Are there examples of algebraic singularities which may be smoothed analytically but not algebraically? It certainly seems possible, but if not, why? Are there conditions under which this becomes true, e.g. what if the singularity is isolated?



Analytically but not algebraically smoothable singularity
For isolated singularities, the answer is **no**.


In fact, it follows by a result of Elkik that any deformation of an isolated singularity is algebraizable. See


R. Elkik, *[Solutions d'équations à coefficients dans un anneau hensélien](https://eudml.org/doc/81927)*, Ann. Sci. Ecole Norm. Sup. **6** (1973), 553-603,


in particular Part IV.


