We prove that all moment varieties of univariate Gaussian mixtures have the
expected dimension. Our approach rests on intersection theory and Terracini's
classification of defective surfaces. The analogous identifiability result is
shown to be false for mixtures of Gaussians in dimension three and higher.
Their moments up to third order define projective varieties that are defective.
Our geometric study suggests an extension of the Alexander-Hirschowitz Theorem
for Veronese varieties to the Gaussian setting.Comment: 18 pages, to appear in International Mathematics Research Notice
