# On the existence of Ulrich bundles on blown-up varieties at a point

## TL;DR

This paper demonstrates the construction of Ulrich bundles on blown-up varieties at a point, extending the existence results to new geometric contexts and exploring applications to algebraic surfaces.

## Contribution

It provides a method to construct Ulrich bundles on blow-ups of varieties at a point, based on suitable very ample divisors, and discusses applications to surface theory.

## Key findings

- Ulrich bundles exist on blow-ups at a point under certain conditions.
- A construction method for Ulrich bundles on blown-up varieties is proposed.
- Applications to minimal models and Kodaira dimension of surfaces are explored.

## Abstract

The objective is to show the construction of an Ulrich vector bundle on the blowing-up $\widetilde X$ of a nonsingular projective variety $X$ at a closed point, where the original variety is embedded by a very ample divisor $H$ and carries an Ulrich vector bundle. In order to achieve this result, we aim to find a suitable very ample divisor on $\widetilde X$, which is dependent on $H$. At the end, we take into consideration some applications to surfaces with regards to minimal models and their Kodaira dimension.