Certain algebraic surfaces of general type with irregularity one and their canonical mappings

In this paper, we show the existence of certain algebraic surfaces of general type with irregularity one, and investigate the canonical mappings of these surfaces. Such a surface has a pencil of nor.-hyperelliptic curves of genus 3 over an elliptic curve, and is obtained as the minimal resolution of a relative quartic hypersurface with at most rational double points as singularities, of the projective plane bundle over an elliptic curve. We use some results on locally free sheaves over elliptic curves by Atiyah and Oda to prove the existence.