$Y$ is semi-smooth;

2.  $f$ is an isomorphism over the semi-smooth open subscheme of $X$;

3.  $f$ maps the singular locus of $Y$ birationally onto the non-normal locus of $X$.

A semi-resolution $f\colon Y\rightarrow X$ is called minimal if no $(-1)$-curve is contracted by $f$, that is, there is no exceptional curve $E$ such that $E^2 =K_Y\cdot E = -1$.

Semi-resolutions always exist and one can also incorporate a boundary [@KollarSMMP 10.5].

\[rem: classification of sings\] Semi-log-canonical surface singularities have been classified in terms of their resolution graphs, at least for reduced boundary [@KSB88].

Let $x\in (X, \Lambda)$ be a Gorenstein slc singularity with minimal log semi-resolution $f\colon Y\to X$. Then it is one of the followings (see [@Kollar-Mori Ch. 4], [@KollarSMMP Sect. 3.3], [@kollar12] and [@LR13]):

Gorenstein lc singularities, $\Lambda=0$

:   In this case $x\in X$ is a canonical singularity, or a simple elliptic respectively cusp singularity. For the latter the resolution graph is a smooth elliptic curve, a nodal rational curve, or a cycle of smooth rational curves (see also [@Lau77] and [@Reid97 Ch. 4]).

Gorenstein lc singularities, $\Lambda\neq 0$

:   Since the boundary is reduced, $\Lambda$ has at most nodes. If $\Lambda$ is smooth so is $X$ 