An inequality on diagonal F -thresholds over standard-graded complete intersection rings

In a recent paper, De Stefani and Núñez-Betancourt proved that for a standard-graded F -pure k -algebraR, its diagonal F -threshold c(R) is always at least ?a(R) , where a(R) is the a-invariant. In this paper, we establisha refinement of this result in the setting of complete intersection rings.

PDF

___

[1] Bhatt B, Singh A. F -thresholds of Calabi-Yau hypersurfaces. Mathematische Annalen 2015; 362: 551-567. doi: 10.1007/s00208-014-1129-0
[2] Blickle M, Mustaţă M, Smith KE. Discreteness and rationality of diagonal F -thresholds. Michigan Mathematical Journal 2008; 57: 43-61. doi: 10.1307/mmj/1220879396
[3] Blickle M, Mustaţă M, Smith KE. F -thresholds of hyper surfaces. Transactions of the American Mathematical Society 2009; 361: 6549-6565. doi: 10.1090/S0002-9947-09-04719-9
[4] Stefani A, Núñez-Betancourt L. F-thresholds of graded rings. Nagoya Mathematical Journal 2018; 229: 141-168. doi: 10.1017/nmj.2016.65
[5] Stefani A, Núñez-Betancourt L, Pérez F. On the existence of F-thresholds and related limits. Transactions of the American Mathematical Society 2018; 370: 6629-6650. doi: 10.1090/tran/7176
[6] Hernandez D. F -pure thresholds of binomial hypersurfaces. Proceedings of the American Mathematical Society 2014; 142 (7): 2227-2242. doi: 10.1090/S0002-9939-2014-11941-1
[7] Huneke C, Mustaţă M, Takagi S, Watanabe KI. Diagonal F -thresholds, tight closure, integral closure, and multiplicity bounds. Michigan Mathematical Journal 2008; 57: 463-483. doi: 10.1307/mmj/1220879419
[8] Huneke C, Takagi S, Watanabe KI. Multiplicity bounds in graded rings. Kyoto Journal of Mathematics 2011; 51 (1): 127-147. doi: 10.1215/0023608X-2010-022
[9] Hirose D, Watanabe K, Yoshida K. F -thresholds versus a-invariants for standard graded toric rings. Communications in Algebra 2014; 42 (6): 2704-2720. doi: 10.1080/00927872.2013.772187
[10] Kustin A, Vraciu A. Socle degrees of Frobenius powers. Illinois Journal of Mathematics 2007; 51 (1): 185-208. doi: 10.1215/ijm/1258735332
[11] Li J. Asymptotic behavior of the socle of Frobenius powers. Illinois Journal of Mathematics 2013; 57 (2): 603-627. doi: 10.1215/ijm/1408453597
[12] Mustaţă M, Takagi S, Watanabe KI. F-thresholds and Bernstein-Sato polynomials. In: Proceedings of the fourth European congress of mathematics, European Mathematical Society, Z̈urich, 2005: pp. 341-364.
[13] Vraciu A. On the degrees of relations on xd1 1 ; ; xdn n ; (x1 + + xn)dn+1 in positive characteristic. Journal of Algebra 2015; 423 (1): 916-949. doi: 10.1016/j.jalgebra.2014.11.004