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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[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.

[13]

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

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[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.

[13]

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

[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.

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