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

Abstract

Kustin--Miller unprojection constructs more complicated Gorenstein rings from simpler ones. Geometrically, it inverts certain projections, and appears in the constructions of explicit birational geometry. However, 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 of the theory, we explicitly construct 7 families of Calabi--Yau 3-folds of high codimensions.

25 pages. v2: Applications in Section 4 expanded

References in corpus (3)

• Fano 3-folds in codimension 4, Tom and Jerry, Part I
• Geometric transitions between Calabi-Yau threefolds related to Kustin-Miller unprojections
• Parallel Kustin--Miller unprojection with an application to Calabi--Yau geometry

Cited by in corpus (7)

• Parallel Kustin--Miller unprojection with an application to Calabi--Yau geometry
• A smoothness test for higher codimensions
• Stellar subdivisions and Stanley-Reisner rings of Gorenstein complexes
• Implementing the Kustin-Miller complex construction
• Unprojection and deformations of tertiary Burniat surfaces
• Diptych varieties. II: Polar varieties
• Birational modifications of surfaces via unprojections