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.
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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.
answered Oct 27, 2017 at 5:43
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$\begingroup$ What a great example... I really appreciate to your answer! $\endgroup$ Oct 28, 2017 at 9:05