Pluricanonical maps of stable log surfaces

Stable surfaces and their log analogues are the type of varieties naturally occurring as boundary points in moduli spaces. We extend classical results of Kodaira and Bombieri to this more general setting: if (X,Δ)(X,Δ) is a stable log surface with reduced boundary (possibly empty) and I   is its global index, then 4I(KX+Δ)4I(KX+Δ) is base-point-free and 8I(KX+Δ)8I(KX+Δ) is very ample.These bounds can be improved under further assumptions on the singularities or invariants, for example, 5(KX+Δ)5(KX+Δ) is very ample if (X,Δ)(X,Δ) has semi-canonical singularities.