Numerical Burniat and irregular surfaces (Q1868206)

The author constructs new examples of surfaces of general type as double planes, that is as double covers of \(\mathbb{P}^2\). More precisely he constructs: a surface with \(p_g=q=0\), \(P_2=4\) as double plane with branch locus of degree \(12\) having five irreducible components; a surface with \(p_g=q=1\), \(P_2=4\) as double plane with branch locus of degree \(12\) having four irreducible components; a surface with \(p_g=q=0\), \(P_2=5\) as double plane with branch locus of degree \(14\) having six irreducible components; a surface with \(p_g=q=0\), \(P_2=4\) as double plane with irreducible branch locus of degree \(22\). The first three examples have a bicanonical map that is not birational, and a torsion group different from zero (more precisely there are non trivial 2-torsion elements in the Picard group arising from irreducible components of the branch locus). All constructions are explicit: the author gives the equations of the branch curves. The results are presented without complete proofs, which will be published elsewhere.