%0 Journal Article %T Postulation of a Union , , of a Given Zero-Dimensional Scheme and Several General Lines %A E. Ballico %J ISRN Geometry %D 2013 %R 10.1155/2013/391721 %X We study the Hilbert function of the union of a given zero-dimensional scheme , , and general lines of , proving, for example, that if , then has maximal rank. 1. Introduction A scheme is said to have maximal rank if for all integers , the restriction map is either injective or surjective, that is, if either or , that is, if imposes the ¡°expected¡± number of conditions to the vector space of the homogeneous degree polynomials in variables. Hartshorne and Hirschowitz proved that for all and , a general union of general lines has maximal rank [1]. Carlini et al. considered several cases in which we allow unions of linear spaces with certain multiplicities (for zero-dimensional subspaces these are general unions of fat points) [2¨C4]. Let denote the set of all unions of disjoint lines. In this paper (following [4]) we consider the case in which is a disjoint union of a given zero-dimensional scheme and a general . Fix a zero-dimensional scheme , , and integers and . Set . In this note we study the Hilbert function of a general union of and disjoint lines. We assume almost nothing on . We only prescribe the integer and an integer . Take a general . Since is fixed, while is general, we have , and hence for all . To study the Hilbert function of a general union of and disjoint lines, we need the following integers and , , defined by the following relations: The scheme has maximal rank if and only if for all integers with and for the minimal integer such that either or and . We first prove the following result, which says that if we may handle general unions of and general lines with respect to homogeneous polynomials in two consecutive degrees (not small with respect to the integer ), then these information propagates to higher degree polynomials. Theorem 1. Fix a zero-dimensional scheme , , and an integer . Set . Assume . If , assume . Let be any nonnegative integer such that for a general . Let be any nonnegative integer such that for a general . Set . Fix an integer , and let be a general element of .(a)For all integers , either or .(b)Assume . Then for all integers , either or . Then we go to a case in which we only assume something about the integer and prove the following result. Theorem 2. Fix positive integers , , and£¿£¿ such that and a zero-dimensional scheme such that and . Set . If either or and , then set . If and , then set . Fix any integer . Let be a general union of and lines. Then has maximal rank. If , then we have the following result. Proposition 3. Fix integers and . Let be a zero-dimensional scheme such that and . Then for each integer , a %U http://www.hindawi.com/journals/isrn.geometry/2013/391721/