Existence of a flat deformation whose all sufficiently small specializations are non-semisimple
Determine whether there exist finite-dimensional complex algebras N and A of the same dimension, with A semisimple, and a formal flat deformation \mathcal N of N such that \mathcal N specializes to an algebra isomorphic to A at t = s0 for some real s0 > 0, yet for every sufficiently small real s > 0 the specialized algebra \mathcal N at t = s is not semisimple (equivalently, has a nonzero nilpotent ideal).
References
It follows that if $\mathcal N$ is a flat deformation of $N$ which deforms to $A$ at $t=s$ then we have the following possibilities (i) $\mathcal N$ is a strongly flat deformation from $N$ into $A'$, for some semisimple $\mathbb C$-algebra $A'$, where $A'$ is uniquely determined by $\mathcal N$, (ii) for all sufficiently small real numbers $s>0$ the algebra $\mathcal N$ specialised at $t=s$ is not semisimple, and hence it has a non-zero nilpotent ideal. We don't know if there are $N,A$ and $\mathcal N$ such that (ii) holds.