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

Abstract

The objective is to show the construction of an Ulrich vector bundle on the blowing-up $\tilde{X}$ 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 $\tilde{X}$, 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.

v2: modified description of Thm.2, beginning of Section 3, Cor.3 and Cor.8. I thank G. Casnati and Y. Kim for their helpful comments

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• On the positivity of the first Chern class of an Ulrich vector bundle