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Symbolic powers in weighted oriented graphs

Mousumi Mandal

Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India

E-mail Address: mousumi@maths.iitkgp.ac.in

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and

Dipak Kumar Pradhan

Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India

E-mail Address: dipakkumar@iitkgp.ac.in

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https://doi.org/10.1142/S0218196721500260Cited by:8 (Source: Crossref)

Abstract

Let $D$ be a weighted oriented graph with the underlying graph $G$ when vertices with non-trivial weights are sinks and $I (D), I (G)$ be the edge ideals corresponding to $D$ and $G,$ respectively. We give an explicit description of the symbolic powers of $I (D)$ using the concept of strong vertex covers. We show that the ordinary and symbolic powers of $I (D)$ and $I (G)$ behave in a similar way. We provide a description for symbolic powers and Waldschmidt constant of $I (D)$ for certain classes of weighted oriented graphs. When $D$ is a weighted oriented odd cycle, we compute $reg (I {(D)}^{(s)} / I {(D)}^{s})$ and prove $reg I {(D)}^{(s)} \leq reg I {(D)}^{s}$ and show that equality holds when there is only one vertex with non-trivial weight.

Communicated by J. McCullough

Keywords:

AMSC: 05C22, 13D02, 13D45, 05C25, 05C38, 05E40

References

1. S. K. Beyarslan, J. Biermann, K. N. Lin and A. O’Keefe, Algebraic invariants of weighted oriented graphs, preprint (2019), arXiv:1910.11773. Google Scholar
2. C. Bocci, S. Cooper, E. Guardo, B. Harbourne, M. Janssen, U. Nagel, A. Seceleanu, A. Van Tuyl and T. Vu, The Waldschmidt constant for squarefree monomial ideals, J. Algebraic Combin. 44(4) (2016) 875–904. Crossref, Web of Science, Google Scholar
3. C. Bocci and B. Harbourne, Comparing powers and symbolic powers of ideals, J. Algebraic Geom. 19(3) (2010) 399–417. Crossref, Web of Science, Google Scholar
4. B. Chakraborty and M. Mandal, Invariants of the symbolic powers of edge ideals, J. Algebra Appl. 19(10) (2020) 2050184. Link, Web of Science, Google Scholar
5. S. Cooper, R. Embree, H. T. Hà and A. H. Hoefel, Symbolic powers of monomial ideals, Proc. Edinb. Math. Soc. (2) 60(1) (2017) 39–55. Crossref, Web of Science, Google Scholar
6. S. Cooper, G. Fatabbi, E. Guardo, A. Lorenzini, J. Migliore, U. Nagel, A. Seceleanu, J. Szpond and A. Van Tuyl, Symbolic powers of codimension two Cohen–Macaulay ideals, Comm. Algebra 48(11) (2020) 4663–4680. Crossref, Web of Science, Google Scholar
7. I. Gitler, C. Valencia and R. H. Villarreal, A note on Rees algebras and the MFMC property, Beiträge Algebra Geom. 48(1) (2007) 141–150. Google Scholar
8. E. Guardo, B. Harbourne and A. Van Tuyl, Asymptotic resurgences for ideals of positive dimensional subschemes of projective space, Adv. Math. 246 (2013) 114–127. Crossref, Web of Science, Google Scholar
9. E. Guardo, B. Harbourne and A. Van Tuyl, Fat lines in $ℙ^{3}$ : Powers vs. symbolic powers, J. Algebra 390 (2013) 221–230. Crossref, Web of Science, Google Scholar
10. E. Guardo, B. Harbourne and A. Van Tuyl, Symbolic powers vs. regular powers of ideals of general points in $ℙ^{1} \times ℙ^{1}$ , Canad. J. Math. 65(4) (2013) 823–842, http://dx.doi.org/10.4153/CJM-2012-045-3. Crossref, Web of Science, Google Scholar
11. Y. Gu, H. T. Hà, J. L. O’Rourke and J. W. Skelton, Symbolic powers of edge ideals of graphs, Comm. Algebra 48(9) (2020) 3743–3760. Crossref, Web of Science, Google Scholar
12. H. T. Hà, K. N. Lin, S. Morey, E. Reyes and R. H. Villarreal, Edge ideals of oriented graphs, Internat. J. Algebra Comput. 29(3) (2019) 535–559. Link, Web of Science, Google Scholar
13. H. T. Hà, N. V. Trung and T. N. Trung, Depth and regularity of powers of sums of ideals, Math. Z. 282(3–4) (2016) 819–838. Crossref, Web of Science, Google Scholar
14. M. Janssen, T. Kamp and J. Vander Woude, Comparing powers of edge ideals, J. Algebra Appl. 18(10) (2019) 1950184. Link, Web of Science, Google Scholar
15. J. Martínez-Bernal, Y. Pitones and R. H. Villarreal, Minimum distance functions of graded ideals and Reed–Muller-type codes, J. Pure Appl. Algebra 221(2) (2017) 251–275. Crossref, Web of Science, Google Scholar
16. Y. Pitones, E. Reyes and J. Toledo, Monomial ideals of weighted oriented graphs, Electron. J. Combin. 26(3) (2019). Crossref, Web of Science, Google Scholar
17. A. Simis, W. Vasconcelos and R. H. Villarreal, On the ideal theory of graphs, J. Algebra 167 (1994) 389–416. Crossref, Web of Science, Google Scholar
18. O. Zariski and P. Samuel, Communications in Algebra, Vol. II (Springer, 1960). Crossref, Google Scholar
19. G. Zhu, L. Xu, H. Wang and Z. Tang, Projective dimension and regularity of edge ideal of some weighted oriented graphs, Rocky Mountain J. Math. 49(4) (2019) 1391–1406. Crossref, Web of Science, Google Scholar