On the k-normality of some projective manifolds

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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 Δ(X,L) ≤ 5 where Δ(X,L) = X + °X -h⁰(L). As a by-product of the proof of the above result the projective normality of a class of surfaces of degree nine in 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 =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.