Critical sets of the total variance can detect all stochastic local operations and classical communication classes of multiparticle entanglement

We present a general algorithm for finding all classes of pure multiparticle states equivalent under stochastic local operations and classical communication (SLOCC). We parametrize all SLOCC classes by the critical sets of the total variance function. Our method works for arbitrary systems of distinguishable and indistinguishable particles. We also show how to calculate the Morse indices of critical points which have the interpretation of the number of independent nonlocal perturbations increasing the variance and hence entanglement of a state. We illustrate our method by two examples.