pair $(X, \Lambda)$ as above a *log surface*; $\Lambda$ is called the (reduced) boundary.

A log surface $(X,\Lambda)$ is said to have *semi-log-canonical (slc)* singularities if it satisfies the following conditions:

1.  $K_X + \Lambda$ is $\mathbb{Q}$-Cartier, that is, $m(K_X+\Lambda)$ is Cartier for some $m\in\mathbb{Z}^{>0}$; the minimal such $m$ is called the (global) index of $(X,\Lambda)$.

2.  The pair $(\bar X, \bar \Lambda+\bar D)$ has log-canonical singularities.

The pair $(X,\Lambda)$ is called stable log surface if in addition $K_X+\Lambda$ is ample. A stable surface is a stable log surface with empty boundary.

By abuse of notation we say $(X, \Lambda)$ is a Gorenstein stable log surface if the index is equal to one, i.e. $K_X+\Lambda$ is an ample Cartier divisor.

Gorenstein slc singularities and semi-resolutions
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Normalizing a demi-normal surface looses all information on the gluing in codimension one. Often it is better to work on a simpler but still non-normal surface.

A surface $X$ is called semi-smooth if every singularity of $X$ is either double normal crossing or a pinch point [^1]. The normalization of a semi-smooth surface is smooth.

A morphism of demi-normal surfaces $f\colon Y\rightarrow X$ is called a semi-resolution if the following conditions are satisfied:

1.  