Fat points, partial intersections and Hamming distance

We use two main techniques, namely, residuation and separators of points, to show that the Hilbert function of a certain fat point set supported on a grid complete intersection is the same as the Hilbert function of a reduced set of points called a partial intersection. As an application, we answer a question of Tohǎneanu and Van Tuyl which relates the minimum Hamming distance of a special linear code and the minimum socle degree of the associated fat point set.

Line arrangements and configurations of points with an unexpected geometric property

We propose here a generalization of the problem addressed by the SHGH conjecture. The SHGH conjecture posits a solution to the question of how many conditions a general union$X$of fat points imposes on the complete linear system of curves in$\mathbb{P}^{2}$of fixed degree$d$, in terms of the occurrence of certain rational curves in the base locus of the linear subsystem defined by$X$. As a first step towards a new theory, we show that rational curves play a similar role in a special case of a generalized problem, which asks how many conditions are imposed by a general union of fat points on linear subsystems defined by imposed base points. Moreover, motivated by work of Di Gennaro, Ilardi and Vallès and of Faenzi and Vallès, we relate our results to the failure of a strong Lefschetz property, and we give a Lefschetz-like criterion for Terao’s conjecture on the freeness of line arrangements.

Homaloidal nets and ideals of fat points II: Subhomaloidal nets

This work is a natural sequel to a previous paper by the authors in that it tackles problems of the same nature. Here, one aims at the ideal theoretic and homological properties of an ideal of general plane fat points for which the corresponding second symbolic power has virtual multiplicities of a proper homaloidal type. For this purpose, one carries a detailed examination of the underlying linear system at the initial degree, where a good deal of the results depends on the method of the classical arithmetic quadratic transformations of Hudson–Nagata. A subsidiary guide to understand these ideals through their initial linear systems has been supplied by questions of birationality with source [Formula: see text] and target higher dimensional spaces. This leads, in particular, to the retrieval of birational maps studied by Geramita–Gimigliano–Pitteloud, including a few of the celebrated Bordiga–White parameterizations.