A construction of equivariant bundles on the space of symmetric forms

Abstract

We construct stable vector bundles on the space $\mathbb{P}(S^d \mathbb{C}^{n+1})$ of symmetric forms of degree $d$ in $n+1$ variables which are equivariant for the action of $\text{SL}\_{n+1}(\mathbb{C})$ 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 $\mathbb{P}^d$, which are moreover equivariant for $\operatorname{SL}\_2(\mathbb{C})$. 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.