# On degeneracy loci of equivariant bi-vector fields on a smooth toric variety

## TL;DR

This paper investigates the structure of degeneracy loci of equivariant bi-vector fields on smooth toric varieties, establishing lower bounds on their dimensions and non-emptiness under certain conditions.

## Contribution

It proves that degeneracy loci of equivariant bi-vector fields on smooth toric varieties have guaranteed non-emptiness and minimum dimension bounds, extending understanding of their geometric properties.

## Key findings

- Degeneracy loci are non-empty under specified conditions.
- Loci have at least one component of dimension ≥ 2k+1.
- Results apply to both compact and non-compact smooth toric varieties.

## Abstract

We study equivariant bi-vector fields on a toric variety. We prove that, on a smooth toric variety of dimension $n$, the locus where the rank of an equivariant bi-vector field is $\leq 2k$ is not empty and has at least a component of dimension $\geq 2k+1$, for all integers $k> 0$ such that $2k < n$. The same is true also for $k=0$, if the toric variety is smooth and compact. While for the non compact case, the locus in question has to be assumed to be non empty.