Non-existence of low rank Ulrich bundles on Veronese varieties

TL;DR

This paper proves that Veronese varieties of dimension four or higher do not admit low-rank Ulrich bundles, extending the understanding of the geometric properties and bundle structures on these varieties.

Contribution

It establishes the non-existence of low-rank Ulrich bundles on high-dimensional Veronese varieties and certain complete intersections, providing new insights into their geometric and bundle-theoretic properties.

Findings

Veronese varieties of dimension ≥ 4 lack Ulrich bundles of rank ≤ 3.

Certain complete intersections of dimension ≥ 4 do not carry low-rank Ulrich bundles.

The results extend non-existence to very general embeddings and specific degrees.

Abstract

We show that Veronese varieties of dimension $n \ge 4$ do not carry any Ulrich bundles of rank $r \le 3$. In order to prove this, we prove that a Veronese embedding of a complete intersection of dimension $m \ge 4$, which if $m=4$ is either $\mathbb P^4$ or has degree $d \ge 2$ and is very general and not of type $(2), (2,2)$, does not carry any Ulrich bundles of rank $r \le 3$.