Conjectures on Cohomology Vanishing of an Ideal Sheaf of a Projective Variety and its Defining Equations

Abstract

The main goal of this survey note is to introduce some interesting and long standing conjectures on cohomology vanishing of an ideal sheaf of a projective algebraic variety and its dening equations to mathematicians with modest background on algebraic geometry. In particular, these conjectures involve the notion of higher order normality, which is useful to distinguish between general and special behaviors of such varieties and their equations. To achieve algebraic understandings and geometric interpretations of these conjectures, it is suggested to undertake a thorough investigation of how the geometry of projective space is reected in the geometry of its algebraic subvarieties, particularly, those of small codimension and to understand vector bundle technics involved in those conjectures. The tools to be used include multisecant lines, higher secant varieties, generic projections, nite schemes, free resolution of the sheaf of ideals of a given variety, vector bundles over a projective space dened over complex numbers.