On the projective normality of Enriques surfaces

Abstract

One of the basic but often difficult tasks in algebraic geometry is to describe the equations of a given smooth projective variety X ⊂ IP in terms of its intrinsic and extrinsic geometry. In particular no general formula is known for the number of generators of the homogeneous ideal of X. Many authors from classical to nowadays, have therefore concentrated their attention on finding sufficient conditions forX to be projectively normal, that is such that the natural restriction maps H (OIPN (j)) → H (OX(j)) are surjective for every j ≥ 0, for then Riemann-Roch and (often) vanishing theorems answer the question. In the case of curves many results are known, starting with Castelnuovo’s [Ca] projective normality of linearly normal curves of genus g and degree at least 2g + 1 (with modern generalization by Mumford [Mu1]) and culminating with Green’s result [G], that if a linearly normal curve of genus g has degree at least 2g + 1+ p then it satisfies property Np [GL2], that is it is projectively normal, its homogeneous ideal is generated by quadrics, the relations among them are