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Comparing powers of edge ideals. (English) Zbl 1423.13106

Summary: Given a nontrivial homogeneous ideal \(I \subseteq k [x_1, x_2, \ldots, x_d]\), a problem of great recent interest has been the comparison of the \(r\) th ordinary power of \(I\) and the \(m\) th symbolic power \(I^{(m)}\). This comparison has been undertaken directly via an exploration of which exponents \(m\) and \(r\) guarantee the subset containment \(I^{(m)} \subseteq I^r\) and asymptotically via a computation of the resurgence \(\rho(I)\), a number for which any \(m / r > \rho(I)\) guarantees \(I^{(m)} \subseteq I^r\). Recently, a third quantity, the symbolic defect, was introduced; as \(I^t \subseteq I^{(t)}\), the symbolic defect is the minimal number of generators required to add to \(I^t\) in order to get \(I^{(t)}\). We consider these various means of comparison when \(I\) is the edge ideal of certain graphs by describing an ideal \(J\) for which \(I^{(t)} = I^t + J\). When \(I\) is the edge ideal of an odd cycle, our description of the structure of \(I^{(t)}\) yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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[1] Bocci, C., Cooper, S., Guardo, E., Harbourne, B., Janssen, M., Nagel, U., Seceleanu, A., Van Tuyl, A. and Vu, T.. The Waldschmidt constant for squarefree monomial ideals, J. Algebraic Combin.44(4) (2016) 875-904. · Zbl 1352.13012
[2] Bocci, C., Cooper, S. and Harbourne, B.. Containment results for ideals of various configurations of points in \(\Bbb P^N\). J. Pure Appl. Algebra218(1) (2014) 65-75. · Zbl 1285.13029
[3] Bocci, C. and Harbourne, B., Comparing powers and symbolic powers of ideals, J. Algebraic Geom.19(3) (2010) 399-417. · Zbl 1198.14001
[4] Cooper, S. M., Embree, R. J. D., Hà, H. T. and Hoefel, A. H., Symbolic powers of monomial ideals, Proc. Edinburgh Math. Soc.60(1) (2017) 39-55. · Zbl 1376.13010
[5] H. Dao, A. De Stefani, E. Grifo, C. Huneke and L. Núñez-Betancourt, Symbolic powers of ideals, preprints (2017), arXiv:1708.03010. · Zbl 1404.13023
[6] Denkert, A. and Janssen, M., Containment problem for points on a reducible conic in \(\Bbb P^2\), J. Algebra394 (2013) 120-138. · Zbl 1316.13027
[7] Dumnicki, M., Szemberg, T. and Tutaj-Gasińska, H., Counterexamples to the \(I^{(3)} \subset I^2\) containment, J. Algebra393 (2013) 24-29. · Zbl 1297.14008
[8] Ein, L., Lazarsfeld, R. and Smith, K., Uniform bounds and symbolic powers on smooth varieties, Invent. Math.144(2) (2001) 241-252. · Zbl 1076.13501
[9] Ellis, S. W. and Wilson, L., Symbolic powers of edge ideals, Rose-Hulman Undergrad. Math. J.5(2) (2004) 1-10.
[10] F. Galetto, A. V. Geramita, Y.-S. Shin and A. Van Tuyl, The symbolic defect of an ideal, preprint (2016), arXiv:1610.00176. · Zbl 1430.13003
[11] Hochster, M. and Huneke, C., Comparison of symbolic and ordinary powers of ideals, Invent. Math.147(2) (2002) 349-369. · Zbl 1061.13005
[12] Simis, A., Vasconcelos, W. V. and Villarreal, R. H., On the ideal theory of graphs, J. Algebra167(2) (1994) 389-416. · Zbl 0816.13003
[13] Van Tuyl, A., A Beginner’s Guide to Edge and Cover Ideals (Springer, Berlin, 2013), pp. 63-94. · Zbl 1310.13003
[14] Villarreal, R. H., Cohen-Macaulay graphs, Manuscripta Math.66(1) (1990) 277-293. · Zbl 0737.13003
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