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The Zamolodchikov $\mathcal{W}_3$-algebra $\mathcal{W}^c_3$ with centralcharge $c$ has full automorphism group $\mathbb{Z}_2$. It was conjectured inthe physics literature over 20 years ago that the orbifold$(\mathcal{W}^c_3)^{\mathbb{Z}_2}$ is of type $\mathcal{W}(2,6,8,10,12)$ forgeneric values of $c$. We prove this conjecture for all $c \neq \frac{559 \pm 7\sqrt{76657}}{95}$, and we show that for these two values, the orbifold is oftype $\mathcal{W}(2,6,8,10,12,14)$. This paper is part of a larger program ofstudying orbifolds and cosets of vertex algebras that depend continuously on aparameter. Minimal strong generating sets for orbifolds and cosets are ofteneasy to find for generic values of the parameter, but determining which valuesare generic is a difficult problem. In the example of$(\mathcal{W}^c_3)^{\mathbb{Z}_2}$, we solve this problem using tools fromalgebraic geometry.

The $\mathbb{Z}_2$-orbifold of the $\mathcal{W}_3$-algebra