January  2016, 23: 8-18. doi: 10.3934/era.2016.23.002

Asymptotic Hilbert polynomial and a bound for Waldschmidt constants

1. 

Jagiellonian University, Faculty of Mathematics and Computer Science, Łojasiewicza 6, PL-30-348 Kraków, Poland, Poland, Poland

Received  December 2015 Revised  April 2016 Published  June 2016

In the paper we give a method to compute an upper bound for the Waldschmidt constants of a wide class of ideals. This generalizes the result obtained by Dumnicki, Harbourne, Szemberg and Tutaj-Gasińska, Adv. Math. 2014, [5]. Our bound is given by a root of a suitable derivative of a certain polynomial associated with the asymptotic Hilbert polynomial.
Citation: Marcin Dumnicki, Łucja Farnik, Halszka Tutaj-Gasińska. Asymptotic Hilbert polynomial and a bound for Waldschmidt constants. Electronic Research Announcements, 2016, 23: 8-18. doi: 10.3934/era.2016.23.002
References:
[1]

Th. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A. L. Knutsen, W. Syzdek and T. Szemberg, A primer on Seshadri constants,, in \emph{Interactions of Classical and Numerical Algebraic Geometry, (2008), 22. doi: 10.1090/conm/496. Google Scholar

[2]

C. Bocci, S. Cooper and B. Harbourne, Containment results for ideals of various configurations of points in $\mathbbP^N$,, \emph{J. Pure Appl. Algebra}, 218 (2014), 65. doi: 10.1016/j.jpaa.2013.04.012. Google Scholar

[3]

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

[4]

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

[5]

M. Dumnicki, B. Harbourne, T. Szemberg and H. Tutaj-Gasińska, Linear subspaces, symbolic powers and Nagata type conjectures,, \emph{Adv. Math.}, 252 (2014), 471. doi: 10.1016/j.aim.2013.10.029. Google Scholar

[6]

M. Dumnicki, T. Szemberg, J. Szpond and H. Tutaj-Gasińska, Symbolic generic initial systems of star configurations,, \emph{J. Pure Appl. Algebra}, 219 (2015). doi: 10.1016/j.jpaa.2014.05.035. Google Scholar

[7]

M. Dumnicki, J. Szpond and H. Tutaj-Gasińska, Asymptotic Hilbert polynomials and limiting shapes,, \emph{J. Pure Appl. Algebra}, 219 (2015), 4446. doi: 10.1016/j.jpaa.2015.02.026. Google Scholar

[8]

L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye and M. Popa, Asymptotic invariants of base loci,, \emph{Ann. Inst. Fourier (Grenoble)}, 56 (2006), 1701. doi: 10.5802/aif.2225. Google Scholar

[9]

L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye and M. Popa, Restricted volumes and base loci of linear series,, \emph{Amer. J. Math.}, 131 (2009), 607. doi: 10.1353/ajm.0.0054. Google Scholar

[10]

L. Ein, R. Lazarsfeld and K. Smith, Uniform bounds and symbolic powers on smooth varieties,, \emph{Invent. Math.}, 144 (2001), 241. doi: 10.1007/s002220100121. Google Scholar

[11]

D. Eisenbud, Commutative Algebra. With a View Toward Algebraic Geometry,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-5350-1. Google Scholar

[12]

H. Esnault and E. Viehweg, Sur une minoration du degré d'hypersurfaces s'annulant en certains points,, \emph{Math. Ann.}, 263 (1983), 75. doi: 10.1007/BF01457085. Google Scholar

[13]

A. Galligo, Á propos du théorème de préparation de Weierstrass,, in \emph{Fonctions de Plusieurs Variables Complexes}, (1974), 543. Google Scholar

[14]

A. V. Geramita, B. Harbourne and J. Migliore, Star configurations in $\mathbbP^n$,, \emph{J. Algebra}, 376 (2013), 279. doi: 10.1016/j.jalgebra.2012.11.034. Google Scholar

[15]

M. L. Green, Generic initial ideals,, in \emph{Six Lectures on Commutative Algebra}, (1998), 119. Google Scholar

[16]

J. Herzog and H. Srinivasan, Bounds for multiplicities,, \emph{Trans. Amer. Math. Soc.}, 350 (1998), 2879. doi: 10.1090/S0002-9947-98-02096-0. Google Scholar

[17]

H. Hochster and C. Huneke, Comparison of symbolic and ordinary powers of ideals,, \emph{Invent. Math.}, 147 (2002), 349. doi: 10.1007/s002220100176. Google Scholar

[18]

S. Mayes, The asymptotic behaviour of symbolic generic initial systems of generic points,, \emph{J. Pure Appl. Alg.}, 218 (2014), 381. doi: 10.1016/j.jpaa.2013.06.002. Google Scholar

[19]

S. Mayes, The limiting shape of the generic initial system of a complete intersection,, \emph{Comm. Algebra}, 42 (2014), 2299. doi: 10.1080/00927872.2012.758271. Google Scholar

[20]

M. Mustaţă, On multiplicities of graded sequences of ideals,, \emph{J. Algebra}, 256 (2002), 229. doi: 10.1016/S0021-8693(02)00112-6. Google Scholar

[21]

S. Sullivant, Combinatorial symbolic powers,, \emph{J. Algebra}, 319 (2008), 115. doi: 10.1016/j.jalgebra.2007.09.024. Google Scholar

show all references

References:
[1]

Th. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A. L. Knutsen, W. Syzdek and T. Szemberg, A primer on Seshadri constants,, in \emph{Interactions of Classical and Numerical Algebraic Geometry, (2008), 22. doi: 10.1090/conm/496. Google Scholar

[2]

C. Bocci, S. Cooper and B. Harbourne, Containment results for ideals of various configurations of points in $\mathbbP^N$,, \emph{J. Pure Appl. Algebra}, 218 (2014), 65. doi: 10.1016/j.jpaa.2013.04.012. Google Scholar

[3]

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

[4]

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

[5]

M. Dumnicki, B. Harbourne, T. Szemberg and H. Tutaj-Gasińska, Linear subspaces, symbolic powers and Nagata type conjectures,, \emph{Adv. Math.}, 252 (2014), 471. doi: 10.1016/j.aim.2013.10.029. Google Scholar

[6]

M. Dumnicki, T. Szemberg, J. Szpond and H. Tutaj-Gasińska, Symbolic generic initial systems of star configurations,, \emph{J. Pure Appl. Algebra}, 219 (2015). doi: 10.1016/j.jpaa.2014.05.035. Google Scholar

[7]

M. Dumnicki, J. Szpond and H. Tutaj-Gasińska, Asymptotic Hilbert polynomials and limiting shapes,, \emph{J. Pure Appl. Algebra}, 219 (2015), 4446. doi: 10.1016/j.jpaa.2015.02.026. Google Scholar

[8]

L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye and M. Popa, Asymptotic invariants of base loci,, \emph{Ann. Inst. Fourier (Grenoble)}, 56 (2006), 1701. doi: 10.5802/aif.2225. Google Scholar

[9]

L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye and M. Popa, Restricted volumes and base loci of linear series,, \emph{Amer. J. Math.}, 131 (2009), 607. doi: 10.1353/ajm.0.0054. Google Scholar

[10]

L. Ein, R. Lazarsfeld and K. Smith, Uniform bounds and symbolic powers on smooth varieties,, \emph{Invent. Math.}, 144 (2001), 241. doi: 10.1007/s002220100121. Google Scholar

[11]

D. Eisenbud, Commutative Algebra. With a View Toward Algebraic Geometry,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-5350-1. Google Scholar

[12]

H. Esnault and E. Viehweg, Sur une minoration du degré d'hypersurfaces s'annulant en certains points,, \emph{Math. Ann.}, 263 (1983), 75. doi: 10.1007/BF01457085. Google Scholar

[13]

A. Galligo, Á propos du théorème de préparation de Weierstrass,, in \emph{Fonctions de Plusieurs Variables Complexes}, (1974), 543. Google Scholar

[14]

A. V. Geramita, B. Harbourne and J. Migliore, Star configurations in $\mathbbP^n$,, \emph{J. Algebra}, 376 (2013), 279. doi: 10.1016/j.jalgebra.2012.11.034. Google Scholar

[15]

M. L. Green, Generic initial ideals,, in \emph{Six Lectures on Commutative Algebra}, (1998), 119. Google Scholar

[16]

J. Herzog and H. Srinivasan, Bounds for multiplicities,, \emph{Trans. Amer. Math. Soc.}, 350 (1998), 2879. doi: 10.1090/S0002-9947-98-02096-0. Google Scholar

[17]

H. Hochster and C. Huneke, Comparison of symbolic and ordinary powers of ideals,, \emph{Invent. Math.}, 147 (2002), 349. doi: 10.1007/s002220100176. Google Scholar

[18]

S. Mayes, The asymptotic behaviour of symbolic generic initial systems of generic points,, \emph{J. Pure Appl. Alg.}, 218 (2014), 381. doi: 10.1016/j.jpaa.2013.06.002. Google Scholar

[19]

S. Mayes, The limiting shape of the generic initial system of a complete intersection,, \emph{Comm. Algebra}, 42 (2014), 2299. doi: 10.1080/00927872.2012.758271. Google Scholar

[20]

M. Mustaţă, On multiplicities of graded sequences of ideals,, \emph{J. Algebra}, 256 (2002), 229. doi: 10.1016/S0021-8693(02)00112-6. Google Scholar

[21]

S. Sullivant, Combinatorial symbolic powers,, \emph{J. Algebra}, 319 (2008), 115. doi: 10.1016/j.jalgebra.2007.09.024. Google Scholar

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