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

Abstract

Given a normal surface singularity ( X , Q ) and a birational morphism to a non-singular surface π : X → S , we investigate the local geometry of the exceptional divisor L of π. We prove that the dimension of the tangent space to L at Q equals the number of exceptional components meeting at Q. Consequences relative to the existence of such birational projections contracting a prescribed number of irreducible curves are deduced. A new characterisation of minimal singularities is obtained in these terms.