On the minimal model of semi-isogenous mixed surfaces

TL;DR

This paper determines the minimal models of semi-isogenous mixed surfaces with specific invariants, extending previous methods to a broader class of surfaces.

Contribution

It develops new techniques for constructing effective divisors on semi-isogenous mixed surfaces, building upon and extending prior work on orbit divisors.

Findings

Identified minimal models for a class of semi-isogenous mixed surfaces with χ=1 and K^2>0.

Extended the orbit divisor method to semi-isogenous mixed surfaces.

Provided a framework for analyzing surfaces of mixed type with specific invariants.

Abstract

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. In order to do this, we further develop the idea of orbit divisors introduced by Frapporti and Lee, to construct effective divisors on surfaces isogenous to a product of mixed type, extending it to the semi-isogenous mixed surfaces.