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Let $X$ be a surface with cyclic quotient singularities, and let $f:X\rightarrow Y$ be a birational morphism with exceptional locus $E$. Assume $E\cong\mathbb{P}^{1}$, $x_{1},x_{2}\in E$ two singular points of $X$, of type $\frac{1}{r}(a_{1},a_{2})$ and $\frac{1}{m}(b_{1},b_{2})$ respectively. Let $y = f(E)\in Y$ be the point on which $E$ is contracted.

What can one say on the type of singularity of $y\in Y$ ?

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You may start with minimal elliptic singularities. Then u have examples of this phenomenon. Other place is to consider quasi-homogeneous singularities and the partial resolution given by their weighted blow up. –  pmath Nov 6 '13 at 11:35
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