On the tautness of rationally contractible curves on a surface

Let a complex curve $A$, that is reducible in general, lie on a nonsingular complex surface $X$, and let a curve $\widetilde A$, that is isomorphic to $A$, lie on a nonsingular surface $\widetilde X$, where the intersection matrices of the components of the curves $A$ and $\widetilde A$ coincide. In this paper we shall study the question of when the isomorphism between the curves $A$ and $\widetilde A$ can be extended to a biholomorphic equivalence of their neighborhoods on the surfaces $X$and $\widetilde X$. We shall prove that this is always possible for curves obtained in the resolution of doubly and triply rational singularities. This implies the tautness (nonvariability) of doubly and triply rational singular points.