  Blow-ups of derivatives and gradient catastrophes for the $n$-dimensional
homogeneous Euler equation are discussed. It is shown that, in the case of
generic initial data, the blow-ups exhibit a fine structure in accordance of
the admissible ranks of certain matrix generated by the initial data. Blow-ups
form a hierarchy composed by $n+1$ levels with the strongest singularity of
derivatives given by $\partial u_i/\partial x_k \sim |\delta
\mathbf{x}|^{-(n+1)/(n+2)}$ along certain critical directions. It is
demonstrated that in the multi-dimensional case there are certain bounded
linear superposition of blow-up derivatives. Particular results for the
potential motion are presented too. Hodograph equations are basic tools of the
analysis.
