Abstract
We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic powers of monomial ideals. In addition, we present some new results on these aspects of the subject.
Keywords
H. Dao—Partially supported by NSA Grant H98230-16-1-0012.
C. Huneke—Partially supported by the NSF Grant 1460638.
L. Núñez-Betancourt—Partially supported by the NSF Grant 1502282.
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Akesseh, S.: Ideal containments under flat extensions. J. Algebra. 492, 44–51 (2017)
Àlvarez Montaner, J., Huneke, C., Núñez-Betancourt, L.: \(D\)-modules, Bernstein–Sato polynomials, and \(F\)-invariants of direct summands. Adv. Math. 321, 298–325 (2017). arXiv:1611.04412
Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Publishing Co., Reading (1969)
Baczyńska, M., Dumnicki, M., Habura, A., Malara, G., Pokora, P., Szemberg, T., Szpond, J., Tutaj-Gasińska, H.: Points fattening on \(\mathbb{P}^1\times \mathbb{P}^1\) and symbolic powers of bi-homogeneous ideals. J. Pure Appl. Algebra 218(8), 1555–1562 (2014)
Berthelot, P.: Altérations de variétés algébriques (d’après A. J. de Jong). Astérisque 241, Exp. No. 815, 5, 273–311 (1997); Séminaire Bourbaki, Vol. 1995/96
Bocci, C., Harbourne, B.: Comparing powers and symbolic powers of ideals. J. Algebr. Geom. 19(3), 399–417 (2010)
Bocci, C., Harbourne, B.: The resurgence of ideals of points and the containment problem. Proc. Am. Math. Soc. 138(4), 175–1190 (2010)
Bocci, C., Cooper, S., Guardo, E., Harbourne, B., Janssen, M., Nagel, U., Seceleanu, A., Van Tuyl, A., Thanh, V.: The Waldschmidt constant for squarefree monomial ideals. J. Algebr. Combin. 44(4), 875–904 (2016)
Böger, E.: Differentielle und ganz-algebraische Abhängigkeit bei Idealen analytischer Algebren. Math. Z. 121, 188–189 (1971)
Brodmann, M.: Asymptotic stability of \({\rm {Ass}}(M/I^{n}M)\). Proc. Am. Math. Soc. 74(1), 16–18 (1979)
Chen, J., Morey, S., Sung, A.: The stable set of associated primes of the ideal of a graph. Rocky Mt. J. Math. 32(1), 71–89 (2002)
Cornuéjols, G.: Combinatorial Optimization: Packing and Covering. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 74. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001)
Cornuéjols, G., Conforti, M.: A decomposition theorem for balanced matrices. Integer Program. Comb. Optim. 74, 147–169 (1990)
Cornuéjols, G., Guenin, B., Margot, F.: The packing property. Math. Program. 89(1, Ser. A), 113–126 (2000)
Dale Cutkosky, S.: Irrational asymptotic behaviour of Castelnuovo–Mumford regularity. J. Reine Angew. Math. 522, 93–103 (2000)
Dumnicki, M.: Containments of symbolic powers of ideals of generic points in \(\mathbb{P}^3\). Proc. Am. Math. Soc. 143(2), 513–530 (2015)
Dumnicki, M., Szemberg, T., Tutaj-Gasińska, H.: Counterexamples to the \(I^{(3)}\subseteq I^2\) containment. J. Algebra 393, 24–29 (2013)
Dumnicki, M., Harbourne, B., Szemberg, T., Tutaj-Gasińska, H.: Linear subspaces, symbolic powers and Nagata type conjectures. Adv. Math. 252, 471–491 (2014)
Dumnicki, M., Harbourne, B., Nagel, U., Seceleanu, A., Szemberg, T., Tutaj-Gasińska, H.: Resurgences for ideals of special point configurations in \(\mathbf{P}^N\) coming from hyperplane arrangements. J. Algebra 443, 383–394 (2015)
Ein, L., Lazarsfeld, R., Smith, K.E.: Uniform bounds and symbolic powers on smooth varieties. Invent. Math. 144(2), 241–252 (2001)
Eisenbud, D.: Commutative Algebra: With a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)
Eisenbud, D., Hochster, M.: A Nullstellensatz with nilpotents and Zariski’s main lemma on holomorphic functions. J. Algebra 58(1), 157–161 (1979)
Eisenbud, D., Mazur, B.: Evolutions, symbolic squares, and fitting ideals. J. Reine Angew. Math. 488, 189–201 (1997)
Fatabbi, G., Harbourne, B., Lorenzini, A.: Inductively computable unions of fat linear subspaces. J. Pure Appl. Algebra 219(12), 5413–5425 (2015)
Francisco, C.A., Hà, H.T., Van Tuyl, A.: Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals. J. Algebra 331, 224–242 (2011)
Francisco, C.A., Mermin, J., Schweig, J.: A survey of Stanley–Reisner theory. Connections Between Algebra, Combinatorics, and Geometry, pp. 209–234. Springer, Berlin (2014)
Gitler, I., Reyes, E., Villarreal, R.H.: Blowup algebras of ideals of vertex covers of bipartite graphs. Algebraic Structures and their Representations. Contemporary Mathematics, vol. 376, pp. 273–279. American Mathematical Society, Providence (2005)
Gitler, I., Valencia, C., Villarreal, R.H.: A note on the Rees algebra of a bipartite graph. J. Pure Appl. Algebra 201(1–3), 17–24 (2005)
Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry
Grifo, E., Huneke, C.: Symbolic powers of ideals defining F-pure and strongly F-regular rings. Int. Mat. Res. Not. (2017). http://doi.org/10.1093/imrn/rnx213
Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV. Inst. Hautes Études Sci. Publ. Math. 32, 361 (1967)
Guardo, E., Harbourne, B., Van Tuyl, A.: Symbolic powers versus regular powers of ideals of general points in \(\mathbb{P}^1\times \mathbb{P}^1\). Canad. J. Math. 65(4), 823–842 (2013)
Harbourne, B., Huneke, C.: Are symbolic powers highly evolved? J. Ramanujan Math. Soc. 28A, 247–266 (2013)
Harbourne, B., Seceleanu, A.: Containment counterexamples for ideals of various configurations of points in \(\mathbf{P}^N\). J. Pure Appl. Algebra 219(4), 1062–1072 (2015)
Harbourne, B., Kapustka, M., Knutsen, A., Syzdek, W., Bauer, T., Di Rocco, S., Szemberg, T.: A primer on Seshadri constants. Contemp. Math. 496, 39–70 (2009)
Herzog, J., Tuân Hoa, L., Trung, N.V.: Asymptotic linear bounds for the Castelnuovo–Mumford regularity. Trans. Am. Math. Soc. 354(5), 1793–1809 (2002)
Hoa, L.T., Trung, T.N.: Partial Castelnuovo–Mumford regularities of sums and intersections of powers of monomial ideals. Math. Proc. Camb. Philos. Soc. 149(2), 229–246 (2010)
Hoa, L.T., Trung, T.N.: Castelnuovo–Mumford regularity of symbolic powers of two-dimensional squarefree monomial ideals. J. Commut. Algebra 8(1), 77–88 (2016)
Hochster, M., Huneke, C.: Tight closure in equal characteristic zero (1999)
Hochster, M.: Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes. Ann. Math. 2(96), 318–337 (1972)
Hochster, M., Huneke, C.: Comparison of symbolic and ordinary powers of ideals. Invent. Math. 147(2), 349–369 (2002)
Hübl, R.: Evolutions and valuations associated to an ideal. J. Reine Angew. Math. 517, 81–101 (1999)
Hübl, R.: Powers of elements and monomial ideals. Commun. Algebra 33(10), 3771–3781 (2005)
Hübl, R., Huneke, C.: Fiber cones and the integral closure of ideals. Collect. Math. 52(1), 85–100 (2001)
Huckaba, S.: Symbolic powers of prime ideals with applications to hypersurface rings. Nagoya Math. J. 113, 161–172 (1989)
Huneke, C.: Uniform bounds in Noetherian rings. Invent. Math. 107(1), 203–223 (1992)
Huneke, C.: Desingularizations and the uniform Artin–Rees theorem. J. Lond. Math. Soc. (2) 62(3), 740–756 (2000)
Huneke, C., Raicu, C.: Introduction to uniformity in commutative algebra. Commutative Algebra and Noncommutative Algebraic Geometry. Vol. I. Mathematical Sciences Research Institute Publications, vol. 67, pp. 163–190. Cambridge University Press, New York (2015)
Huneke, C., Ribbe, J.: Symbolic squares in regular local rings. Math. Z. 229(1), 31–44 (1998)
Huneke, C., Ulrich, B.: Powers of licci ideals. Commutative Algebra (Berkeley, CA, 1987). Mathematical Sciences Research Institute Publications, vol. 15, pp. 339–346. Springer, New York (1989)
Huneke, C., Katz, D., Validashti, J.: Uniform equivalente of symbolic and adic topologies. Ill. J. Math. 53(1), 325–338 (2009)
Huneke, C., Katz, D., Validashti, J.: Uniform symbolic topologies and finite extensions. J. Pure Appl. Algebra 219(3), 543–550 (2015)
Huy Tài Hà and Susan Morey: Embedded associated primes of powers of squarefree monomial ideals. J. Pure Appl. Algebra 214(4), 301–308 (2010)
Izumi, S.: A measure of integrity for local analytic algebras. Publ. Res. Inst. Math. Sci. 21(4), 719–735 (1985)
Kunz, E.: On Noetherian rings of characteristic \(p\). Am. J. Math. 98(4), 999–1013 (1976)
Kunz, E.: Kähler Differentials. Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig (1986)
Kurano, K., Roberts, P.C.: The positivity of intersection multiplicities and symbolic powers of prime ideals. Compos. Math. 122(2), 165–182 (2000)
Lam, H.M., Trung, N.V.: Associated primes of powers of edge ideals and ear decompositions of graphs (2015). arXiv:1506.01483
Ma, L., Schwede, K.: Perfectoid multiplier/test ideals in regular rings and bounds on symboic powers (2017). arxiv:1705.02300
Matsumura, H. Commutative algebra. 2nd edn. Mathematics Lecture note series, vol. 56, pp. Xv+313. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., (1980). ISBN: 0-8053-7026-9
Mazur, B.: Deformations of Galois representations and Hecke algebras. Harvard Course Notes (1994)
McAdam, S.: Quintasymptotic primes and four results of Schenzel. J. Pure Appl. Algebra 47(3), 283–298 (1987)
Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)
Minh, N.C., Trung, N.V.: Corrigendum to Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley–Reisner ideals [Adv. Math. 226(2), 1285–1306 (2011)] [mr2737785]. Adv. Math. 228(5), 2982–2983 (2011)
Minh, N.C., Trung, N.V.: Regularity of symbolic powers and Arboricity of matroids (2017). arXiv:1702.04491
Minh, N.C., Trung, N.V.: Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley–Reisner ideals. Adv. Math. 226(2), 1285–1306 (2011)
More, A.A.: A note on the Eisenbud–Mazur conjecture. J. Pure Appl. Algebra 219(7), 2740–2755 (2015)
Morey, S., Villarreal, R.H.: Edge ideals: algebraic and combinatorial properties. Progress in Commutative Algebra 1, pp. 85–126. de Gruyter, Berlin (2012)
Nagata, M.: Local Rings. Interscience (1962)
Rees, D.: Degree functions in local rings. Proc. Camb. Philos. Soc. 57, 1–7 (1961)
Roberts, P.C.: Multiplicities and Chern classes in local algebra. Cambridge Tracts in Mathematics, vol. 133. Cambridge University Press, Cambridge (1998)
Sather-Wagstaff, S.: Intersections of symbolic powers of prime ideals. J. Lond. Math. Soc. (2) 65(3), 560–574 (2002)
Scheja, G., Storch, U.: Über differentielle Abhängigkeit bei Idealen analytischer Algebren. Math. Z. 114, 101–112 (1970)
Schenzel, P.: Symbolic powers of prime ideals and their topology. Proc. Am. Math. Soc. 93(1), 15–20 (1985)
Schenzel, P.: Finiteness of relative Rees rings and asymptotic prime divisors. Math. Nachr. 129, 123–148 (1986)
Serre, J.-P.: Local Algebra. Springer Monographs in Mathematics. Springer, Berlin (2000). (Translated from the French by CheeWhye Chin and revised by the author)
Sullivant, S.: Combinatorial symbolic powers. J. Algebra 319(1), 115–142 (2008)
Swanson, I.: Powers of ideals. Primary decompositions, Artin–Rees lemma and regularity. Math. Ann. 307(2), 299–313 (1997)
Swanson, I.: Linear equivalence of topologies. Math. Zeitschrift 234, 755–775 (2000)
Szemberg, T., Szpond, J.: On the containment problem (2016). arXiv:1601.01308
Taylor, R., Wiles, A.: Ring-theoretic properties of certain Hecke algebras. Ann. Math. (2) 141(3), 553–572 (1995)
Terai, N., Trung, N.V.: Cohen-Macaulayness of large powers of Stanley–Reisner ideals. Adv. Math. 229(2), 711–730 (2012)
Trung, N.V., Tuan, T.M.: Equality of ordinary and symbolic powers of Stanley–Reisner ideals. J. Algebra 328, 77–93 (2011)
Varbaro, M.: Symbolic powers and matroids. Proc. Am. Math. Soc. 139, 2357–2366 (2011)
Waldschmidt, M.: Propriétés arithmétiques de fonctions de plusieurs variables. II. Séminaire Pierre Lelong (Analyse) année 1975/76. Lecture Notes in Mathematics, vol. 578, pp. 108–135. Springer, Berlin (1977)
Walker, R.M.: Rational singularities and uniform symbolic topologies. Ill. J. Math. 60(2), 541–550 (2016)
Walker, R.M.: Uniform Harbourne–Huneke bounds via flat extensions (2016). arXiv:1608.02320
Walker, R.M.: Uniform symbolic topologies via multinomial expansions (2017). arXiv:1703.04530
Wiles, A.: Modular elliptic curves and Fermat’s last theorem. Ann. Math. (2) 141(3), 443–551 (1995)
Yassemi, S.: Weakly associated primes under change of rings. Commun. Algebra 26(6), 2007–2018 (1998)
Zariski, O.: A fundamental lemma from the theory of holomorphic functions on an algebraic variety. Ann. Mat. Pura Appl. 4(29), 187–198 (1949)
Acknowledgements
We thank Jeff Mermin for many helpful conversations concerning the packing problem, and in particular for discussions leading to Remark 4.19 and Corollary 4.20. We thank Jonathan Montaño, Andrew Conner, Jack Jeffries, and Robert Walker for helpful comments. Part of this work was done when the second and fifth authors were at the University of Virginia. They wish to thank this institution for its hospitality. Finally, the fifth author thanks the organizing committee for the ‘Brazil-Mexico 2nd meeting on Singularities’ in Salvador, Bahia, Brazil, where this project was initiated.
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Dao, H., De Stefani, A., Grifo, E., Huneke, C., Núñez-Betancourt, L. (2018). Symbolic Powers of Ideals. In: Araújo dos Santos, R., Menegon Neto, A., Mond, D., Saia, M., Snoussi, J. (eds) Singularities and Foliations. Geometry, Topology and Applications. NBMS BMMS 2015 2015. Springer Proceedings in Mathematics & Statistics, vol 222. Springer, Cham. https://doi.org/10.1007/978-3-319-73639-6_13
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