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Quaternion rational surfaces. (English) Zbl 1445.14082
Let \(f(s,u)\) and \(g(t,v)\) two one-to-one real projective parametrizations of rational space curves. The quaternion multiplication \(f(s,u)*g(t,v)=h(s,u;t,v)\) yields an one-to-one parametrization of a real rational surface called quaternion rational surface. The surface is assumed to have no base points. Quaternions have the important property that every unit represents a rotation and the multiplication of two quaternions corresponds to the composition of rotations; this allows changing quaternion rational surfaces (location, orientation and shape) by manipulating the two rational space curves that generate the surface. As the authors say in the abstract, since the structure of the graded minimal free resolution of a rational surface is, in general, unknown, the main goal of the paper is, under certain hypotheses, to construct the graded minimal free resolution of a quaternion rational surface generated by two rational space curves. They give explicit formulas for the maps of these graded minimal free resolutions by using \(\mu\)-bases of the generating rational curves, and create the generating sets for the first and second syzygy modules in the graded minimal free resolutions. Necessary and sufficient conditions for a real rational surface to be a quaternion rational surface are also provided. Finally, the authors find the relationsheep between the ideal generated by the parametrization and the ideal generated by the moving planes; they prove that the ideal generated by the first syzygy module expressed in terms of moving planes is the same as the ideal generated by the parametrization in the affine ring.
MSC:
14Q05 Computational aspects of algebraic curves
14Q10 Computational aspects of algebraic surfaces
13D02 Syzygies, resolutions, complexes and commutative rings
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