On the minimal model of semi-isogenous mixed surfaces

The aim of this paper is to determine minimal models of the semi-isogenous mixed surfaces with $\chi =1$ and $K^2>0$ constructed by Cancian and Frapporti (Math. Nachr., 291(2-3): 264-283, 2018). In order to do this, we further develop the idea of orbit divisors introduced by Frapporti and Lee (Pac. J. Math., 318(2): 233-247, 2022), to construct effective divisors on surfaces isogenous to a product of mixed type, extending it to the semi-isogenous mixed surfaces.