Plucker-Clebsch formula in higher dimension
2011
Let $S\subset\mathbb{P}^r$ ($r\geq 5$) 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 $\mathbb{P}^4$. In the present paper we prove
that $ \delta_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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