A containment result in 𝑃ⁿ and the Chudnovsky Conjecture

Dumnicki, Marcin; Tutaj-Gasińska, Halszka

doi:10.1090/proc/13582

A containment result in $P^n$ and the Chudnovsky Conjecture
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by Marcin Dumnicki and Halszka Tutaj-Gasińska PDF

Proc. Amer. Math. Soc. 145 (2017), 3689-3694 Request permission

Abstract:

In this paper we prove the containment $I^{(nm)}\subset M^{(n-1)m}I^m$, for a radical ideal $I$ of $s$ general points in $\mathbb {P}^n$, where $s\geq 2^n$. As a corollary we get that the Chudnovsky Conjecture holds for a very general set of at least $2^n$ points in $\mathbb {P}^n$.

References

Alessandro Arsie and Jon Eivind Vatne, A note on symbolic and ordinary powers of homogeneous ideals, Ann. Univ. Ferrara Sez. VII (N.S.) 49 (2003), 19–30 (English, with English and Italian summaries). MR 2164987
Cristiano Bocci and Brian Harbourne, The resurgence of ideals of points and the containment problem, Proc. Amer. Math. Soc. 138 (2010), no. 4, 1175–1190. MR 2578512, DOI 10.1090/S0002-9939-09-10108-9
Karen A. Chandler, Regularity of the powers of an ideal, Comm. Algebra 25 (1997), no. 12, 3773–3776. MR 1481564, DOI 10.1080/00927879708826084
G. V. Chudnovsky, Singular points on complex hypersurfaces and multidimensional Schwarz lemma, Seminar on Number Theory, Paris 1979–80, Progr. Math., vol. 12, Birkhäuser, Boston, Mass., 1981, pp. 29–69. MR 633888
Harm Derksen and Jessica Sidman, A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements, Adv. Math. 172 (2002), no. 2, 151–157. MR 1942401, DOI 10.1016/S0001-8708(02)00019-1
Marcin Dumnicki, An algorithm to bound the regularity and nonemptiness of linear systems in $\Bbb P^n$, J. Symbolic Comput. 44 (2009), no. 10, 1448–1462. MR 2543429, DOI 10.1016/j.jsc.2009.04.005
Marcin Dumnicki, Symbolic powers of ideals of generic points in $\Bbb P^3$, J. Pure Appl. Algebra 216 (2012), no. 6, 1410–1417. MR 2890510, DOI 10.1016/j.jpaa.2011.12.010
Marcin Dumnicki, Containments of symbolic powers of ideals of generic points in $\Bbb P^3$, Proc. Amer. Math. Soc. 143 (2015), no. 2, 513–530. MR 3283641, DOI 10.1090/S0002-9939-2014-12273-8
Anthony V. Geramita, Alessandro Gimigliano, and Yves Pitteloud, Graded Betti numbers of some embedded rational $n$-folds, Math. Ann. 301 (1995), no. 2, 363–380. MR 1314592, DOI 10.1007/BF01446634
Lawrence Ein, Robert Lazarsfeld, and Karen E. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math. 144 (2001), no. 2, 241–252. MR 1826369, DOI 10.1007/s002220100121
Hélène Esnault and Eckart Viehweg, Sur une minoration du degré d’hypersurfaces s’annulant en certains points, Math. Ann. 263 (1983), no. 1, 75–86 (French). MR 697332, DOI 10.1007/BF01457085
Laurent Evain, On the postulation of $s^d$ fat points in $\Bbb P^d$, J. Algebra 285 (2005), no. 2, 516–530. MR 2125451, DOI 10.1016/j.jalgebra.2004.09.034
Brian Harbourne and Craig Huneke, Are symbolic powers highly evolved?, J. Ramanujan Math. Soc. 28A (2013), 247–266. MR 3115195
Melvin Hochster and Craig Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2002), no. 2, 349–369. MR 1881923, DOI 10.1007/s002220100176
Masayoshi Nagata, On the $14$-th problem of Hilbert, Amer. J. Math. 81 (1959), 766–772. MR 105409, DOI 10.2307/2372927
H. Skoda, Estimations $L^{2}$ pour l’opérateur $\overline \partial$ et applications arithmétiques, Journées sur les Fonctions Analytiques (Toulouse, 1976) Lecture Notes in Math., Vol. 578, Springer, Berlin, 1977, pp. 314–323 (French). MR 0460723
Michel Waldschmidt, Propriétés arithmétiques de fonctions de plusieurs variables. II, Séminaire Pierre Lelong (Analyse) année 1975/76, Lecture Notes in Math., Vol. 578, Springer, Berlin, 1977, pp. 108–135 (French). MR 0453659