as a fiber. It has a unique 1-section $\Delta_0$ whose self-intersection is $-d$.

-   We use ’$\equiv$’ to denote linear equivalent relation of divisors.

-   If $D$ is a Cartier divisor on $X$, then we denote $\Phi_{|D|}\colon X\dashrightarrow \mathbb{P}:=|D|^*$ as the rational map defined by the linear system $|D|$.

-   A line $l$ on a variety $X$ with respect to $\mathcal{O}_X(1)$ is a rational curve such that $l\cdot \mathcal{O}_X(1)=1$.

Preliminaries {#prelim}
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Let $X$ be a demi-normal surface, i.e. $X$ satisfies $S_2$ and is at worst ordinary double at any generic point of codimension 1. Denote $\pi\colon  \bar X \to X$ as the normalisation map of $X$. The conductor ideal $ \mathrm{\mathcal{H}om}_{\mathcal{O}_X}(\pi_*\mathcal{O}_{\bar{X}}, \mathcal{O}_X)$ is an ideal sheaf both on $X$ and $\bar{X}$ and hence defines subschemes $D\subset X \text{ and } \bar D\subset \bar X$, both reduced and of pure codimension 1; we often refer to $D$ as the non-normal locus of $X$.

Let $\Lambda$ be a reduced curve on $X$ whose support does not contain any irreducible component of $D$. Then the strict transform $\bar \Lambda$ in the normalization is well defined. We have $\pi^*(K_X+\Lambda)=K_{\bar{X}}+\bar D+\bar \Lambda$ and $(K_X+\Lambda)^2 = (K_{\bar X}+\bar D+\bar \Lambda)^2$.

\[defin: slc\] We call a 