It is about deformation theory on algebraic surfaces.
If there are two singular points on an algebraic surfaces, is it possible that two singular points collapse to a new point as the surface deforms?
Also, if it is possible, I am wondering that the new point might be nonsingular.



Can two singular points collapse to a new singular point?
Yes, it is possible, as shown by the following simple example. 


Think of a double cover $S$ of $\mathbb{P}^2$ branched on two smooth conics intersecting transversally: it has four singularities of type $A\_1$ (locally, they are of the form $x^2+y^2+z^2=0$). 


When you deform the branch locus to two smooth conics that are tangent at two points, the singularities of $S$ collide in pairs and you obtain a double cover $S\_0$ having two singularities of type $A\_3$ (locally, they are of the form $x^2+y^2+z^4=0$).


In any case, the resulting points after collision will be singular, since the multiplicity cannot decrease under specialization.


