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.

Resurgence and Waldschmidt constant of the ideal of a fat almost collinear subscheme in ℙ2

Let Zn = p0 + p1 + ··· + pn be a configuration of points in ℙ2, where all points pi except p0 lie on a line, and let I(Zn) be its corresponding homogeneous ideal in 𝕂 [ℙ2]. The resurgence and the Waldschmidt constant of I(Zn) in [5] have been computed. In this note, we compute these two invariants for the defining ideal of a fat point subscheme Zn,c = cp0 + p1 +··· + pn, i.e. the point p0 is considered with multiplicity c. Our strategy is similar to [5].

A Presentation of the Kähler Differential Module for a Fat Point Scheme in ℙn1 × … × ℙnk

Let 핐 be a fat point scheme in ℙn1 × … × ℙnk over a field K of characteristic zero. In this paper we introduce the multi-graded Kähler differential module for 핐 and we establish a short exact sequence of this module in terms of the thickening of 핐.