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...
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2021-01-01
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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 |
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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. |
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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 |
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