H^0(X, K_X+\Lambda)}|_{\tilde{C}}$ on $\tilde{C}$.

\[fibersection\] Let $(X,\Lambda)$ be a reducible Gorenstein stable log surface such that $X=X_1\cup X_2$ and $C:=X_1\cap 