Skew-symmetric matrices and Palatini scrolls

TL;DR

This paper establishes a birational equivalence between certain Grassmannians of skew-symmetric forms and the Hilbert scheme of Palatini scrolls, revealing deep geometric relationships for specific parameter ranges.

Contribution

It proves a birational correspondence between Grassmannians of skew-symmetric forms and Hilbert schemes of Palatini scrolls, extending known geometric connections.

Findings

Grassmannian of m-dimensional subspaces is birational to the Hilbert scheme of Palatini scrolls for m>3, k>m-2.

For m=3, the Grassmannian is birational to pairs of smooth plane curves and stable rank-2 bundles.

The results generalize the understanding of the geometry of Palatini scrolls and their moduli.

Abstract

We prove that, for m greater than 3 and k greater than m-2, the Grassmannian of m-dimensional subspaces of the space of skew-symmetric forms over a vector space of dimension 2k is birational to the Hilbert scheme of Palatini scrolls in P^(2k-1). For m=3 and k greater than 3, this Grassmannian is proved to be birational to the set of pairs (E,Y), where Y is a smooth plane curve of degree k and E is a stable rank-2 bundle on Y whose determinant is O(k-1).