On syzygies of Segre embeddings of P1×P1

Abstract

We construct a nonminimal graded free resolution of Segre embeddings of ${\mathbb{P}}^1\times {\mathbb{P}}^1,$ although we don’t compute all maps. We use this to prove an explicit formula for certain nonzero entries in the graded Betti table, at the end of the first row. We work over an arbitrary field k.

Keywords:

• Syzygies
• Segre

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D02

Acknowledgments

This research was part of my PhD which was funded by the Flemish Research Council (Fonds voor Wetenschappelijk Onderzoek, FWO). At the time of writing this article I worked at the Department of Mathematics of the KU Leuven. I also want to thank the anonymous referee from the previous submission for his/her helpful comments.

Notes

1 This is not exactly a double complex as the composition of horizontal maps is not zero.

Additional information

Funding

This research was part of my PhD which was funded by the Flemish Research Council (Fonds voor Wetenschappelijk Onderzoek, FWO)