Ulrich Bundles on Projective Spaces

Abstract

We assume that $\mathcal{E}$ is a rank $r$ Ulrich bundle for $(P^n,\mathcal{O}(d))$. 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 $0\le j\le n$, and every Ulrich bundle $\mathcal{E}$ has a resolution in terms of $n$ of the trivial bundle over $P^n$. As a corollary, we can give a necessary and sufficient condition for Ulrich bundles if $n\le 3$, which can be used to find some new examples, i.e., rank $2$ bundles for $(P^3,\mathcal{O}(2))$ and rank $3$ bundles for $(P^2,\mathcal{O}(3))$.

11 pages, 1 figure

References in corpus (1)

• Ulrich bundles on surfaces with $q=p_g=0$