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This article investigates the monodromy conjecture for a space monomial curvethat appears as the special fiber of an equisingular family of curves with aplane branch as generic fiber. Roughly speaking, the monodromy conjecturestates that every pole of the motivic, or related, Igusa zeta function inducesan eigenvalue of monodromy. As the poles of the motivic zeta functionassociated with such a space monomial curve have been determined in earlierwork, it remains to study the eigenvalues of monodromy. After reducing theproblem 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'Campoformula in terms of this resolution to compute the zeta function of monodromy.Combining all results, we prove the monodromy conjecture for this class ofmonomial curves.

The monodromy conjecture for a space monomial curve with a plane semigroup