Parallel Kustin-Miller unprojection with an application to Calabi-Yau geometry

Kustin-Miller unprojection is a process by which one can construct interesting new Gorenstein rings starting from simpler ones. Geometrically, it inverts certain projections and appears in the constructions of explicit birational geometry. It is often desirable to perform not only one but a series of unprojections. The main aim of the present paper is to develop a theory, which we call parallel Kustin-Miller unprojection, that applies when all the unprojection ideals of a series of unprojections correspond to ideals already present in the initial ring. As an application, we explicitly construct seven families of Calabi-Yau 3-folds of high codimensions.