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  2017, 24: 123-128. doi: 10.3934/era.2017.24.013

The containment problem and a rational simplicial arrangement

Justyna Szpond and  Grzegorz Malara

Department of Mathematics, Pedagogical University of Cracow, Podchorążych 2, PL-30-084 Kraków, Poland

Received  August 22, 2017 Revised  October 17, 2017 Published  March 2018

Fund Project: Research of Malara was partially supported by National Science Centre, Poland, grant 2016/21/N/ST1/01491. Research of Szpond was partially supported by National Science Centre, Poland, grant 2014/15/B/ST1/02197.

Since Dumnicki, Szemberg, and Tutaj-Gasińska gave in 2013 in [11] 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 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 [18, Theorem A, Problem 1]. In [14, Problem 5.1] the authors asked if there are other rational examples. This has motivated our research. The purpose of this note is to flag the existence of a new example of a set of 49 rational points with the same non-containment property for powers of its homogeneous ideal. Here we establish the existence and justify it computationally. A more conceptual proof, based on Seceleanu's criterion [22] will be published elsewhere [19].

Keywords: containment problem, line arrangements, simplicial arrangements, symbolic powers.
Mathematics Subject Classification: Primary: 13F20. Secondary: 14N20, 13A15, 52C35.
Citation: Justyna Szpond, Grzegorz Malara. The containment problem and a rational simplicial arrangement. Electronic Research Announcements, 2017, 24: 123-128. doi: 10.3934/era.2017.24.013
References:
[1]

M. Artebani and I. Dolgachev, The Hesse pencil of plane cubic curves, L'Enseignement Mathématique. Revue Internationale. 2e Série, 55 (2009), 235-273.  doi: 10.4171/LEM/55-3-3.

[2]

Th. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Lundman, P. Pokora and T. Szemberg, Bounded negativity and arrangements of lines, Int. Math. Res. Not., 19 (2015), 9456-9471. 

[3]

Th. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Seceleanu and T. Szemberg, Negative curves on symmetric blowups of the projective plane, resurgences and Waldschmidt constants, to appaear in Int. Math. Res. Not.

[4]

T. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A. L. Knutsen, W. Syzdek and T. Szemberg, A primer on Seshadri constants, in Interactions of Classical and Numerical Algebraic Geometry, Contemporary Mathematics, 496, Amer. Math. Soc., Providence, RI, 2009, 33-70.

[5]

T. Bauer, B. Harbourne, A. L. Knutsen, A. Küronya, S. Müller-Stach, X. Roulleau and T. Szemberg, Negative curves on algebraic surfaces, Duke Math. J., 162 (2013), 1877-1894.  doi: 10.1215/00127094-2335368.

[6]

M. Cuntz, Simplicial arrangements with up to 27 lines, Discrete Comput Geom, 48 (2012), 682-701.  doi: 10.1007/s00454-012-9423-7.

[7]

A. Czapliński, A. Główka, G. Malara, M. Lampa-Baczynska, P. Łuszcz-Świdecka, P. Pokora and J. Szpond, A counterexample to the containment $I^{(3)}\subset I^2$ over the reals, Adv. Geom., 16 (2016), 77-82. 

[8]

W. Decker, G. -M. Greuel, G. Pfister and H. Schönemann, Singular 4-0-2—A computer algebra system for polynomial computations, http://www.singular.uni-kl.de, (2015).

[9]

P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math., 17 (1972), 273-302.  doi: 10.1007/BF01406236.

[10]

M. Dumnicki, B. Harbourne, U. Nagel, A. Seceleanu, T. Szemberg and H. Tutaj-Gasińska, Resurgences for ideals of special point configurations in $\mathbb{P}^N$ coming from hyperplane arrangements, J. Algebra, 443 (2015), 383-394.  doi: 10.1016/j.jalgebra.2015.07.022.

[11]

M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Counterexamples to the $I^{(3)} \subset I^2$ containment, J. Algebra, 393 (2013), 24-29. 

[12]

L. Ein, R. Lazarsfeld and K. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math., 144 (2001), 241-252.  doi: 10.1007/s002220100121.

[13]

D. Eisenbud, Commutative Algebra. With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995.

[14]

Ƚ. Farnik, J. Kabat, M. Lampa-Baczyńska and H. Tutaj-Gasińska, On the parameter space of Böröczky configurations, arXiv: 1706.09053.

[15]

B. Grünbaum, A catalogue of simplicial arrangements in the real projective plane, Ars Math. Contemp., 2 (2009), 1-25. 

[16]

B. Harbourne and A. Seceleanu, Containment counterexamples for ideals of various configurations of points in $\mathbb{P}^N$, J. Pure Appl. Algebra, 219 (2015), 1062-1072.  doi: 10.1016/j.jpaa.2014.05.034.

[17]

M. Hochster and C. Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math., 147 (2002), 349-369.  doi: 10.1007/s002220100176.

[18]

M. Lampa-Baczyńska and J. Szpond, From Pappus Theorem to parameter spaces of some extremal line point configurations and applications, Geom. Dedicata, 188 (2017), 103-121.  doi: 10.1007/s10711-016-0207-8.

[19]

G. Malara and J. Szpond, Weyl groupoids, simplicial arrangements and the containment problem, preprint, 2017.

[20]

E. Melchior, Über Vielseite der projektiven Ebene, Deutsche Math., 5 (1941), 461-475. 

[21]

U. Nagel and A. Seceleanu, Ordinary and symbolic Rees algebras for ideals of Fermat point configurations, J. Algebra, 468 (2016), 80-102.  doi: 10.1016/j.jalgebra.2016.08.011.

[22]

A. Seceleanu, A homological criterion for the containment between symbolic and ordinary powers of some ideals of points in $\mathbb{P}^2$, J. Pure Appl. Alg., 219 (2015), 4857-4871.  doi: 10.1016/j.jpaa.2015.03.009.

[23]

T. Szemberg and J. Szpond, On the containment problem, Rend. Circ. Mat. Palermo, Ⅱ. Ser, 66 (2017), 233-245.  doi: 10.1007/s12215-016-0281-7.

show all references

References:
[1]

M. Artebani and I. Dolgachev, The Hesse pencil of plane cubic curves, L'Enseignement Mathématique. Revue Internationale. 2e Série, 55 (2009), 235-273.  doi: 10.4171/LEM/55-3-3.

[2]

Th. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Lundman, P. Pokora and T. Szemberg, Bounded negativity and arrangements of lines, Int. Math. Res. Not., 19 (2015), 9456-9471. 

[3]

Th. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Seceleanu and T. Szemberg, Negative curves on symmetric blowups of the projective plane, resurgences and Waldschmidt constants, to appaear in Int. Math. Res. Not.

[4]

T. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A. L. Knutsen, W. Syzdek and T. Szemberg, A primer on Seshadri constants, in Interactions of Classical and Numerical Algebraic Geometry, Contemporary Mathematics, 496, Amer. Math. Soc., Providence, RI, 2009, 33-70.

[5]

T. Bauer, B. Harbourne, A. L. Knutsen, A. Küronya, S. Müller-Stach, X. Roulleau and T. Szemberg, Negative curves on algebraic surfaces, Duke Math. J., 162 (2013), 1877-1894.  doi: 10.1215/00127094-2335368.

[6]

M. Cuntz, Simplicial arrangements with up to 27 lines, Discrete Comput Geom, 48 (2012), 682-701.  doi: 10.1007/s00454-012-9423-7.

[7]

A. Czapliński, A. Główka, G. Malara, M. Lampa-Baczynska, P. Łuszcz-Świdecka, P. Pokora and J. Szpond, A counterexample to the containment $I^{(3)}\subset I^2$ over the reals, Adv. Geom., 16 (2016), 77-82. 

[8]

W. Decker, G. -M. Greuel, G. Pfister and H. Schönemann, Singular 4-0-2—A computer algebra system for polynomial computations, http://www.singular.uni-kl.de, (2015).

[9]

P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math., 17 (1972), 273-302.  doi: 10.1007/BF01406236.

[10]

M. Dumnicki, B. Harbourne, U. Nagel, A. Seceleanu, T. Szemberg and H. Tutaj-Gasińska, Resurgences for ideals of special point configurations in $\mathbb{P}^N$ coming from hyperplane arrangements, J. Algebra, 443 (2015), 383-394.  doi: 10.1016/j.jalgebra.2015.07.022.

[11]

M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Counterexamples to the $I^{(3)} \subset I^2$ containment, J. Algebra, 393 (2013), 24-29. 

[12]

L. Ein, R. Lazarsfeld and K. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math., 144 (2001), 241-252.  doi: 10.1007/s002220100121.

[13]

D. Eisenbud, Commutative Algebra. With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995.

[14]

Ƚ. Farnik, J. Kabat, M. Lampa-Baczyńska and H. Tutaj-Gasińska, On the parameter space of Böröczky configurations, arXiv: 1706.09053.

[15]

B. Grünbaum, A catalogue of simplicial arrangements in the real projective plane, Ars Math. Contemp., 2 (2009), 1-25. 

[16]

B. Harbourne and A. Seceleanu, Containment counterexamples for ideals of various configurations of points in $\mathbb{P}^N$, J. Pure Appl. Algebra, 219 (2015), 1062-1072.  doi: 10.1016/j.jpaa.2014.05.034.

[17]

M. Hochster and C. Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math., 147 (2002), 349-369.  doi: 10.1007/s002220100176.

[18]

M. Lampa-Baczyńska and J. Szpond, From Pappus Theorem to parameter spaces of some extremal line point configurations and applications, Geom. Dedicata, 188 (2017), 103-121.  doi: 10.1007/s10711-016-0207-8.

[19]

G. Malara and J. Szpond, Weyl groupoids, simplicial arrangements and the containment problem, preprint, 2017.

[20]

E. Melchior, Über Vielseite der projektiven Ebene, Deutsche Math., 5 (1941), 461-475. 

[21]

U. Nagel and A. Seceleanu, Ordinary and symbolic Rees algebras for ideals of Fermat point configurations, J. Algebra, 468 (2016), 80-102.  doi: 10.1016/j.jalgebra.2016.08.011.

[22]

A. Seceleanu, A homological criterion for the containment between symbolic and ordinary powers of some ideals of points in $\mathbb{P}^2$, J. Pure Appl. Alg., 219 (2015), 4857-4871.  doi: 10.1016/j.jpaa.2015.03.009.

[23]

T. Szemberg and J. Szpond, On the containment problem, Rend. Circ. Mat. Palermo, Ⅱ. Ser, 66 (2017), 233-245.  doi: 10.1007/s12215-016-0281-7.

Figure 1.  Affine part of the simplicial arrangement $A(25, 2)$
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