It is well known by G. Horrocks’ theorem [Proc. Lond. Math. Soc., III. Ser. 14, 689–713 (1964; Zbl 0126.16801) and ibid. 14, 714–718 (1964; Zbl 0132.28103)] that a rank 2 vector bundle \(E\) on \(\mathbb{P}^r\) is arithmetically Cohen-Macaulay (ACM), i.e. \(h^i(E(n))= 0\), for all \(n\) and \(0< i< r\), if and only if it splits.
If the bundle is defined over a smooth threefold, Horrocks’ theorem does not hold any more, even on a smooth general surface of low degree in \(\mathbb{P}^4\).
Nevertheless the authors proved in a previous paper [Int. J. Math. 15, No. 4, 341–359 (2004; Zbl 1059.14054)] that Horrocks’ theorem holds on a general sextic surface in \(\mathbb{P}^4\). The present paper investigates further the splitting of ACM bundles, focusing on general hypersurfaces in \(\mathbb{P}^5\). The main result of the paper is in fact the following one.
Theorem. A rank 2 bundle \(E\) on a general hypersurface \(X_r\) in \(\mathbb{P}^5\) of degree \(r= 3,4,5,6\) splits if and only if \(h^i(E(n))= 0\), for all \(n\) and \(i= 1,2\).
Since rank 2 bundles give rise, via the Serre correspondence, to subcanonical surfaces, the following extension of the classical first theorem of G. Gherardelli for curves [Atti Accad. Italia, Rend., Cl. Sci. Fis. Mat. Nat., VII. Ser. 4, 128–132 (1943; Zbl 0061.35802)] follows directly.
Corollary. A surface \(S\) contained in a general hypersurface \(X\) in \(\mathbb{P}^5\) of degree \(r= 3,4,5,6\) is a complete intersection if and only if its canonical class is \({\mathcal O}_S(e)\), for some \(e\) (\(S\) is subcanonical) and \(h^i ({\mathcal I}_{S/S})= 0\) for all \(n\) and \(i= 1,2\) (\(S\) is arithmetically normal), \({\mathcal I}_{S/X}\) being the ideal sheaf of \(S\) in \(X\).