A general projective hypersurface \(X\subset\mathbb P^4\) has Picard group equal to \(\mathbb Z\), so that all surfaces contained in \(X\) are complete intersections.
Much less is known on curves contained in a general hypersurface \(X\). There are examples of such curves which are even arithmetically Cohen-Macaulay, but not complete intersections. So it is natural to ask about conditions forcing a curve in a general hypersurface to be a complete intersection. An example of such a criterion was found by G. Gherardelli for curves \(C\) in \(\mathbb P^3\): \(C\) is a complete intersection if and only if it is arithmetically Gorenstein. In the theory of vector bundles, Gherardelli’s theorem is equivalent to the celebrated Horrocks’ slitting criterion, for rank \(2\) vector bundles \(E\) on \(\mathbb P^3\): \(E\) splits if and only if \(H^1(E(n))=0\) for all \(n\). In other terms, splitting rank \(2\) bundles are those with no intermediate cohomology.
Gherardelli’s and Horrocks’ criteria fail on general hypersurfaces of low degree in \(\mathbb P^4\), as well as on particular hypersurfaces of any degree. However, there was evidence in the literature that as soon as the degree of \(X\) is high and \(X\) is general enough, the two criteria could hold.
This is the main achievement in the paper under review. The authors use a careful examination of deformations of rank \(2\) bundles and curves in families of hypersurfaces, and prove that indeed on a general hypersurface \(X\) of any degree \(d\geq 6\), a curve \(C\) is a complete intersection if and only if it is arithmetically Gorenstein. Equivalently, a rank \(2\) bundle splits if and only if its intermediate cohomology vanishes.