PROJECTIVE NORMALITY OF VARIETIES OF SMALL DEGREE ALBERTO ALZATI, MARINA BERTOLINI, AND GIAN MARIO BESANA June 5, 1997 Abstract. The projective normality of linearly normal smooth complex varieties of degree d ≤ 8 is investigated. The complete list of non projectively normal such manifolds is given; all of them are shown to be not 2-normal. Key Words: Algebraic Varieties - Projective Normality. 1. Introduction The classification of projective smooth complex varieties of small degree was accomplished by Ionescu [19], [20], Okonek [22], [23], [24], Alexander [3] and very recently completed by Abo, Decker and Sasakura [2]. Our interest is on the projective normality of these varieties. A projective variety X ⊂ P N is k-normal if τ k : H 0 (P N , O P N (k)) → H 0 (X, O X (k)) is surjective. X is linearly normal if it is 1-normal and it is projectively normal, (p.n.), if τ k is surjective for all k ≥ 1. It is known that all linearly normal varieties of degree d ≤ 5 are p.n., see [13], [21]. In his classification papers [19], [20] Ionescu already established the projective normality and more generally the arithmetical normality in some cases for 6 ≤ d ≤ 8. In this work the projective normality of varieties of degree 6 ≤ d ≤ 8 is thoroughly investigated. Andreatta [1], followed by a generalization by Ein and Lazarsfeld [10], posed the problem of classifying n dimensional varieties (X, L) polarized with a very ample line bundle L, such that the adjoint linear system |L| = |K X +(n - 1)L| gives an embedding which is not projectively normal. Andreatta and Ballico [4] gave examples of surfaces (S, L) with the above behavior, where deg S = 10 under the adjoint embedding. We conducted a detailed check of the non projectively normal varieties found in this work and none of them qualifies as an example for the above problem. The study of projective normality is conducted by combining Fujita’s work on polarized varieties with ad hoc techniques. A detailed analysis of the geometry of the embedding is necessary for the case of a surface of degree 8 in P 5 which is a scroll over a curve of genus g =2, see subsection 5.1. Among the cases left open in [19] and [20] there were scrolls over an elliptic curve. The projective normality of 2 dimensional such scrolls was established by Homma, [17], [18]. In a different context Butler [9] obtained results which essentially give the projective normality 1