Syzygies of Abelian and Bielliptic Surfaces in P⁴

So far only six families of smooth irregular surfaces are known to exist in P^4 (up to pullbacks by suitable finite covers of P^4). These are the elliptic quintic scrolls, the minimal abelian and bielliptic surfaces (of degree 10), two different families of non-minimal abelian surfaces of degree 15, and one family of non-minimal bielliptic surfaces of degree 15. The main purpose of the paper is to describe the structure of the Hartshorne-Rao modules and the syzygies for each of these smooth irregular surfaces in P^4, providing at the same time a unified construction method (via syzygies) for these families of surfaces.