Special syzygies of rational surfaces generated by dual quaternions

In this paper, we first generate a family of rational surfaces in affine 3-space from three rational space curves by dual quaternion multiplication utilizing dual quaternions as a tool to represent rigid transformations. We provide an algorithm to compute all the base points of the homogeneous tensor product parametrization of this family of surfaces. Our main focus is the syzygies of these surfaces. We discover two sets of special syzygies, and show that the syzygy module and a μ-basis of this surface can be extracted from either set of special syzygies. Finally, we describe the structure of a free resolution of the module generated by these special syzygies, and use this free resolution to classify the minimal free resolutions of this module.