A classification theorem is given of projective threefolds that are covered by the lines of a two-dimensional family, but not by a higher dimensional family. Precisely, if X is such a threefold, let Σ denote the Fano scheme of lines on X and µ the number of lines contained in X and passing through a general point of X. Assume that Σ is generically reduced. Then µ ≤ 6. Moreover, X is birationally a scroll over a surface (µ = 1), or X is a quadric bundle, or X belongs to a finite list of threefolds of degree at most 6. The smooth varieties of the third type are precisely the Fano threefolds with −K X = 2H X .

A classification theorem is given of smooth threefolds of P 5 covered by a family of dimension at least three of plane integral curves of degree d 2. It is shown that for such a threefold X there are two possibilities:(1) X is any threefold contained in a hyperquadric;(2) d 3 and X is either the Bordiga or the Palatini scroll.1.-Preliminaries and threefolds not of isolated type.Let X ⊂ P 5 be an integral projective variety of dimension 3, and degree d. We will always assume that X is non-degenerate, i.e. it is not contained in any hyperplane.We suppose that X contains an algebraic family F of dimension at least 3 of plane, integral curves. It is immediate to remark that, if the union of the curves of F does not cover X, then this union is a surface containing a 3-dimensional family of plane curves, hence a union of surfaces of P 3 . We will always exclude this situation, so from now on we assume that the curves of F cover X. Our aim is to classify such varieties X.If the curves of F are lines, then the answer is classical and is given by the following theorem.

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It is known that the smooth rational threefolds of P5 having a rational non-special surface of P4 as general hyperplane section have degree d = 3 , . . . , 7 . We study such threefolds X from the point of view of linear systems of surfaces in P 3 , looking in each case for an explicit description of a birational map from P3 t o X . For d = 3 , . . . 6 we prove that there exists a line L on X such that the projection map of X centered at L is birational; we completely describe the base loci B of the linear systems found in this way and give a description of any such threefold X as a suitable blowing-down of the blowing-up of P3 along B. If d = 7, i.e., if X is a Palatini scroll, we prove that, conversely, a similar projection never exists.

I-34100-Trieste (Italy). This paper deals with the following problem. Robbiano showed in [9] that standard bases, Grobner bases, Macaulay bases are all instances of the same general situation. In this paper, we develop this philosophy from the point of view of the Rees algebra R of a ring A w.r.t. a filtration F given on A. The ring R plays a fine job between A and the graded ring G associated to (A, F). The use of R and the properties of termorderings and their relate Grobner bases led naturally to the definition of Grobner filtrations in general commutative rings.