On smooth surfaces in projective four-space lying on quartic hypersurfaces with isolated singularities

We prove that a smooth surface, non of general type, in projective four-space, which lies on a quartic hypersurface with isolated singularities has degree at most 27 (in fact we prove a slightly more general result)

On Buchsbaum bundles on quadric hypersurfaces

Let $E$ be an indecomposable rank two vector bundle on the projective space \PP^n, n \ge 3, over an algebraically closed field of characteristic zero. It is well known that $E$ is arithmetically Buchsbaum if and only if $n=3$ and $E$ is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q_n\subset\PP^{n+1}, $n\ge 3$. We give in fact a full classification and prove that $n$ must be at most 5. As to $k$-Buchsbaum rank two vector bundles on $Q_3$, $k\ge2$, we prove two boundedness results.Comment: 22 pages, no figur