On varieties with Ulrich twisted tangent bundles

TL;DR

This paper investigates varieties with tangent bundles twisted by a line bundle that are Ulrich, establishing bounds on the twisting parameter and classifying certain cases, advancing understanding of Ulrich bundles on varieties.

Contribution

It provides a sharp bound for the twisting parameter on curves, classifies pairs for the case k=1, and shows the non-existence of k=2 for higher dimensions.

Findings

Bound for k in curves is sharp.

Classified pairs (X, O_X(1)) for k=1.

Proved k=2 does not occur for n ≥ 4.

Abstract

We study varieties $X \subseteq \mathbb P^N$ of dimension $n$ such that $T_X(k)$ is an Ulrich vector bundle for some $k \in \mathbb Z$. First we give a sharp bound for $k$ in the case of curves. Then we show that $k \le n+1$ if $2 \le n \le 12$. We classify the pairs $(X,\mathcal O_X(1))$ for $k=1$ and we show that, for $n \ge 4$, the case $k=2$ does not occur.