Log canonical models of foliated surfaces

We study log canonical models of foliated surfaces of general type. In particular, we show that log canonical models of general type and their minimal partial du Val resolutions are bounded. Moreover, we show the valuative criteria of separatedness and properness and a property related to local-closedness for the moduli functor ${\mathcal{S}}_P^{sm}$ which parametrizes the stable smoothable foliated surface pairs. On the way, we also show a result on the invariance of plurigenera.