Quadric surfaces in the Pfaffian hypersurface in ℙ¹⁴

Abstract

We study smooth quadric surfaces in the Pfaffian hypersurface in ${\mathbb{P}}^{14}$ parameterizing 6 × 6 skew-symmetric matrices of rank at most 4, not intersecting its singular locus. Such surfaces correspond to quadratic systems of skew-symmetric matrices of size 6 and constant rank 4, and give rise to a globally generated vector bundle E on the quadric. We analyse these bundles and their geometry, relating them to linear congruences in ${\mathbb{P}}^5$.

Keywords:

• Skew-symmetric matrices
• constant rank
• pfaffian hypersurface
• quadric surface
• globally generated vector bundles

2020 Mathematics Subject Classifications:

• 15A30
• 14N05
• 14F05
• 14M15

Acknowledgments

The second and third named author are supported by PRIN 2017SSNZAW. The third named author is also supported by FRA of the University of Trieste. The first named author has been partially supported by MIUR grant Dipartimenti di Eccellenza 2018-2022 (E11G18000350001). All authors are members of INdAM–GNSAGA. We thank the referee for the careful reading and for the keen observations made, that allowed us to improve the article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The second and third named author are supported by PRIN 2017SSNZAW. The third named author is also supported by Università degli Studi di Trieste project FRA-BEORCHIA-18, n. j961c1800138001. The first named author has been partially supported by MIUR grant Dipartimenti di Eccellenza 2018–2022 [grant number E11G18000350001]. All authors are members of INdAM–GNSAGA.