# American Institute of Mathematical Sciences

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 Interactions of Classical and Numerical Algebraic Geometry, Proceedings of a conference in honor of A. J. Sommese, held at Notre Dame, May 22-24 2008 (eds. D. J. Bates, G.-M. Besana, S. Di Rocco and C. W. Wampler), Contemporary Mathematics, 496, American Mathematical Society, Providence, RI, 2009, 362pp. doi: 10.1090/conm/496. [2] C. Bocci, S. Cooper and B. Harbourne, Containment results for ideals of various configurations of points in $\mathbbP^N$, J. Pure Appl. Algebra, 218 (2014), 65-75. doi: 10.1016/j.jpaa.2013.04.012. [3] 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. [4] 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. [5] M. Dumnicki, B. Harbourne, T. Szemberg and H. Tutaj-Gasińska, Linear subspaces, symbolic powers and Nagata type conjectures, Adv. Math., 252 (2014), 471-491. doi: 10.1016/j.aim.2013.10.029. [6] M. Dumnicki, T. Szemberg, J. Szpond and H. Tutaj-Gasińska, Symbolic generic initial systems of star configurations, J. Pure Appl. Algebra, 219 (2015), 1073–-1081. doi: 10.1016/j.jpaa.2014.05.035. [7] M. Dumnicki, J. Szpond and H. Tutaj-Gasińska, Asymptotic Hilbert polynomials and limiting shapes, J. Pure Appl. Algebra, 219 (2015), 4446-4457. doi: 10.1016/j.jpaa.2015.02.026. [8] L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye and M. Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble), 56 (2006), 1701-1734. doi: 10.5802/aif.2225. [9] L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye and M. Popa, Restricted volumes and base loci of linear series, Amer. J. Math., 131 (2009), 607-651. doi: 10.1353/ajm.0.0054. [10] 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. [11] D. Eisenbud, Commutative Algebra. With a View Toward Algebraic Geometry, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-5350-1. [12] H. Esnault and E. Viehweg, Sur une minoration du degré d'hypersurfaces s'annulant en certains points, Math. Ann., 263 (1983), 75-86. doi: 10.1007/BF01457085. [13] A. Galligo, Á propos du théorème de préparation de Weierstrass, in Fonctions de Plusieurs Variables Complexes, Lecture Notes in Math., Vol. 409, Springer, Berlin, 1974, 543-579. [14] A. V. Geramita, B. Harbourne and J. Migliore, Star configurations in $\mathbbP^n$, J. Algebra, 376 (2013), 279-299. doi: 10.1016/j.jalgebra.2012.11.034. [15] M. L. Green, Generic initial ideals, in Six Lectures on Commutative Algebra, Progr. Math., 166, Birkhäuser, Basel, 1998, 119-186. [16] J. Herzog and H. Srinivasan, Bounds for multiplicities, Trans. Amer. Math. Soc., 350 (1998), 2879-2902. doi: 10.1090/S0002-9947-98-02096-0. [17] H. Hochster and C. Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math., 147 (2002), 349-369. doi: 10.1007/s002220100176. [18] S. Mayes, The asymptotic behaviour of symbolic generic initial systems of generic points, J. Pure Appl. Alg., 218 (2014), 381-390. doi: 10.1016/j.jpaa.2013.06.002. [19] S. Mayes, The limiting shape of the generic initial system of a complete intersection, Comm. Algebra, 42 (2014), 2299-2310. doi: 10.1080/00927872.2012.758271. [20] M. Mustaţă, On multiplicities of graded sequences of ideals, J. Algebra, 256 (2002), 229-249. doi: 10.1016/S0021-8693(02)00112-6. [21] S. Sullivant, Combinatorial symbolic powers, J. Algebra, 319 (2008), 115-142. doi: 10.1016/j.jalgebra.2007.09.024.

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##### 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 Interactions of Classical and Numerical Algebraic Geometry, Proceedings of a conference in honor of A. J. Sommese, held at Notre Dame, May 22-24 2008 (eds. D. J. Bates, G.-M. Besana, S. Di Rocco and C. W. Wampler), Contemporary Mathematics, 496, American Mathematical Society, Providence, RI, 2009, 362pp. doi: 10.1090/conm/496. [2] C. Bocci, S. Cooper and B. Harbourne, Containment results for ideals of various configurations of points in $\mathbbP^N$, J. Pure Appl. Algebra, 218 (2014), 65-75. doi: 10.1016/j.jpaa.2013.04.012. [3] 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. [4] 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. [5] M. Dumnicki, B. Harbourne, T. Szemberg and H. Tutaj-Gasińska, Linear subspaces, symbolic powers and Nagata type conjectures, Adv. Math., 252 (2014), 471-491. doi: 10.1016/j.aim.2013.10.029. [6] M. Dumnicki, T. Szemberg, J. Szpond and H. Tutaj-Gasińska, Symbolic generic initial systems of star configurations, J. Pure Appl. Algebra, 219 (2015), 1073–-1081. doi: 10.1016/j.jpaa.2014.05.035. [7] M. Dumnicki, J. Szpond and H. Tutaj-Gasińska, Asymptotic Hilbert polynomials and limiting shapes, J. Pure Appl. Algebra, 219 (2015), 4446-4457. doi: 10.1016/j.jpaa.2015.02.026. [8] L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye and M. Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble), 56 (2006), 1701-1734. doi: 10.5802/aif.2225. [9] L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye and M. Popa, Restricted volumes and base loci of linear series, Amer. J. Math., 131 (2009), 607-651. doi: 10.1353/ajm.0.0054. [10] 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. [11] D. Eisenbud, Commutative Algebra. With a View Toward Algebraic Geometry, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-5350-1. [12] H. Esnault and E. Viehweg, Sur une minoration du degré d'hypersurfaces s'annulant en certains points, Math. Ann., 263 (1983), 75-86. doi: 10.1007/BF01457085. [13] A. Galligo, Á propos du théorème de préparation de Weierstrass, in Fonctions de Plusieurs Variables Complexes, Lecture Notes in Math., Vol. 409, Springer, Berlin, 1974, 543-579. [14] A. V. Geramita, B. Harbourne and J. Migliore, Star configurations in $\mathbbP^n$, J. Algebra, 376 (2013), 279-299. doi: 10.1016/j.jalgebra.2012.11.034. [15] M. L. Green, Generic initial ideals, in Six Lectures on Commutative Algebra, Progr. Math., 166, Birkhäuser, Basel, 1998, 119-186. [16] J. Herzog and H. Srinivasan, Bounds for multiplicities, Trans. Amer. Math. Soc., 350 (1998), 2879-2902. doi: 10.1090/S0002-9947-98-02096-0. [17] H. Hochster and C. Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math., 147 (2002), 349-369. doi: 10.1007/s002220100176. [18] S. Mayes, The asymptotic behaviour of symbolic generic initial systems of generic points, J. Pure Appl. Alg., 218 (2014), 381-390. doi: 10.1016/j.jpaa.2013.06.002. [19] S. Mayes, The limiting shape of the generic initial system of a complete intersection, Comm. Algebra, 42 (2014), 2299-2310. doi: 10.1080/00927872.2012.758271. [20] M. Mustaţă, On multiplicities of graded sequences of ideals, J. Algebra, 256 (2002), 229-249. doi: 10.1016/S0021-8693(02)00112-6. [21] S. Sullivant, Combinatorial symbolic powers, J. Algebra, 319 (2008), 115-142. doi: 10.1016/j.jalgebra.2007.09.024.
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