A construction of equivariant bundles on the space of symmetric forms

TL;DR

This paper constructs new stable, equivariant vector bundles on symmetric form spaces, providing explicit resolutions and examples that reach theoretical bounds, advancing understanding in algebraic geometry.

Contribution

It introduces a method to construct stable equivariant vector bundles on symmetric form spaces with explicit resolutions, including new examples on projective spaces.

Findings

Constructed stable equivariant vector bundles with length 2 resolutions.

Provided examples reaching Westwick's upper bound for matrix rank.

Extended the class of known equivariant vector bundles on projective spaces.

Abstract

We construct stable vector bundles on the space of symmetric forms of degree d in n+1 variables which are equivariant for the action of SL_{n+1}(C), and admit an equivariant free resolution of length 2. For n=1, we obtain new examples of stable vector bundles of rank d-1 on P^d, which are moreover equivariant for SL_2(C). The presentation matrix of these bundles attains Westwick's upper bound for the dimension of vector spaces of matrices of constant rank and fixed size.