Steiner Configurations Ideals: Containment and Colouring

Given a homogeneous ideal <inline-formula><math display="inline"><semantics><mrow><mi>I</mi><mo>⊆</mo><mi>k</mi><mo>[</mo><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><m...

Full description

Bibliographic Details
Main Authors: Edoardo Ballico, Giuseppe Favacchio, Elena Guardo, Lorenzo Milazzo, Abu Chackalamannil Thomas
Format: Article
Language:English
Published: MDPI AG 2021-01-01
Series:Mathematics
Subjects:
Online Access:https://www.mdpi.com/2227-7390/9/3/210
_version_ 1797409034170531840
author Edoardo Ballico
Giuseppe Favacchio
Elena Guardo
Lorenzo Milazzo
Abu Chackalamannil Thomas
author_facet Edoardo Ballico
Giuseppe Favacchio
Elena Guardo
Lorenzo Milazzo
Abu Chackalamannil Thomas
author_sort Edoardo Ballico
collection DOAJ
description Given a homogeneous ideal <inline-formula><math display="inline"><semantics><mrow><mi>I</mi><mo>⊆</mo><mi>k</mi><mo>[</mo><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub><mo>]</mo></mrow></semantics></math></inline-formula>, the Containment problem studies the relation between symbolic and regular powers of <i>I</i>, that is, it asks for which pairs <inline-formula><math display="inline"><semantics><mrow><mi>m</mi><mo>,</mo><mi>r</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><msup><mi>I</mi><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msup><mo>⊆</mo><msup><mi>I</mi><mi>r</mi></msup></mrow></semantics></math></inline-formula> holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne–Huneke Conjecture, and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in <inline-formula><math display="inline"><semantics><msubsup><mi mathvariant="double-struck">P</mi><mi>k</mi><mi>n</mi></msubsup></semantics></math></inline-formula>. We can also show that the ideal of a Complement of a Steiner Configuration of points has expected resurgence, that is, its resurgence is strictly less than its big height, and it also satisfies Chudnovsky and Demailly’s Conjectures. Moreover, given a hypergraph <i>H</i>, we also study the relation between its colourability and the failure of the containment problem for the cover ideal associated to <i>H</i>. We apply these results in the case that <i>H</i> is a Steiner System.
first_indexed 2024-03-09T04:08:27Z
format Article
id doaj.art-26661a01b26a4eed98fca63fcc79d762
institution Directory Open Access Journal
issn 2227-7390
language English
last_indexed 2024-03-09T04:08:27Z
publishDate 2021-01-01
publisher MDPI AG
record_format Article
series Mathematics
spelling doaj.art-26661a01b26a4eed98fca63fcc79d7622023-12-03T14:03:32ZengMDPI AGMathematics2227-73902021-01-019321010.3390/math9030210Steiner Configurations Ideals: Containment and ColouringEdoardo Ballico0Giuseppe Favacchio1Elena Guardo2Lorenzo Milazzo3Abu Chackalamannil Thomas4Dipartimento di Matematica, via Sommarive, 14, 38123 Povo, ItalyDISMA-Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, ItalyDipartimento di Matematica e Informatica, Viale A. Doria, 6, 95100 Catania, ItalyDipartimento di Matematica e Informatica, Viale A. Doria, 6, 95100 Catania, ItalyDepartment of Mathematics, Tulane University, New Orleans, LA 70118, USAGiven a homogeneous ideal <inline-formula><math display="inline"><semantics><mrow><mi>I</mi><mo>⊆</mo><mi>k</mi><mo>[</mo><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub><mo>]</mo></mrow></semantics></math></inline-formula>, the Containment problem studies the relation between symbolic and regular powers of <i>I</i>, that is, it asks for which pairs <inline-formula><math display="inline"><semantics><mrow><mi>m</mi><mo>,</mo><mi>r</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><msup><mi>I</mi><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msup><mo>⊆</mo><msup><mi>I</mi><mi>r</mi></msup></mrow></semantics></math></inline-formula> holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne–Huneke Conjecture, and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in <inline-formula><math display="inline"><semantics><msubsup><mi mathvariant="double-struck">P</mi><mi>k</mi><mi>n</mi></msubsup></semantics></math></inline-formula>. We can also show that the ideal of a Complement of a Steiner Configuration of points has expected resurgence, that is, its resurgence is strictly less than its big height, and it also satisfies Chudnovsky and Demailly’s Conjectures. Moreover, given a hypergraph <i>H</i>, we also study the relation between its colourability and the failure of the containment problem for the cover ideal associated to <i>H</i>. We apply these results in the case that <i>H</i> is a Steiner System.https://www.mdpi.com/2227-7390/9/3/210monomial idealsideals of pointssymbolic powers of idealsWaldschmidt constantSteiner systems
spellingShingle Edoardo Ballico
Giuseppe Favacchio
Elena Guardo
Lorenzo Milazzo
Abu Chackalamannil Thomas
Steiner Configurations Ideals: Containment and Colouring
Mathematics
monomial ideals
ideals of points
symbolic powers of ideals
Waldschmidt constant
Steiner systems
title Steiner Configurations Ideals: Containment and Colouring
title_full Steiner Configurations Ideals: Containment and Colouring
title_fullStr Steiner Configurations Ideals: Containment and Colouring
title_full_unstemmed Steiner Configurations Ideals: Containment and Colouring
title_short Steiner Configurations Ideals: Containment and Colouring
title_sort steiner configurations ideals containment and colouring
topic monomial ideals
ideals of points
symbolic powers of ideals
Waldschmidt constant
Steiner systems
url https://www.mdpi.com/2227-7390/9/3/210
work_keys_str_mv AT edoardoballico steinerconfigurationsidealscontainmentandcolouring
AT giuseppefavacchio steinerconfigurationsidealscontainmentandcolouring
AT elenaguardo steinerconfigurationsidealscontainmentandcolouring
AT lorenzomilazzo steinerconfigurationsidealscontainmentandcolouring
AT abuchackalamannilthomas steinerconfigurationsidealscontainmentandcolouring