On the k-normality of some projective manifolds

A long standing conjecture, known to us as the Eisenbud Goto conjecture, states that an n-dimensional variety embedded with degree $d$ in the $N$- dimensional projective space is $(d-(N-n)+1)$-regular in the sense of Castelnuovo-Mumford. In this work the conjecture is proved for all smooth varieties $X$ embedded by the complete linear system associated with a very ample line bundle $L$ such that $\Delta (X,L) \le 5$ where $\Delta (X,L) = \dim{X} + \deg{X} -h^0(L).$ As a by-product of the proof of the above result the projective normality of a class of surfaces of degree nine in $\Pin{5}$ which was left as an open question in a previous work of the second author and S. Di Rocco alg-geom/9710009 is established. The projective normality of scrolls $X =\Proj{E}$ over a curve of genus 2 embedded by the complete linear system associated with the tautological line bundle assumed to be very ample is investigated. Building on the work of Homma and Purnaprajna and Gallego alg-geom/9511013, criteria for the projective normality of three-dimensional quadric bundles over elliptic curves are given, improving some results due to D. Butler.