  We consider the problem of identifying a mixture of Gaussian distributions
with same unknown covariance matrix by their sequence of moments up to certain
order. Our approach rests on studying the moment varieties obtained by taking
special secants to the Gaussian moment varieties, defined by their natural
polynomial parametrization in terms of the model parameters. When the order of
the moments is at most three, we prove an analogue of the Alexander-Hirschowitz
theorem classifying all cases of homoscedastic Gaussian mixtures that produce
defective moment varieties. As a consequence, identifiability is determined
when the number of mixed distributions is smaller than the dimension of the
space. In the two component setting we provide a closed form solution for
parameter recovery based on moments up to order four, while in the one
dimensional case we interpret the rank estimation problem in terms of secant
varieties of rational normal curves.
