# A construction of equivariant bundles on the space of symmetric forms

### Ada Boralevi

Politecnico di Torino, Italy### Daniele Faenzi

Université de Bourgogne et Franche Comté, Dijon, France### Paolo Lella

Politecnico di Milano, Italy

## 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.

## Cite this article

Ada Boralevi, Daniele Faenzi, Paolo Lella, A construction of equivariant bundles on the space of symmetric forms. Rev. Mat. Iberoam. 38 (2022), no. 3, pp. 761–782

DOI 10.4171/RMI/1307