The pure $Y=X^d$ truncated moment problem

Let $\beta \equiv {\beta }^{(2n)}$ be a real bivariate sequence of degree $2n$. We study the existence of representing measures for $\beta$ supported in the curve $y=x^d$ ($d\ge 1$) in the case when all column dependence relations in the moment matrix $M_n(\beta )$ are generated by the relation $Y=X^d$. We prove that the core variety of $\beta$, $\mathcal{C}\mathcal{V}(L_{\beta })$, is nonempty (equivalently, representing measures exist) if and only if $C$, the partially defined core matrix of $\beta$, admits a positive, recursively generated completion $C[A]$. Moreover, $\mathcal{C}\mathcal{V}(L_{\beta })$ is the entire curve $y=x^d$ if and only if there is a positive definite completion $C[A]$. In the remaining case, if there is a measure, it is unique and finitely atomic. For $d=3$, we use these results to compute the core variety of $\beta$ and give new characterizations of the existence of representing measures, which complement a result of the first-named author.