Non-existence of small resolutions for the singularity $y^2=u^2+v^2+w^3$

A $3$-dimensional hypersurface singularity of type $$y^2=u^2+v^2+w^k$$ admits a small resolution if and only if $k$ is even. If $k$ is odd the corresponding singularity is factorial, so there is no small resolution.

See the paper by R. Friedman Simultaneous resolution of 3-fold double points, Mathematische Annalen 274 (1986), p. 675.

This answer to a more general question might also be relevant for similar questions.

It is worth noting the following result by S. Katz:

If the singularity $X$ defined by $xy-g(z,t)$ is isolated (so $g=0$ is reduced) and $cA_n$ (meaning a general hyperplane section is a type $A_n$ singularity) then $X$ has a small resolution iff the curve $g=0$ has $n+1$ distinct branches.

The statement you want is the case $n=1$: $z^2+t^k=0$ has $2$ branches iff $k$ even.

I learned about this result in this paper, where the authors proved the non-commutative(!) version of Katz's theorem (see Theorem 5.5).