Tom and Jerry triples with an application to Fano 3-folds

Abstract

Unprojection is a theory due to Reid which constructs more complicated rings starting from simpler data. In the present work, we develop a new method of unprojection. Starting from a codimension 3 ideal defined by the Pfaffians of a 5 × 5 skewsymmetric matrix, we describe a number of unprojection formats that construct Gorenstein rings of codimension 6. We give two applications that construct families of Fano 3-folds embedded anticanonically in codimension 6 that existing methods do not realize.

Keywords:

• Birational geometry
• Fano 3-folds
• Gorenstein rings
• unprojection

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 14M05
• 14J45
• Secondary: 13H10
• 14E99

Acknowledgments

I would like to express my deep gratitude to my supervisor Stavros Papadakis for his valuable guidance and ideas which contributed immensely in the present work. As well as for the important suggestions which have improved the presentation of the paper. I also thank an anonymous referee for contributing to the further improvement of the material of the paper. I benefited from experiments with the computer algebra program Macaulay2 [19].

Additional information

Funding

This work is part of a Univ. of Ioannina Ph.D. thesis, financially supported by the Special Account for Research Funding (E.L.K.E) of University of Ioannina (UOI) under the program with code 82561 and title “Program of financial support for Ph.D. students and postdoctoral researchers.”