Rank two bundles on P^n with isolated cohomology

Abstract

The purpose of this paper is to study minimal monads associated to a rank two vector bundle $\mathcal{E}$ on ${\mathbb{P}}^n$. In particular, we study situations where $\mathcal{E}$ has $H_{\ast }^i(\mathcal{E})=0$ for $1<i<n-1$, except for one pair of values $(k,n-k)$. We show that on ${\mathbb{P}}^8,$ if $H_{\ast }^3(\mathcal{E})=H_{\ast }^4(\mathcal{E})=0$, then $\mathcal{E}$ must be decomposable. More generally, we show that for $n\ge 4k$, there is no indecomposable bundle $\mathcal{E}$ for which all intermediate cohomology modules except for $H_{\ast }^1,H_{\ast }^k,H_{\ast }^{n-k},H_{\ast }^{n-1}$ are zero.

arXiv comment14 pages, no figures