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Given a normal surface singularity $(X, Q)$ and a birational morphism to anon- singular surface $\pi : X \to S$, we investigate the local geometry of theexceptional divisor $L$ of $\pi$. We prove that the dimension of the tangentspace to $L$ at $Q$ equals the number of exceptional components meeting at $Q$.Consequences relative to the existence of such birational projectionscontracting a prescribed number of irreducible curves are deduced. A newcharacterization of minimal singularities is obtained in these terms.

On the exceptional locus of the birational projections of normal surface singularity into a plane