Plucker-Clebsch formula in higher dimension

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

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