Instanton bundles vs Ulrich bundles on projective spaces

Abstract

We relate the existence of rank $r$ Ulrich bundles on a Veronese 3-fold $\left(\mathbb{P}^3, \mathcal{O}_{\mathbb{P}^3}(d)\right)$ with the existence of rank $r$ instanton bundles on $\mathbb{P}^3$. This relation will allow us to prove the existence of rank $r$ Ulrich bundles on $\left(\mathbb{P}^3, \mathcal{O}_{\mathbb{P}^3}(d)\right)$ for certain values of $(d, r)$. For instance, we explicitly determine the integers $r$ such that rank $r$ Ulrich bundles on $\mathbb{P}^3$ for the Veronese embedding $\mathcal{O}_{\mathbb{P}^3}(3)$ exist and, in particular, we solve the first open case of Conjecture 4.1.