On the first infinitesimal neighborhood of a linear configuration of points in P^2

Given a reduced 0-dimensional subscheme X={P1,…,Ps} of P2, it is a well-studied but open question to determine the possible Hilbert functions of its first infinitesimal neighborhood (or double point scheme) Z defined by IZ=I2P1n?nI2Ps. If one fixes the Hilbert function of the support X of Z, the determination of the set D(h) of all possible Hilbert functions for Z seems to be a very difficult problem. The authors show that D(h) contains a unique minimal element hmin (with respect to the obvious partial ordering). Moreover, they construct a specific element of D(h) for every h. Similarly, the authors study the set Betti D(h) of all collections of graded Betti numbers of schemes Z whose support has Hilbert function h. They give an algorithm to describe a member of Betti D(h) for every h. To achieve these results, the authors use the type vector T=(d1,…,dr) of h and special sets of points, called linear configurations, derived from this type vector. These sets are characterized by having di points Xi on a line Li where no point of Xi is on Lj for i?j. Although it is not true that every double point scheme whose support is a linear configuration with Hilbert function h has the same Hilbert function, the authors show that it has the same regularity which is also the maximal possible. Moreover, the authors characterize the Hilbert functions h for which double point schemes supported on a linear configuration having Hilbert function h share the same graded Betti numbers. Further methods of proof are Horace's method [see J. E. Alexander and A. Hirschowitz, J. Algebraic Geom. 4 (1995), no. 2, 201--222; MR1311347 (96f:14065)] and the technique of basic double linkage.