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Central limit theorems in the geometry of numbers
The containment problem and a rational simplicial arrangement
Department of Mathematics, Pedagogical University of Cracow, Podchorążych 2, PL-30-084 Kraków, Poland |
Since Dumnicki, Szemberg, and Tutaj-Gasińska gave in 2013 in [
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. |

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