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A Survey on the Eisenbud-Green-Harris Conjecture

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Women in Commutative Algebra

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 29))

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Abstract

The Eisenbud-Green-Harris (EGH) conjecture offers a generalization of the famous Macaulay’s theorem about the Hilbert functions of homogeneous ideals in a polynomial ring K[x1, …, xn]. In this survey paper, we provide a good compilation of results on the EGH conjecture that have been obtained so far. We discuss these results in terms of their approaches.

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References

  1. Abedelfatah, A.: On the Eisenbud-Green-Harris conjecture. Proc. Amer. Math. Soc. 143, no. 1, 105–115 (2015)

    Article  MathSciNet  Google Scholar 

  2. Abedelfatah, A.:Hilbert functions of monomial ideals containing a regular sequence. Israel J. Math. 214, no. 2, 857–865 (2016)

    Google Scholar 

  3. Bigatti, A.: Upper bounds for the betti numbers of a given Hilbert function. Comm. Algebra 21, no. 7, 2317–2334 (1993)

    Article  MathSciNet  Google Scholar 

  4. Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics. 39, Cambridge University Press, Cambridge (1993)

    Google Scholar 

  5. Caviglia, G., Constantinescu, A., Varbaro, M.: On a conjecture by Kalai. Israel J. Math. 204, no. 1, 469–475 (2014)

    Article  MathSciNet  Google Scholar 

  6. Caviglia, G., De Stefani, A.: A Cayley-Bacharach theorem for points in \(\mathbb {P}^n\). Bull. Lond. Math. Soc., 53, no. 4, 1185–1195 (2021)

    Google Scholar 

  7. Caviglia, G. De Stefani, A.: The Eisenbud-Green-Harris conjecture for fast-growing degree sequences. Preprint. arXiv:2007.15467v2 (2020)

  8. Caviglia, G., Kummini, M.: Poset embeddings of Hilbert functions and Betti numbers. J. Algebra 410, 244–257 (2014)

    Article  MathSciNet  Google Scholar 

  9. Caviglia, G., Maclagan, D.: Some cases of the Eisenbud-Green-Harris conjecture. Math. Res. Lett. 15, no. 3, 427–433 (2008)

    Article  MathSciNet  Google Scholar 

  10. Caviglia, G., Sammartano, A.: On the lex-plus-powers conjecture. Adv. Math. 340, 284–299 (2018)

    Article  MathSciNet  Google Scholar 

  11. Chen, R.-X.: Some special cases of the Eisenbud-Green-Harris conjecture. Illinois J. Math. 56, no. 3, 661–675 (2012)

    Article  MathSciNet  Google Scholar 

  12. Chong, K.F.E.: An Application of liaison theory to the Eisenbud-Green-Harris conjecture. J. Algebra 445, 221–231 (2016)

    Article  MathSciNet  Google Scholar 

  13. Clements, G., Lindström, B.: A generalization of a combinatorial theorem of Macaulay. J. Combinatorial Theory 7, 230–238 (1969)

    Article  MathSciNet  Google Scholar 

  14. Cooper, S. M.: Subsets of complete intersections and the EGH conjecture. Progress in Commutative Algebra 1, de Gruyter, Berlin 167–198 (2012)

    Google Scholar 

  15. Davis, E. D., Geramita A. V., Orecchia, F.: Gorenstein algebras and the Cayley– Bacharach theorem. Proc. Amer. Math. Soc. 93, no. 4, 593–597 (1985)

    Article  MathSciNet  Google Scholar 

  16. Eisenbud, D., Green, M., Harris, J.: Higher Castelnuovo theory. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992), Astérisque 218, 187–202 (1993)

    Google Scholar 

  17. Eisenbud, D., Green, M., Harris, J.: Cayley-Bacharach theorems and conjectures. Bull. Amer. Math. Soc. (N.S.) 33, no. 3, 295–324 (1996)

    Google Scholar 

  18. Francisco, C.: Almost complete intersections and the lex-plus-powers conjecture. J. Algebra 276, no. 2, 737–760 (2004)

    Article  MathSciNet  Google Scholar 

  19. Francisco, C. A., Richert, B. P.: Lex-plus-powers ideals. Syzygies and Hilbert functions, Lect. Notes Pure Appl. Math. 254, 113–144 (2007)

    Google Scholar 

  20. Gasharov, V.: Hilbert functions and homogeneous generic forms II. Compositio Math. 116, no. 2, 167–172 (1999)

    Article  MathSciNet  Google Scholar 

  21. Geramita, A., Kreuzer, M.: On the uniformity of zero-dimensional complete intersections. J. Algebra, 391, 82–92 (2013)

    Article  MathSciNet  Google Scholar 

  22. Green, M.: Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann. In Algebraic curves and projective geometry (Trento, 1988), Lecture Notes in Math. 1389, 76–86. Springer, Berlin (1989)

    Google Scholar 

  23. Greene, C., Kleitman, D.: Proof techniques in the theory of finite sets. Studies in combinatorics, MAA Stud. Math., Math. Assoc. America 17, 22–79 (1978)

    Google Scholar 

  24. Gotzmann, G.: Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes. Math. Z. 158, 61–70 (1978)

    Article  MathSciNet  Google Scholar 

  25. Gunturkun, S., Hochster, M.: The Eisenbud-Green-Harris conjecture for defect two quadratic ideals. Math. Res. Letters 27, no. 5, 1341–1365 (2020)

    Article  MathSciNet  Google Scholar 

  26. Harima, T., Wachi, A., Watanabe, J.: The EGH conjecture and the Sperner property of complete intersections. Proc. Amer. Math. Soc. 145, no. 4, 1497–1503 (2017)

    Article  MathSciNet  Google Scholar 

  27. Herzog, J., Popescu, D.: Hilbert functions and generic forms. Compositio Math. 113, no. 1, 1–22 (1998)

    Article  MathSciNet  Google Scholar 

  28. Hulett, H.A.: Maximum betti numbers of homogeneous ideals with a given Hilbert function. Comm. Algebra 21, no.7, 2335–2350 (1993)

    Article  MathSciNet  Google Scholar 

  29. Jorgenson, G.: Secant indices of projective varieties. Preprint arXiv:2003.08481 (2020)

  30. Katona, G.: A theorem for finite sets. Theory of Graphs (P. Erdös and G. Katona, eds.), Academic Press, New York 187–207 (1968)

    Google Scholar 

  31. Kruskal, J.: The number of simplices in a complex. Mathematical Optimization Techniques (R. Bellman, ed.), University of California Press, Berkeley/Los Angeles 251–278 (1963)

    Google Scholar 

  32. Laplagne, S., Valdettaro, M.: Strictly positive polynomials in the boundary of the SOS cone. Preprint arXiv:2012.05951 (2020)

  33. Grayson, D. R., Stillman, M. E.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/

  34. Macaulay, F.: Some properties of enumeration in the theory of modular systems. Proc. London Math. Soc. 26, 531–555 1927)

    Article  MathSciNet  Google Scholar 

  35. Mermin, J., Murai, S.: The lex-plus-powers conjecture holds for pure powers. Adv. Math. 226, no. 4, 3511–3539 (2011)

    Article  MathSciNet  Google Scholar 

  36. Pardue, K.: Deformation classes of graded modules and maximal betti numbers. Illinois J. Math. 40, no.4, 564–585 (1996)

    Article  MathSciNet  Google Scholar 

  37. Richert, B. P.: A study of the lex plus powers conjecture. J. Pure Appl. Algebra 186, no. 2, 169–183 (2004)

    Article  MathSciNet  Google Scholar 

  38. Richert, B. P., Sabourin, S.: The residuals of lex plus powers ideals and the Eisenbud-Green-Harris conjecture. Illinois J. Math. 52, no. 4, 1355–1384 (2008)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author thanks Mel Hochster for introducing and proposing to work on the EGH conjecture during her postdoctoral research. She is deeply grateful for all of their conversations. The author thanks the referee for their valuable feedback and comments. She also thanks Martin Kreuzer for pointing out their work.

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Authors and Affiliations

  1. Amherst College, Department of Mathematics and Statistics, Amherst, MA, USA

    Sema Güntürkün

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  1. Sema Güntürkün
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Correspondence to Sema Güntürkün .

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Editors and Affiliations

  1. Department of Mathematics, Syracuse University, Syracuse, NY, USA

    Claudia Miller

  2. Department of Mathematics, Fairfield University, Fairfield, CT, USA

    Janet Striuli

  3. Department of Mathematics, University of Kansas, Lawrence, KS, USA

    Emily E. Witt

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Güntürkün, S. (2021). A Survey on the Eisenbud-Green-Harris Conjecture. In: Miller, C., Striuli, J., Witt, E.E. (eds) Women in Commutative Algebra. Association for Women in Mathematics Series, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-030-91986-3_13

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