关于包含问题与零散单纯形线构形

On the containment hierarchy for simplicial ideals

The purpose of this note is to study containment relations and asymptotic invariants for ideals of fixed codimension skeletons (simplicial ideals) determined by arrangements of $n + 1$ general hyperplanes in the $n-$dimensional projective space over an arbitrary field.

New phenomena in the containment problem for simplicial arrangements

In this note we consider two simplicial arrangements of lines and ideals $I$ of intersection points of these lines. There are $127$ intersection points in both cases and the numbers $t_i$ of points lying on exactly $i$ configuration lines (points of multiplicity $i$) coincide. We show that in one of these examples the containment $I^{(3)} \subseteq I^2$ holds, whereas it fails in the other. We also show that the containment fails for a subarrrangement of $21$ lines. The interest in the containment relation between $I^{(3)}$ and $I^2$ for ideals of points in $¶^2$ is motivated by a question pos…

The containment problem and a rational simplicial arrangement

Since Dumnicki, Szemberg and Tutaj-Gasińska gave in 2013 in [9] the first example of a set of points in the complex projective plane such that for its homogeneous ideal I the containment of the third symbolic power in the second ordinary power fails, there has been considerable interest in searching for further examples with this property and investigating into the nature of such examples. Many examples, defined over various fields, have been found but so far there has been essentially just one example found of 19 points defined over the rationals, see [16, Theorem A, Problem 1]. In [12, Probl…

关于加一生成线排列的组合学

On combinatorics of plus-one generated line arrangements

本文研究加一生成线排列的组合约束，构造解码多项式，利用Klein和Wiman反射排列给出新例子，并发现27条线内9个最小加一生成的零星单纯排列。