Asymptotic resurgence via integral closures

DiPasquale, Michael; Francisco, Christopher; Mermin, Jeffrey; Schweig, Jay

doi:10.1090/tran/7835

Asymptotic resurgence via integral closures
HTML articles powered by AMS MathViewer

by Michael DiPasquale, Christopher A. Francisco, Jeffrey Mermin and Jay Schweig PDF

Trans. Amer. Math. Soc. 372 (2019), 6655-6676 Request permission

Abstract:

Given an ideal in a polynomial ring, we show that the asymptotic resurgence studied by Guardo, Harbourne, and Van Tuyl can be computed using integral closures. As a consequence, the asymptotic resurgence of an ideal is the maximum of finitely many ratios involving Waldschmidt-like constants (which we call skew Waldschmidt constants) defined in terms of Rees valuations. We use this to prove that the asymptotic resurgence coincides with the resurgence if the ideal is normal (that is, all of its powers are integrally closed).

For a monomial ideal, the skew Waldschmidt constants have an interpretation involving the symbolic polyhedron defined by Cooper, Embree, Hà, and Hoefel. Using this intuition, we provide several examples of squarefree monomial ideals whose resurgence and asymptotic resurgence are different.

References

M. Asgharzadeh, On the rigidity of symbolic powers, arXiv:1611.06528 [math.AC] (2016).
Cristiano Bocci, Susan Cooper, Elena Guardo, Brian Harbourne, Mike Janssen, Uwe Nagel, Alexandra Seceleanu, Adam Van Tuyl, and Thanh Vu, The Waldschmidt constant for squarefree monomial ideals, J. Algebraic Combin. 44 (2016), no. 4, 875–904. MR 3566223, DOI 10.1007/s10801-016-0693-7
Cristiano Bocci, Susan M. Cooper, and Brian Harbourne, Containment results for ideals of various configurations of points in $\mathbf {P}^N$, J. Pure Appl. Algebra 218 (2014), no. 1, 65–75. MR 3120609, DOI 10.1016/j.jpaa.2013.04.012
T. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Seceleanu, and T. Szemberg, Negative curves on symmetric blowups of the projective plane, resurgences and Waldschmidt constants, arXiv:1609.08648 [math.AG] (2016).
Thomas Bauer, Sandra Di Rocco, Brian Harbourne, MichałKapustka, Andreas Knutsen, Wioletta Syzdek, and Tomasz Szemberg, A primer on Seshadri constants, Interactions of classical and numerical algebraic geometry, Contemp. Math., vol. 496, Amer. Math. Soc., Providence, RI, 2009, pp. 33–70. MR 2555949, DOI 10.1090/conm/496/09718
Cristiano Bocci and Brian Harbourne, Comparing powers and symbolic powers of ideals, J. Algebraic Geom. 19 (2010), no. 3, 399–417. MR 2629595, DOI 10.1090/S1056-3911-09-00530-X
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
Susan M. Cooper, Robert J. D. Embree, Huy Tài Hà, and Andrew H. Hoefel, Symbolic powers of monomial ideals, Proc. Edinb. Math. Soc. (2) 60 (2017), no. 1, 39–55. MR 3589840, DOI 10.1017/S0013091516000110
Gérard Cornuéjols, Combinatorial optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 74, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. Packing and covering. MR 1828452, DOI 10.1137/1.9780898717105
Barbara Costa, Aron Simis, and Zaqueu Ramos, A theorem about Cremona maps and symbolic Rees algebras, Internat. J. Algebra Comput. 24 (2014), no. 8, 1191–1212. MR 3296363, DOI 10.1142/S0218196714500532
Hailong Dao, Alessandro De Stefani, Eloísa Grifo, Craig Huneke, and Luis Núñez-Betancourt, Symbolic powers of ideals, Singularities and foliations. geometry, topology and applications, Springer Proc. Math. Stat., vol. 222, Springer, Cham, 2018, pp. 387–432. MR 3779569, DOI 10.1007/978-3-319-73639-6_{1}3
M. Dumnicki, B. Harbourne, U. Nagel, A. Seceleanu, T. Szemberg, and H. Tutaj-Gasińska, Resurgences for ideals of special point configurations in $\mathbf {P}^N$ coming from hyperplane arrangements, J. Algebra 443 (2015), 383–394. MR 3400406, DOI 10.1016/j.jalgebra.2015.07.022
Marcin Dumnicki, Tomasz Szemberg, and Halszka Tutaj-Gasińska, Counterexamples to the $I^{(3)}\subset I^2$ containment, J. Algebra 393 (2013), 24–29. MR 3090054, DOI 10.1016/j.jalgebra.2013.06.039
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
Y. Gu, H. T. Hà, J. L. O’Rourke, and J. W. Skelton, Symbolic powers of edge ideals of graphs, arXiv:1805.03428 [math.AC] (2018).
Elena Guardo, Brian Harbourne, and Adam Van Tuyl, Asymptotic resurgences for ideals of positive dimensional subschemes of projective space, Adv. Math. 246 (2013), 114–127. MR 3091802, DOI 10.1016/j.aim.2013.05.027
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
Huy Tài Hà and Kuei-Nuan Lin, Normal 0-1 polytopes, SIAM J. Discrete Math. 29 (2015), no. 1, 210–223. MR 3304257, DOI 10.1137/130948240
Brian Harbourne and Alexandra Seceleanu, Containment counterexamples for ideals of various configurations of points in $\mathbf {P}^N$, J. Pure Appl. Algebra 219 (2015), no. 4, 1062–1072. MR 3282125, DOI 10.1016/j.jpaa.2014.05.034
Dmitry Kozlov, Combinatorial algebraic topology, Algorithms and Computation in Mathematics, vol. 21, Springer, Berlin, 2008. MR 2361455, DOI 10.1007/978-3-540-71962-5
Masayoshi Nagata, On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 33 (1960/61), 271–293. MR 126444, DOI 10.1215/kjm/1250775912
Hidefumi Ohsugi and Takayuki Hibi, Normal polytopes arising from finite graphs, J. Algebra 207 (1998), no. 2, 409–426. MR 1644250, DOI 10.1006/jabr.1998.7476
Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432
Aron Simis, Wolmer V. Vasconcelos, and Rafael H. Villarreal, The integral closure of subrings associated to graphs, J. Algebra 199 (1998), no. 1, 281–289. MR 1489364, DOI 10.1006/jabr.1997.7171
Irena Swanson, Linear equivalence of ideal topologies, Math. Z. 234 (2000), no. 4, 755–775. MR 1778408, DOI 10.1007/s002090050007