2013, 20: 51-54. doi: 10.3934/era.2013.20.51

New results on fat points schemes in $\mathbb{P}^2$

1. 

Jagiellonian University, Institute of Mathematics, Łojasiewicza 6, PL-30-348 Kraków, Poland, Poland

2. 

Instytut Matematyki UP, Podchorążych 2, PL-30-084 Kraków, Poland

Received  February 2013 Published  April 2013

The purpose of this note is to announce two results, Theorem A and Theorem B below, concerning geometric and algebraic properties of fat points in the complex projective plane. Their somewhat technical proofs are available in [10] and will be published elsewhere. Here we present only main ideas which are fairly transparent.
Citation: Marcin Dumnicki, Tomasz Szemberg, Halszka Tutaj-Gasińska. New results on fat points schemes in $\mathbb{P}^2$. Electronic Research Announcements, 2013, 20: 51-54. doi: 10.3934/era.2013.20.51
References:
[1]

C. Bocci and B. Harbourne, Comparing powers and symbolic powers of ideals, J. Algebraic Geometry, 19 (2010), 399-417. doi: 10.1090/S1056-3911-09-00530-X.  Google Scholar

[2]

C. Bocci and B. Harbourne, The resurgence of ideals of points and the containment problem, Proc. Amer. Math. Soc., 138 (2010), 1175-1190. doi: 10.1090/S0002-9939-09-10108-9.  Google Scholar

[3]

C. Ciliberto, Geometric aspects of polynomial interpolation in more variables and of Waring's problem, in "European Congress of Mathematics, Vol. I" (Barcelona, 2000), Progr. Math., 201, Birkhäuser, Basel, (2001), 289-316.  Google Scholar

[4]

C. Ciliberto and R. Miranda, Degenerations of planar linear systems, J. Reine Angew. Math., 501 (1998), 191-220.  Google Scholar

[5]

C. Ciliberto and R. Miranda, Linear systems of plane curves with base points of equal multiplicity, Trans. Amer. Math. Soc., 352 (2000), 4037-4050. doi: 10.1090/S0002-9947-00-02416-8.  Google Scholar

[6]

M. Dumnicki, Cutting diagram method for systems of plane curves with base points, Ann. Polon. Math., 90 (2007), 131-143. doi: 10.4064/ap90-2-3.  Google Scholar

[7]

M. Dumnicki, Symbolic powers of ideals of generic points in $\mathbbP^3$, J. Pure Appl. Alg., 216 (2012), 1410-1417. doi: 10.1016/j.jpaa.2011.12.010.  Google Scholar

[8]

M. Dumnicki and W. Jarnicki, New effective bounds on the dimension of a linear system in $\mathbbP^2$, J. Symb. Comp., 42 (2007), 621-635. doi: 10.1016/j.jsc.2007.01.004.  Google Scholar

[9]

M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, A counter-example to a question of Huneke and Harbourne, preprint, arXiv:1301.7440, (2013). Google Scholar

[10]

M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, A vanishing theorem and symbolic powers of planar point ideals, preprint, arXiv:1302.0871, (2013). Google Scholar

[11]

A. Gimigliano, Our thin knowledge of fat points, in "The Curves Seminar at Queen's, Vol. VI" (Kingston, ON, 1989), Exp. No. B, Queen's Papers in Pure and Appl. Math., 83, Queen's Univ., Kingston, ON, (1989), 50 pp.  Google Scholar

[12]

A. Gimigliano, B. Harbourne and M. Idá, Betti numbers for fat point ideals in the plane: A geometric approach, Trans. Amer. Math. Soc., 361 (2009), 1103-1127. doi: 10.1090/S0002-9947-08-04599-6.  Google Scholar

[13]

B. Harbourne and C. Huneke, Are symbolic powers highly evolved?,, to appear in J. Ramanujan Math. Soc., ().   Google Scholar

[14]

S. Yang, Linear systems in $\mathbbP^2$ with base points of bounded multiplicity, J. Algebraic Geom., 16 (2007), 19-38. doi: 10.1090/S1056-3911-06-00447-4.  Google Scholar

show all references

References:
[1]

C. Bocci and B. Harbourne, Comparing powers and symbolic powers of ideals, J. Algebraic Geometry, 19 (2010), 399-417. doi: 10.1090/S1056-3911-09-00530-X.  Google Scholar

[2]

C. Bocci and B. Harbourne, The resurgence of ideals of points and the containment problem, Proc. Amer. Math. Soc., 138 (2010), 1175-1190. doi: 10.1090/S0002-9939-09-10108-9.  Google Scholar

[3]

C. Ciliberto, Geometric aspects of polynomial interpolation in more variables and of Waring's problem, in "European Congress of Mathematics, Vol. I" (Barcelona, 2000), Progr. Math., 201, Birkhäuser, Basel, (2001), 289-316.  Google Scholar

[4]

C. Ciliberto and R. Miranda, Degenerations of planar linear systems, J. Reine Angew. Math., 501 (1998), 191-220.  Google Scholar

[5]

C. Ciliberto and R. Miranda, Linear systems of plane curves with base points of equal multiplicity, Trans. Amer. Math. Soc., 352 (2000), 4037-4050. doi: 10.1090/S0002-9947-00-02416-8.  Google Scholar

[6]

M. Dumnicki, Cutting diagram method for systems of plane curves with base points, Ann. Polon. Math., 90 (2007), 131-143. doi: 10.4064/ap90-2-3.  Google Scholar

[7]

M. Dumnicki, Symbolic powers of ideals of generic points in $\mathbbP^3$, J. Pure Appl. Alg., 216 (2012), 1410-1417. doi: 10.1016/j.jpaa.2011.12.010.  Google Scholar

[8]

M. Dumnicki and W. Jarnicki, New effective bounds on the dimension of a linear system in $\mathbbP^2$, J. Symb. Comp., 42 (2007), 621-635. doi: 10.1016/j.jsc.2007.01.004.  Google Scholar

[9]

M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, A counter-example to a question of Huneke and Harbourne, preprint, arXiv:1301.7440, (2013). Google Scholar

[10]

M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, A vanishing theorem and symbolic powers of planar point ideals, preprint, arXiv:1302.0871, (2013). Google Scholar

[11]

A. Gimigliano, Our thin knowledge of fat points, in "The Curves Seminar at Queen's, Vol. VI" (Kingston, ON, 1989), Exp. No. B, Queen's Papers in Pure and Appl. Math., 83, Queen's Univ., Kingston, ON, (1989), 50 pp.  Google Scholar

[12]

A. Gimigliano, B. Harbourne and M. Idá, Betti numbers for fat point ideals in the plane: A geometric approach, Trans. Amer. Math. Soc., 361 (2009), 1103-1127. doi: 10.1090/S0002-9947-08-04599-6.  Google Scholar

[13]

B. Harbourne and C. Huneke, Are symbolic powers highly evolved?,, to appear in J. Ramanujan Math. Soc., ().   Google Scholar

[14]

S. Yang, Linear systems in $\mathbbP^2$ with base points of bounded multiplicity, J. Algebraic Geom., 16 (2007), 19-38. doi: 10.1090/S1056-3911-06-00447-4.  Google Scholar

[1]

Marian Gidea, Yitzchak Shmalo. Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6123-6148. doi: 10.3934/dcds.2018264

[2]

Nicola Abatangelo, Sven Jarohs, Alberto Saldaña. Positive powers of the Laplacian: From hypersingular integrals to boundary value problems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 899-922. doi: 10.3934/cpaa.2018045

[3]

Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139

[4]

Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems & Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267

[5]

Tianxiao Wang. Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I. Mathematical Control & Related Fields, 2019, 9 (2) : 385-409. doi: 10.3934/mcrf.2019018

[6]

Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems & Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215

[7]

Jaime Arango, Adriana Gómez. Critical points of solutions to elliptic problems in planar domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 327-338. doi: 10.3934/cpaa.2011.10.327

[8]

Dmitry Kleinbock, Xi Zhao. An application of lattice points counting to shrinking target problems. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 155-168. doi: 10.3934/dcds.2018007

[9]

Monica Motta, Caterina Sartori. Asymptotic problems in optimal control with a vanishing Lagrangian and unbounded data. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4527-4552. doi: 10.3934/dcds.2015.35.4527

[10]

Leonardo Mora. Homoclinic bifurcations, fat attractors and invariant curves. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1133-1148. doi: 10.3934/dcds.2003.9.1133

[11]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2947-2969. doi: 10.3934/dcds.2020392

[12]

Jianping Wang, Mingxin Wang. Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4721-4736. doi: 10.3934/dcdsb.2020121

[13]

Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725

[14]

Jacek Serafin. A faithful symbolic extension. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1051-1062. doi: 10.3934/cpaa.2012.11.1051

[15]

Manli Song, Jinggang Tan. Hardy inequalities for the fractional powers of the Grushin operator. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4699-4726. doi: 10.3934/cpaa.2020192

[16]

Song Shao, Xiangdong Ye. Non-wandering sets of the powers of maps of a star. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1175-1184. doi: 10.3934/dcds.2003.9.1175

[17]

Ling Yun Wang, Wei Hua Gui, Kok Lay Teo, Ryan Loxton, Chun Hua Yang. Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications. Journal of Industrial & Management Optimization, 2009, 5 (4) : 705-718. doi: 10.3934/jimo.2009.5.705

[18]

Jiawei Chen, Shengjie Li, Jen-Chih Yao. Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. Journal of Industrial & Management Optimization, 2020, 16 (2) : 707-724. doi: 10.3934/jimo.2018174

[19]

Jim Wiseman. Symbolic dynamics from signed matrices. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 621-638. doi: 10.3934/dcds.2004.11.621

[20]

Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419

2019 Impact Factor: 0.5

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (0)

[Back to Top]