Marked Godeaux surfaces with special bicanonical fibers (Q6597194)

Numerical Godeaux surfaces are the projective algebraic surfaces of general type with the smallest possible invariants, namely \(K_X^2 = \chi(\mathcal O_X) = 1\). They have been intensively studied but their classification has remained elusive so far.\N\NIn [\textit{F.-O. Schreyer} and \textit{I. Stenger}, Trans. Am. Math. Soc. 376, No. 5, 3419--3443 (2023; Zbl 1518.14052)] the authors have employed homological ideas to give an explict construction of marked Godeaux surface, where the bi-canonical map has four simple base points and these are ordered. In this paper, they continue their investigation by looking at special elements in the bi-canonical pencil. In this way, they identify the known families with torsion \(\mathbb Z/5\) and \(\mathbb Z/3\) in terms of their construction.\N\NMore work needs to be done to exclude the existence of unknown exotic families, and they include some experimental results in this direction.