Projective normality and normal presentation for adjoint linear series on minimal varieties

Abstract

In this thesis, we study projective normality and normal presentation of adjoint linear series associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorems, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. Next we concentrate on varieties with trivial canonical bundle. In the first part, we prove an effective projective normality result for an ample line bundle on regular smooth four-folds with trivial canonical bundle. In the second part, we emphasize on the projective normality of multiples of ample and globally generated line bundles on certain classes of known examples (up to deformation) of projective hyperk\"ahler varieties. As a corollary we show that excepting two extremal cases in dimensions $4$ and $6$, a general curve section of any ample and globally generated linear system on the above mentioned examples is non-hyperelliptic. This thesis is based on the articles \cite{MR19} and \cite{MR20} both of which are joint works with Jayan Mukherjee.