Resurgences for ideals of special point configurations in PN coming from hyperplane arrangements

M. Dumnicki; B. Harbourne; U. Nagel; A. Seceleanu; T. Szemberg; H. Tutaj-Gasińska

doi:10.1016/j.jalgebra.2015.07.022

Resurgences for ideals of special point configurations in P^N coming from hyperplane arrangements

M. Dumnicki, B. Harbourne, U. Nagel, A. Seceleanu, T. Szemberg, H. Tutaj-Gasińska

Mathematics

Research output: Contribution to journal › Article › peer-review

41 Scopus citations

Abstract

Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers; see for example [3,7,13,16,18-20] to cite just a few. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence [3,15].There have been exciting new developments in this area recently. It had been expected for several years that I^(Nr-N+1)⊆I^r should hold for the ideal I of any finite set of points in P^N for all r > 0, but in the last year various counterexamples have now been constructed (see [11,17,8]), all involving point sets coming from hyperplane arrangements. In the present work, we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal.

Original language	English
Pages (from-to)	383-394
Number of pages	12
Journal	Journal of Algebra
Volume	443
DOIs	https://doi.org/10.1016/j.jalgebra.2015.07.022
State	Published - Dec 1 2015

Bibliographical note

Publisher Copyright:
© 2015 Elsevier Inc.

Funding

The second author's work on this project was sponsored by the National Security Agency under Grant/Cooperative agreement “Topics in Algebra and Geometry, Curves and Containments” Number H98230-13-1-0213 . The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notice. The work of the third author was partially supported by the National Security Agency under Grant Number H98230-12-1-0247 . The first, fifth and sixth author's research was partially supported by the National Science Centre , Poland, grant 2014/15/B/ST1/02197 . We also thank the referee for helpful comments and corrections.

Funders	Funder number
Narodowym Centrum Nauki	2014/15/B/ST1/02197
National Security Agency	H98230-13-1-0213, H98230-12-1-0247

Keywords

Fat points
Homogeneous ideals
Polynomial rings
Projective space
Symbolic powers

ASJC Scopus subject areas

Algebra and Number Theory

Access to Document

10.1016/j.jalgebra.2015.07.022

Cite this

@article{41d02ebac18a4ed78369c459fbbce56c,

title = "Resurgences for ideals of special point configurations in PN coming from hyperplane arrangements",

abstract = "Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers; see for example [3,7,13,16,18-20] to cite just a few. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence [3,15].There have been exciting new developments in this area recently. It had been expected for several years that I(Nr-N+1)⊆Ir should hold for the ideal I of any finite set of points in PN for all r > 0, but in the last year various counterexamples have now been constructed (see [11,17,8]), all involving point sets coming from hyperplane arrangements. In the present work, we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal.",

keywords = "Fat points, Homogeneous ideals, Polynomial rings, Projective space, Symbolic powers",

author = "M. Dumnicki and B. Harbourne and U. Nagel and A. Seceleanu and T. Szemberg and H. Tutaj-Gasi{\'n}ska",

note = "Publisher Copyright: {\textcopyright} 2015 Elsevier Inc.",

year = "2015",

month = dec,

day = "1",

doi = "10.1016/j.jalgebra.2015.07.022",

language = "English",

volume = "443",

pages = "383--394",

}

TY - JOUR

T1 - Resurgences for ideals of special point configurations in PN coming from hyperplane arrangements

AU - Dumnicki, M.

AU - Harbourne, B.

AU - Nagel, U.

AU - Seceleanu, A.

AU - Szemberg, T.

AU - Tutaj-Gasińska, H.

PY - 2015/12/1

Y1 - 2015/12/1

N2 - Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers; see for example [3,7,13,16,18-20] to cite just a few. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence [3,15].There have been exciting new developments in this area recently. It had been expected for several years that I(Nr-N+1)⊆Ir should hold for the ideal I of any finite set of points in PN for all r > 0, but in the last year various counterexamples have now been constructed (see [11,17,8]), all involving point sets coming from hyperplane arrangements. In the present work, we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal.

AB - Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers; see for example [3,7,13,16,18-20] to cite just a few. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence [3,15].There have been exciting new developments in this area recently. It had been expected for several years that I(Nr-N+1)⊆Ir should hold for the ideal I of any finite set of points in PN for all r > 0, but in the last year various counterexamples have now been constructed (see [11,17,8]), all involving point sets coming from hyperplane arrangements. In the present work, we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal.

KW - Fat points

KW - Homogeneous ideals

KW - Polynomial rings

KW - Projective space

KW - Symbolic powers

UR - http://www.scopus.com/inward/record.url?scp=84939496724&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939496724&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2015.07.022

DO - 10.1016/j.jalgebra.2015.07.022

M3 - Article

AN - SCOPUS:84939496724

SN - 0021-8693

VL - 443

SP - 383

EP - 394

JO - Journal of Algebra

JF - Journal of Algebra

ER -