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2021 The monodromy conjecture for a space monomial curve with a plane semigroup
Jorge Martín-Morales, Willem Veys, Lena Vos
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Publ. Mat. 65(2): 529-597 (2021). DOI: 10.5565/PUBLMAT6522105

Abstract

This article investigates the monodromy conjecture for a space monomial curve that appears as the special fiber of an equisingular family of curves with a plane branch as generic fiber. Roughly speaking, the monodromy conjecture states that every pole of the motivic, or related, Igusa zeta function induces an eigenvalue of monodromy. As the poles of the motivic zeta function associated with such a space monomial curve have been determined in earlier work, it remains to study the eigenvalues of monodromy. After reducing the problem to the curve seen as a Cartier divisor on a generic embedding surface, we construct an embedded $\mathbb Q$-resolution of this pair and use an A'Campo formula in terms of this resolution to compute the zeta function of monodromy. Combining all results, we prove the monodromy conjecture for this class of monomial curves.

Funding Statement

The first author is partially supported by MTM2016-76868-C2-2-P from the Departamento de Industria e Innovación del Gobierno de Aragón and Fondo Social Europeo E22 17R Grupo Consolidado Álgebra y Geometría, and by FQM-333 from Junta de Andalucía. The second author is partially supported by the Research Foundation - Flanders (FWO) project G.0792.18N. The third author is supported by a PhD Fellowship of the Research Foundation - Flanders (no. 71587).

Citation

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Jorge Martín-Morales. Willem Veys. Lena Vos. "The monodromy conjecture for a space monomial curve with a plane semigroup." Publ. Mat. 65 (2) 529 - 597, 2021. https://doi.org/10.5565/PUBLMAT6522105

Information

Received: 7 January 2020; Revised: 13 November 2020; Published: 2021
First available in Project Euclid: 21 June 2021

Digital Object Identifier: 10.5565/PUBLMAT6522105

Subjects:
Primary: 14E15
Secondary: 14H20, 14J17, 32S40

Rights: Copyright © 2021 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.65 • No. 2 • 2021
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