Let $SsubsetPs^r$ ($rgeq 5$) be a nondegenerate, irreducible, smooth, complex, projective surface of degree $d$. Let $delta_S$ be the number of double points of a general projection of $S$ to $Ps^4$. In the present paper we prove that $ delta_Sleq{binom {d-2} {2}}$, with equality if and only if $S$ is a rational scroll. Extensions to higher dimensions are discussed.
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