Steiner Configurations Ideals: Containment and Colouring

Edoardo Ballico, Giuseppe Favacchio, Elena Guardo, Lorenzo Milazzo and Abu Chackalamannil Thomas
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Edoardo Ballico: Dipartimento di Matematica, via Sommarive, 14, 38123 Povo, Italy
Giuseppe Favacchio: DISMA-Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy
Elena Guardo: Dipartimento di Matematica e Informatica, Viale A. Doria, 6, 95100 Catania, Italy
Lorenzo Milazzo: Dipartimento di Matematica e Informatica, Viale A. Doria, 6, 95100 Catania, Italy
Abu Chackalamannil Thomas: Department of Mathematics, Tulane University, New Orleans, LA 70118, USA

Mathematics, 2021, vol. 9, issue 3, 1-15

Abstract: Given a homogeneous ideal I ⊆ k [ x 0 , … , x n ] , the Containment problem studies the relation between symbolic and regular powers of I , that is, it asks for which pairs m , r ∈ N , I ( m ) ⊆ I r 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 P k n . 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 H , we also study the relation between its colourability and the failure of the containment problem for the cover ideal associated to H . We apply these results in the case that H is a Steiner System.

Keywords: monomial ideals; ideals of points; symbolic powers of ideals; Waldschmidt constant; Steiner systems (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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