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 ${\mathrm{SL}}_2(\mathbb{C})$. 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.