On equations of double planes with $p_g=q=1$

TL;DR

This paper details methods to compute equations of certain minimal Du Val double planes with specific invariants, including explicit constructions for cases with high bicanonical map complexity, using Magma software.

Contribution

It introduces a computational approach to derive equations of Du Val double planes with given invariants and constructs an example with a non-composed bicanonical map.

Findings

Computed equations for minimal Du Val double planes with p_g=q=1 and K^2=2,...,8

Constructed a double plane with K^2=8 and non-composed bicanonical map

Demonstrated use of Magma for algebraic surface computations

Abstract

This paper describes how to compute equations of plane models of minimal Du Val double planes of general type with $p_g=q=1$ and $K^2=2,...,8.$ A double plane with $K^2=8$ having bicanonical map not composed with the associated involution is also constructed. The computations are done using the algebra system Magma.