Skip to main content
Log in

Resurgence numbers of fiber products of projective schemes

  • Published:
Collectanea Mathematica Aims and scope Submit manuscript

Abstract

We investigate the resurgence and asymptotic resurgence numbers of fiber products of projective schemes. Particularly, we show that while the asymptotic resurgence number of the k-fold fiber product of a projective scheme remains unchanged, its resurgence number could strictly increase.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akesseh, S.: Ideal containments under flat extensions. J. Algebra 492, 44–51 (2017)

    Article  MathSciNet  Google Scholar 

  2. Bocci, C., Cooper, S.M., Harbourne, B.: Containment results for ideals of various configurations of points in \({\mathbb{P}}^N\). J. Pure Appl. Algebra 218(1), 65–75 (2014)

    Article  MathSciNet  Google Scholar 

  3. Bocci, C., Harbourne, B.: Comparing powers and symbolic powers of ideals. J. Algebr. Geom. 19, 399–417 (2010)

    Article  MathSciNet  Google Scholar 

  4. Cooper, S.M., Embree, R., Hà, H.T., Hoefel, A.H.: Symbolic powers of monomial ideals. Proc. Edinb. Math. Soc. (2) 60(1), 39–55 (2017)

    Article  MathSciNet  Google Scholar 

  5. Czapliński, A., Główka, A., Malara, G., Lampa-Baczyńska, M., Łuszcz-Świdecka, P., Pokora, P., Szpond, J.: A counterexample to the containment \(I^{(3)} \subset I^2\) over the reals. Adv. Geom. 16(1), 77–82 (2016)

    Article  MathSciNet  Google Scholar 

  6. Denkert, A., Janssen, M.: Containment problem for points on a reducible conic in \({\mathbb{P}}^2\). J. Algebra 394, 120–138 (2013)

    Article  MathSciNet  Google Scholar 

  7. Dipasquale, M., Francisco, C.A., Mermin, J., Schweig, J.: Asymptotic resurgence via integral closures. Trans. Am. Math. Soc. 372(9), 6655–6676 (2019)

    Article  MathSciNet  Google Scholar 

  8. Dumnicki, M., Harbourne, B., Nagel, U., Seceleanu, A., Szemberg, T., Tutaj-Gasińska, H.: Resurgences for ideals of special point configurations in \({\mathbb{P}}^N\) coming from hyperplane arrangements. J. Algebra 443, 383–394 (2015)

    Article  MathSciNet  Google Scholar 

  9. Dumnicki, M., Szemberg, T., Tutaj-Gasińska, H.: Counterexamples to the \(I^{(3)} \subset I^2\) containment. J. Algebra 393, 24–29 (2013)

    Article  MathSciNet  Google Scholar 

  10. Dumnicki, M., Tutaj-Gasińska, H.: A containment result in \({\mathbb{P}}^n\) and the Chudnovsky conjecture. Proc. Am. Math. Soc. 145, 3689–3694 (2017)

    Article  Google Scholar 

  11. Ein, L., Lazarsfeld, R., Smith, K.E.: Uniform bounds and symbolic powers on smooth varieties. Invent. Math. 144, 241–252 (2001)

    Article  MathSciNet  Google Scholar 

  12. Eisenbud, D.: Commutative Algebra: With a View Toward Algebraic Geometry. Springer, Berlin (1995)

    Book  Google Scholar 

  13. Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. https://faculty.math.illinois.edu/Macaulay2/

  14. Guardo, E., Harbourne, B., Van Tuyl, A.: Asymptotic resurgence for ideals of positive dimensional subschemes of projective space. Adv. Math. 246, 114–127 (2013)

    Article  MathSciNet  Google Scholar 

  15. Hà, H.T., Nguyen, H.D., Trung, N.V., Trung, T.N.: Symbolic powers of sums of ideals. Math. Z. 294(3–4), 1499–1520 (2020)

    Article  MathSciNet  Google Scholar 

  16. Harbourne, B., Huneke, C.: Are symbolic powers highly evolved? J. Ramanujan Math. Soc. 28A, 247–266 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Harbourne, B., Seceleanu, A.: Containment counterexamples for ideals of various configurations of points in \({\mathbb{P}}^N\). J. Pure Appl. Algebra 219, 1062–1072 (2015)

    Article  MathSciNet  Google Scholar 

  18. Hartshorne, R.: Algebraic Geometry. Springer, Berlin (1977)

    Book  Google Scholar 

  19. Hochster, M., Huneke, C.: Comparison of symbolic and ordinary powers of ideals. Invent. Math. 147, 349–369 (2002)

    Article  MathSciNet  Google Scholar 

  20. Huneke, C., Swanson, I.: Integral closure of ideals, rings, and modules. In: London Mathematical Society Lecture Note Series, 336. Cambridge University Press, Cambridge (2006)

  21. Keane, R., Küronya, A., McMahon, E.: An elementary approach to containment relations between symbolic and ordinary powers of certain monomial ideals. Commun. Algebra 45(11), 4583–4587 (2017)

    Article  MathSciNet  Google Scholar 

  22. Lampa-Baczyńska, M., Malara, G.: On the containment hierarchy for simplicial ideals. J. Pure Appl. Algebra 219(12), 5402–5412 (2015)

    Article  MathSciNet  Google Scholar 

  23. Seceleanu, A.: A homological criterion for the containment between symbolic and ordinary powers of some ideals of points in \({\mathbb{P}}^2\). J. Pure Appl. Algebra 219(11), 4857–4871 (2015)

    Article  MathSciNet  Google Scholar 

  24. Szemberg, T., Szpond, J.: On the containment problem. Rend. Circ. Mat. Palermo (2) 66(2), 233–245 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank Louiza Fouli and Paolo Mantero for pointing out a mistake in the first draft of the paper. The authors also thank Elena Guardo and Alexandra Seceleanu for many stimulating discussions on related topics. Part of this work was done while the third author visited the other authors at Tulane University—the authors thank Tulane University for its hospitality. The second author is partially supported by Louisiana Board of Regents (Grant #LEQSF(2017-19)-ENH-TR-25). The computational commutative algebra package Macaulay 2, [13], has helped us in testing containment between symbolic and ordinary powers which are key to several results in this paper. The authors thank an anonymous referee for many useful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Huy Tài Hà.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bisui, S., Hà, H.T., Jayanthan, A.V. et al. Resurgence numbers of fiber products of projective schemes. Collect. Math. 72, 605–614 (2021). https://doi.org/10.1007/s13348-020-00302-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13348-020-00302-5

Keywords

Mathematics Subject Classification

Navigation