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 $\beta$, $\mathcal{CV}(L_{\beta})$, 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, $\mathcal{CV}(L_{\beta})$ 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.