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.
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).
"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