Demailly\u27s Conjecture and the Containment Problem

We investigate Demailly’s Conjecture for a general set of sufficiently many points. Demailly’s Conjecture generalizes Chudnovsky’s Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly’s bound, and prove that a general version of that containment holds for generic determinantal ideals and defining ideals of star configurations

Chudnovsky's Conjecture and the stable Harbourne-Huneke containment

In this paper, we investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky's Conjecture and the stable version of the Harbourne--Huneke containment conjectures for a general set of sufficiently many points.Comment: Comments welcome! In v2, the introduction has been rewritte

RandomPoints package for Macaulay2

We present {\tt RandomPoints}, a package in \emph{Macaulay2} designed mainly to identify rational and geometric points in a variety over a finite field. We provide tools to estimate the dimension of a variety. We also present methods to obtain non-vanishing minors of a given size in a given matrix, by evaluating the matrix at a point.Comment: 10 pages, comments welcome. Package by Sankhaneel Bisui, Zhan Jiang, Sarasij Maitra, Th\'ai Th\`anh Nguy\^en, Frank-Olaf Schreyer, Karl Schwede. The current version can be found here https://github.com/Macaulay2/Workshop-2020-Cleveland/blob/FastLinAlg/FastLinAlg/M2/RandomPoints.m