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. 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. 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, 201 (2001), 289.

[4]

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

[5]

C. Ciliberto and R. Miranda, Linear systems of plane curves with base points of equal multiplicity,, Trans. Amer. Math. Soc., 352 (2000), 4037. 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. 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. 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. 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, (2013).

[10]

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

[11]

A. Gimigliano, Our thin knowledge of fat points,, in, 83 (1989).

[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. 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. 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. 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. 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, 201 (2001), 289.

[4]

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

[5]

C. Ciliberto and R. Miranda, Linear systems of plane curves with base points of equal multiplicity,, Trans. Amer. Math. Soc., 352 (2000), 4037. 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. 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. 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. 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, (2013).

[10]

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

[11]

A. Gimigliano, Our thin knowledge of fat points,, in, 83 (1989).

[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. 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. doi: 10.1090/S1056-3911-06-00447-4.

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