Quadric surfaces in the Pfaffian hypersurface in $\mathbb{P}^{14}$

TL;DR

This paper investigates smooth quadric surfaces within a specific Pfaffian hypersurface in projective space, exploring their geometric properties and associated vector bundles, and relating these to line congruences in five-dimensional projective space.

Contribution

It provides a detailed analysis of the geometry of quadric surfaces in the Pfaffian hypersurface and their connection to vector bundles and line congruences, a novel exploration in this context.

Findings

Characterization of smooth quadric surfaces in the Pfaffian hypersurface.

Analysis of the associated globally generated vector bundles.

Relation between these surfaces and linear congruences of lines in $P^5$.

Abstract

We study smooth quadric surfaces in the Pfaffian hypersurface in $\mathbb{P}^{14}$ parameterising $6 \times 6$ skew-symmetric matrices of rank at most 4, not intersecting the Grassmannian $\mathbb{G}(1,5)$. 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 of lines in $\mathbb{P}^5$.