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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Asymptotic resurgence via integral closures


Authors: Michael DiPasquale, Christopher A. Francisco, Jeffrey Mermin and Jay Schweig
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 14C20, 13B22; Secondary 13F20, 13A18
DOI: https://doi.org/10.1090/tran/7835
Published electronically: April 4, 2019
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Abstract: Given an ideal in a polynomial ring, we show that the asymptotic resurgence studied by Guardo, Harbourne, and Van Tuyl can be computed using integral closures. As a consequence, the asymptotic resurgence of an ideal is the maximum of finitely many ratios involving Waldschmidt-like constants (which we call skew Waldschmidt constants) defined in terms of Rees valuations. We use this to prove that the asymptotic resurgence coincides with the resurgence if the ideal is normal (that is, all of its powers are integrally closed).

For a monomial ideal, the skew Waldschmidt constants have an interpretation involving the symbolic polyhedron defined by Cooper, Embree, Hà, and Hoefel. Using this intuition, we provide several examples of squarefree monomial ideals whose resurgence and asymptotic resurgence are different.


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Additional Information

Michael DiPasquale
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078-1058
Email: Michael.DiPasquale@colostate.edu

Christopher A. Francisco
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078-1058
Email: chris.francisco@okstate.edu

Jeffrey Mermin
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078-1058
Email: mermin@math.okstate.edu

Jay Schweig
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078-1058
Email: jay.schweig@okstate.edu

DOI: https://doi.org/10.1090/tran/7835
Received by editor(s): August 3, 2018
Received by editor(s) in revised form: February 12, 2019
Published electronically: April 4, 2019
Additional Notes: The content of this paper arose out of various attempts to prove the conjecture which we encountered at the 2017 BIRS-CMO workshop in Oaxaca. We are grateful to BIRS-CMO and the organizers and participants of the Oaxaca workshop and for the inspiring discussions and problems we encountered there. We are especially indebted to Adam Van Tuyl for introducing this conjecture in Oaxaca and for his insight into this problem.
Article copyright: © Copyright 2019 American Mathematical Society