Buchsbaum bundles on \(\mathbb{P}^n\) (Q1582741)

A rank-two bundle on \(\mathbb{P}^n\) is said to be \(k\)-Buchsbaum if its first cohomology module is annihilated by the \(k\)-th power of the irrelevant ideal. Buchsbaum varieties of codimension two (in the sense of Chang) correspond to bundles which satisfy a stronger condition than 1-Buchsbaum (in fact, they have both all the intermediate cohomology modules and all the intermediate cohomology modules of all possible restrictions to linear subspaces 1-Buchsbaum). In this paper the authors show that there is no indecomposable rank two bundle on \(\mathbb{P}^n\) \((n>3)\) whose first cohomology module is 1-Buchsbaum (the case \(n=3\) -- the null-correlation bundle -- is well-known) and that there is no indecomposable rank two vector bundle on \(\mathbb{P}^n\) \((n>4)\) whose first cohomology module is 2-Buchsbaum [the case \(n=3\) was solved by \textit{Ph. Ellia} and \textit{A. Sarti}, in: ``Commuttaive algebra and algebraic geometry'', Proc. Ferrara Meeting Honor M. Fiorentini, Ferrara, Lect. Notes Pure Appl. Math. 206, 81-92 (1999; Zbl 0960.14026) and the case \(n=4\) is open].