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Ulrich Bundles on Projective Spaces (1703.06424v1)
Published 19 Mar 2017 in math.AG
Abstract: We assume that $\mathcal{E}$ is a rank $r$ Ulrich bundle for $(Pn, \mathcal{O}(d))$. The main result of this paper is that $\mathcal{E}(i)\otimes \Omega{j}(j)$ has natural cohomology for any integers $i \in \mathbb{Z}$ and $0 \leq j \leq n$, and every Ulrich bundle $\mathcal{E}$ has a resolution in terms of $n$ of the trivial bundle over $Pn$. As a corollary, we can give a necessary and sufficient condition for Ulrich bundles if $n \leq 3$, which can be used to find some new examples, i.e., rank $2$ bundles for $(P3, \mathcal{O}(2))$ and rank $3$ bundles for $(P2, \mathcal{O}(3))$.